RELATIONS BETWEEN THRESHOLD CONSTANTS FOR YAMABE TYPE BORDISM INVARIANTS

BERND AMMANN AND NADINE GROSSE

Abstract. In the work of Ammann, Dahl and Humbert it has turned out that the Yamabe invariant on closed manifolds is a bordism invariant below a certain positive threshold constant. A similar result holds for a spinorial analogon. These threshold constants are characterized through Yamabe-type equations on products of spheres with rescaled hyperbolic spaces. We give variational characterizations of these threshold constants, and our investigations lead to an explicit positive lower bound for the spinorial threshold constants.

1. Introduction

The smooth Yamabe invariant, also called Schoen’s σ-constant, of a closed manifold M^{m} is defined as

σ^{∗}(M) := sup inf ∫_{M} scal^{g}dvol_{g} ∈ ( − ∞, σ^{∗}(𝕊^{m}) = m(m − 1)vol(𝕊^{m})^{\frac{2}{m}} ],

where the supremum runs over all conformal classes [g_{0}] on M, and the infimum goes over all metrics g in [g_{0}] such that (M, g) has volume one. The smooth Yamabe invariant is an important geometric quantity. In particular, σ^{∗}(M) > 0 if and only if M admits a metric of positive scalar curvature. However,the smooth Yamabe invariant is quite mysterious. It is only known for very few examples, e.g. the sphere, cp. Remark 2.4, σ^{∗}(𝕋^{m}) = 0, [20, Cor. 2.5], σ^{∗}(ℝℙ^{3}) = 2^{−2/3}σ^{∗}(𝕊^{3}), [17, Cor. 2.3], σ^{∗}(ℂP^{2}) = 12\sqrt{2}π, [33, 26], and σ^{∗}(Γ \ ℍ^{3}) = −6v_{Γ}^{2/3} where v_{Γ} is the volume of the compact quotient Γ \ ℍ^{3} with respect to the hyperbolic metric, [13], [31, Sect. II.8]. In particular, there is no known example of a manifold of dimension m ≥ 5 with σ^{∗}(M) ∉ {0, σ^{∗}(𝕊^{m})}. In [5] M. Dahl, E. Humbert, and the first author proved a surgery formula for the smooth Yamabe invariant, cp. Theorem 2.5. In particular it says that if a manifold N^{m} is obtained from a closed manifold M^{m} by a surgery of codimension m−k ≥ 3 and σ^{∗}(M) is below a certain positive threshold constant Λ_{m,k}, then σ^{∗}(N) ≥ σ^{∗}(M). The threshold constants Λ_{m,k} appearing in this result are certain Yamabe-type invariants for special noncompact model spaces 𝕄_{c}^{m,k} which are products of rescaled hyperbolic spaces and spheres, see Subsection 2.2 for the precise definition. In particular, it follows that the smooth Yamabe invariant is a bordism invariant in the following sense: Suppose that M and N are connected closed smooth spin manifolds of dimension m ≥ 5 with fundamental group Γ, representing the same element in Ω_{m}^{spin}(BΓ), then 0 ≤ σ^{∗}(M) < Λ_{m} := min_{k=2,...,m−3} Λ_{m,k} implies σ^{∗}(M) = σ^{∗}(N), [5, Sec. 1.4]. Thus, if sufficiently many manifolds with σ^{∗}(M) ∈ (0, Λ_{m}) exist, one obtains a rich and interesting subgroup in the bordism group Ω_{m}^{spin}(BΓ) and similar versions hold in oriented bordism classes. In order to understand the structure of subgroup it is essentially to get as much knowledge about the surgery constants Λ_{m,k} as possible.

Date: September 2, 2022.

2010 Mathematics Subject Classification. 53C21 (Primary) 35J60, 53C27, 57R65 (Secondary).

Key words and phrases. Dirac operator, Yamabe constant, Yamabe invariant, conformal Hijazi inequality.

If current conjectures about explicit lower bounds for Λ_{m,k}, see [5, Sec. 1.4], turn out to be true, then the supremum in the smooth Yamabe invariant of ℂℙ^{3} is not attained by the Fubini-Study metric. The current article will not give explicit positive lower bound for Λ_{m,k}, but it will provide many relations to a spinorial analogue of the problem. The smooth Yamabe invariant σ^{∗}(M) has a spinorial analogue σ_{spin}^{∗}(M), cf. Subsection 2.2. For closed manifolds the Hijazi inequality gives σ_{spin}^{∗}(M) ≥ σ^{∗}(M). As in the Yamabe case there is a surgery formula for σ_{spin}^{∗}, cp. Theorem 2.6, and again a threshold constant Λ_{m,k}^{spin} appears. However, in contrast to the (non-spinorial) Yamabe invariant codimension 2 is allowed as well. This has implications for the smooth Yamabe constant as well: If M and N are arbitrary closed spin manifolds (not necessarily simply connected), and if M is spin-bordant to N, then σ_{spin}^{∗}(M) < Λ_{m,k}^{spin} for k = 0, …, m − 2 implies σ_{spin}^{∗}(M) = σ_{spin}^{∗}(N). In particular, σ_{spin}^{∗}(M) ≥ σ^{∗}(N). Finding interesting manifolds with σ_{spin}^{∗}(M) < Λ_{m,k}^{spin} consists of two parts. First one has to obtain explicit positive lower bounds for Λ_{m,k}^{spin} which is the main subject of the present article and then finding examples for M which is not covered here. As the threshold constant are defined as (spinorial) Yamabe-type invariants of noncompact model spaces, one expects in view of the Hijazi inequality that Λ_{m,k}^{spin} ≥ Λ_{m,k}. This question is quite subtle because on noncompact manifolds there are several ways to define Yamabetype invariants which are sometimes related and sometimes unrelated to each other. One goal of the article is to clarify these relations. The structure of the article is as follows: In Section 2 we fix notations, summarize some preliminaries and recall existing results. In particular, we define the model spaces and several versions of the spinorial and non-spinorial Yamabe-type invariants for noncompact manifolds. This allows us to summarize the results of the article in Section 3. These results are proved in the remaining sections. In particular, in Section 4 we provide regularity statement for the Euler-Lagrange equation of the spinorial Yamabe functional, which is a nonlinear Dirac eigenvalue equation. For more details we refer to the end of Section 4.

2. Preliminaries

Throughout the article we assume that the reader is familiar with the basic facts about the solution of the Yamabe problem on closed manifolds by Trudinger, Aubin, Schoen and Yau. There are many beautifully written introductions in the literature, e.g. [34], [28].

2.1. Notations. In the article a spin manifold always means a manifold admitting a spin structure together with a fixed choice of spin structure. The notion of a (topological) spin structure can be defined for arbitrary oriented manifolds. A topological spin structure is a principal bundle P for the non-trivial double cover of GL_{+}(n, ℝ), together with an equivariant map ϑ : P → P_{GL_{+}(n,ℝ)}M to the GL_{+}(n, ℝ)- principal bundle P_{GL_{+}(n,ℝ)}M of positively oriented frames. Topological spin structures exist, as soon as an obstruction, the second Stiefel-Whitney class w_{2} of its tangent bundle vanishes. For defining the spinor bundle, we should pass to metric spin structures. By restricting P_{GL_{+}(n,ℝ)}M to (positively oriented) orthonormal frames we obtain the SO(n)-principal bundle P_{SO(n)}M and the Spin(n)-principal bundle P_{Spin(n)}M := ϑ^{−1}(P_{SO(n)}M). Furthermore ϑ restricts to an equivariant map P_{Spin(n)}(M) → P_{SO(n)}M. A metric spin structure is a Spin(n)-principal bundle together with such an equivariant map. Above we have seen, how we obtain a metric spin structure from a topological one, and this construction actually yields a bijection from the set of equivalence classes of topological spin structures to the set of equivalence classes of metric spin structures. Thus one can identify topological and metric spin structures, and they are simply called spin structures. Applying the associated bundle construction for the spinor representation s : Spin(n) → GL($) we obtain the spinor bundle S_{M} := P_{Spin(n)}M ×_{s} $. In case the underlying manifold M and its metric are fixed, we shortly write S = S_{M}. The space of spinors, i.e., sections of S, is denoted by Γ(S). The space of smooth compactly supported sections is called C_{c}^{∞}(M, S). The hermitian metric on fibers of S is written as ⟨., .⟩, the corresponding norm as |.|. We write (., .)_{g} for the L^{2}-product of spinors. We denote by D: C_{c}^{∞}(M, S) → C_{c}^{∞}(M, S) the Dirac operator on (M, g). In case several manifolds or metrics are involved, we sometimes specify its affiliation, i.e., D^{g}, D^{M} or D^{M,g}. Analogously we proceed for other operators and quantities. The sphere 𝕊^{1} carries two spin structures, one of them, the so-called bounding spin structure is obtained by restricting the unique spin structure on the two-dimensional disk to its boundary. The kernel of the Dirac operator for this spin structure is trivial. The sphere 𝕊^{1} with the other spin structure represents the non-trivial spin-bordism class in dimension 1. In the article we will always assume that 𝕊^{1} is equipped with the bounding spin structure, unless stated otherwise. A Riemannian manifold is of bounded geometry if it is complete, its injectivity radius is bounded from below and the curvature tensor and all derivatives are bounded. The ball around x ∈ M with radius ε w.r.t. the metric g on M is written as B_{ε}^{M,g}(x) = B_{ε}(x) ⊂ M. In the article we need several Sobolev and Schauder spaces: For s ∈ [1, ∞] we write ∥.∥_{L^{s}(g)} for the L^{s}-norm on (M, g). In case the underlying metric is clear from the context we abbreviate shortly by ∥.∥_{s}. Let H_{k}^{s} denote both the space of distributions on M and the one of distributional sections in S_{M} that have finite H_{k}^{s}norm given by

∥φ∥_{H_{k}^{s}}^{s} = ∑_{i=0}^{k} ∥(∇)^{i}φ∥_{L^{s}}^{s}.

Here ∇ denotes the covariant derivative on M and S_{M}, respectively, dependent whether φ is a distribution on M and a distributional section in S_{M}, respectively. H_{k,loc}^{s} means that any restriction of the distribution to a compact subset has to be in H_{k}^{s} of that subset. The space of i-times continuously differentiable functions on M is denoted by C^{i}(M), and C^{i,α} denotes the corresponding Schauder space for α ∈ (0, 1].

2.2. The model spaces 𝕄_{c}^{m,k}. Let 0 ≤ k ≤ m − 1 and c ∈ [0, 1]. (𝕄_{c}^{m,k} = ℍ_{c}^{k+1} × 𝕊^{m−k−1}, g_{c} = g_{ℍ_{c}^{k+1}} + σ^{m−k−1}) where σ^{m−k−1} denotes the standard metric on 𝕊^{m−k−1} and (ℍ_{c}^{k+1}, g_{ℍ_{c}^{k+1}}) is the rescaled hyperbolic space with scalar curvature −c^{2}k(k + 1) if c ∈ (0, 1] and the Euclidean space if c = 0. We introduce coordinates on ℍ_{c}^{k+1} by equipping ℝ^{k+1} with the metric g_{ℍ_{c}^{k+1}} = dr^{2}+f(r)^{2}σ^{k} where

f_{c}(r) := sinh_{c}(r) := \begin{cases}\frac{1}{c} sinh(cr) & if c ≠ 0 \\ r & if c = 0. \end{cases}

The manifold (𝕄_{1}^{m,k} = ℍ^{k+1} × 𝕊^{m−k−1}, g_{1} = g_{ℍ^{k+1}} + σ^{m−k−1} = sinh^{2} t σ^{k} + dt^{2} + σ^{m−k−1}) is conformal to (𝕊^{m} \ 𝕊^{k}, σ^{m}), [5, Prop. 3.1],

𝔲: ℍ^{k+1} × 𝕊^{m−k−1} → 𝕊^{m} \ 𝕊^{k}, g_{1} = f^{2}u^{∗}σ^{m} where f = f(t) = cosh^{2} t.

2.3. Regularity theory. We recall the standard estimates:

Theorem 2.1. Let (M^{m}, g) be a Riemannian spin manifold of bounded geometry. Let R > 0 be smaller than the injectivity radius of M, and let r ∈ (0, R).

(i) (Inner L^{s}-estimate, [19, proof of Thm. 8.8], spin version [2, proof of Thm. 3.2.1 and 3.2.3]) Let φ ∈ H_{1,loc}^{s} be a solution of Dφ = ψ for ψ ∈ H_{k,loc}^{s}. Then, there exists a constant C = C(s, r, R) such that for all x ∈ M

∥φ∥_{H_{k+1}^{s}(B_{r}(x))} ≤ C(∥φ∥_{L^{s}(B_{R}(x))} + ∥ψ∥_{H_{k}^{s}(B_{R}(x))})

\begin{matrix}a & p & p^{∗} & q & q^{∗} \\ 4\frac{m−1}{m−2} & \frac{2m}{m−2} & \frac{2m}{m+2} & \frac{2m}{m−1} & \frac{2m}{m+1} \end{matrix}

Figure 1. Some constants in the article for m-dimensional manifolds

(ii) (Embedding into C^{0,γ}) Let m < s and 0 ≤ γ ≤ 1 − \frac{m}{s}. By the spin version of the proof of [19, Sect. 7.8 (Thm. 7.26)] there exists a constant C = C(s, r) such that H_{1}^{s}(B_{R}(x)) is continuously embedded in C^{0,γ}(\overline{B_{r}(x)}) for all x ∈ M.

(iii) (Schauder estimates) [2, Corollary 3.1.14] There is a constant C = C(r, R, k) > 0 such that for α > 0, ψ ∈ C^{k,α} with Dφ = ψ weakly it holds for all x ∈ M

∥φ∥_{C^{k+1,α}(B_{r}(x))} ≤ C(∥φ∥_{C^{k}(B_{R}(x))} + ∥ψ∥_{C^{k,α}(B_{R}(x))}).

(iv) (Sobolev Embedding into L^{p}, [19, Thm. 7.26]) Let k, ℓ ∈ ℝ, k ≥ ℓ and s, t ∈ (1, ∞) with k −(m/s) ≥ ℓ−(m/t), then the restriction map H_{k}^{s}(B_{R}(x), S) → H_{ℓ}^{t}(B_{r}(x), S) is continuous for all x ∈ M and r > 0. For fixed R > r > 0 the operator norm of these restriction maps can be chosen uniformly in x.

2.4. L^{s}-invertibility of Dirac operators. For a complete manifold M, we define the norm ∥φ∥H̃_{1}^{s} := ∥φ∥_{s} + ∥Dφ∥_{s} for 1 ≤ s < ∞. For 1 ≤ s < ∞, let H̃_{1}^{s} = H̃_{1}^{s}(M, S) be the completion of C_{c}^{∞}(M, S) w.r.t. the norm ∥φ∥H̃_{1}^{s}. Then D_{s}: H̃_{1}^{s} := dom D_{s} ⊂ L^{s} → L^{s} is a closed extension of the Dirac operator. By [10, Lem. B.2] we have (D_{s})^{∗} = D_{s^{∗}} for 1 < s < ∞ and s^{−1} + (s^{∗})^{−1} = 1. Note that on manifolds of bounded geometry and 1 < s < ∞ the H_{1}^{s}-norm and the graph norms H̃_{1}^{s} are equivalent, [10, Lem. A.2]. General properties of the L^{s}-spectrum of Dirac operators can be found in [10, App. B]. Here, we only cite the result on L^{s}-invertibility of our model spaces.

Proposition 2.2. [10, Thm. 1.1] Let 1 ≤ s ≤ ∞. The Dirac operator D: L^{s} → L^{s} on 𝕄_{c}^{m,k} is L^{s}-invertible if λ_{1} = \frac{m−k−1}{2} > ck |1/s − 1/2|.

2.5. Yamabe type constants and Yamabe type invariants. Let (M^{m}, g) be a complete m-dimensional Riemannian spin manifold. By Δ^{g} we denote the Laplacian on (M, g) and by scal^{g} its scalar curvature. Let L^{g} = aΔ^{g} + scal^{g} be the conformal Laplacian where a = 4\frac{m−1}{m−2}. We recall the following definitions

Definition 2.3. Functionals

F(v) := \frac{∫_{M} vL^{g}v dvol_{g}}{∥v∥_{L^{\frac{2m}{m−2}} (g)}^{2}} , F^{spin}(φ) := \frac{∥D^{g}φ∥_{L^{\frac{2m}{m+1}} (g)}^{2}}{(D^{g}φ, φ)_{g}}

For further use we define for the rest of the paper p := \frac{2m}{m−2} and q := \frac{2m}{m−1}. The corresponding Euler-Lagrange equations for normalized solutions are the so-called Yamabe equation [34]

L^{g}v = μv^{\frac{m+2}{m−2}} , ∥v∥_{L^{p= \frac{2m}{m−2}}} = 1

and the spinorial brother [1]

D^{g}φ = λ|φ|^{\frac{2}{m−1}} φ, ∥φ∥_{L^{q= \frac{2m}{m−1}}} = 1.

Yamabe type constants defined by compactly supported test functions.

Q^{∗}(M, g) := inf {F(v) | v ∈ C_{c}^{∞}(M) \ {0}} ,
λ_{min}^{+,∗}(M, g) := inf {F^{spin}(φ) | φ ∈ C_{c}^{∞}(M, S), (D^{g}φ, φ)_{g} > 0}

Yamabe type constants defined over solutions. Q̃(M, g) is the Yamabe invariant “defined over the solutions”, i.e.,

Q̃(M, g) := inf{μ_{v} | v ∈ Ω^{(1)}(M, g)}

where Ω^{(1)}(M, g) is the set of all nonnegative functions v ∈ C^{2}(M) ∩ L^{∞}(M) ∩ L^{2}(M, g) satisfying L^{g}v = μ_{v}v^{p−1} for a real number μ_{v} and with ∥v∥_{L^{p}(M,g)} = 1 (p = \frac{2m}{m−2} as always). Analogously, we introduce a quantity corresponding to λ_{min}^{+,∗}(M, g) defined using the solutions of the Euler-Lagrange equation of F^{spin}:

λ̃_{min}^{+}(M, g) := inf{λ ∈ (0, ∞) | ∃φ ∈ L^{∞} ∩ L^{2} ∩ C^{1} :
0 < ∥φ∥_{L^{\frac{2m}{m−1}} (M,g)} ≤ 1, D^{g}φ = λ|φ|^{\frac{2}{m−1}} φ\}.  (1)

We will see in the next section why these different quantities are geometrically relevant.

Renormalized spinorial invariants We also introduce renormalized versions of λ_{min}^{+,∗} and λ̃_{min}^{+}:

Q_{spin}^{∗}(M, g) = 4\frac{m − 1}{m}λ_{min}^{+,∗}(M, g)^{2}, Q̃_{spin}(M, g) = 4\frac{m − 1}{m}λ̃_{min}^{+}(M, g)^{2}.  (2)

This renormalization will make simpler.

Yamabe type invariants for compact manifolds. Now we define the smooth Yamabe invariant σ^{∗}(M) as

σ^{∗}(M) := sup_{g} Q^{∗}(M, g)

where the supremum runs over all Riemannian metrics on M. Thus, σ^{∗} only depends on the diffeomorphism type of M. Note that the smooth Yamabe invariant is positive if and only if M admits a metric of positive scalar curvature. A similar spinorial Yamabe invariant τ^{+}(M) was introduced in [12, 11]. It is

τ^{+}(M) := \begin{cases}sup_{g∈R^{inv}(M)} λ_{min}^{+,∗}(M, g) & if R^{inv}(M) ≠ ∅ \\ 0 & if R^{inv}(M) = ∅, \end{cases}

where R^{inv}(M) is the set of Riemannian metrics on M such that D^{g} is invertible. The definition of τ^{+} is slightly different to the original one in [12, 11], but obviously equivalent. Note that for connected closed manifolds one knows from [3] that R^{inv}(M) ≠ ∅ if and only if the index of M^{m} in KO_{m} vanishes. Thus, τ^{+}(M) is positive if and only if this index vanishes. The invariant τ^{+}(M) only depends on both the diffeomorphism type of M and its spin structure. These Yamabe type invariants will be considered in this article only in the case that M is compact. In this case the solution of the classical Yamabe problem [34] implies Q̃(M, g) = Q^{∗}(M, g), and similar results in the spin case [1] implies λ̃_{min}^{+}(M, g) = λ_{min}^{+,∗}(M, g). Thus, we also see

σ^{∗}(M) = sup_{g} Q̃(M, g), τ^{+}(M) = \begin{cases}sup_{g∈R^{inv}(M)} λ̃_{min}^{+}(M, g) & if R^{inv}(M) ≠ ∅ \\ 0 & if R^{inv}(M) = ∅. \end{cases}

Similar to above we also define (for M compact) a renormalized version

σ_{spin}^{∗}(M) := 4\frac{m − 1}{m}τ^{+}(M)^{2}.

We want to remark that σ^{∗}(M) was considered for non-compact manifolds in [25].

The Λ-invariants. We define

Λ̃_{m,k} := inf_{c∈[0,1]} Q̃(𝕄_{c}^{m,k}) and Λ_{m,k}^{∗} := inf_{c∈[0,1]} Q^{∗}(𝕄_{c}^{m,k}).

These invariants are important because of their relation to the invariant Λ_{m,k} that contributes to the Surgery Theorem 2.5. We have Λ_{m,k} = Λ̃_{m,k} unless m = k − 3 ≥ 7 or m = k − 2 [6, Thm. 3.1 and proof of Corollary 3.2]. The idea behind the notation is that the invariant with ∗ is the infimum of a functional, the invariant with ∼ is defined using solutions of the Euler-Lagrange equation and the invariant without such decoration is the invariant in the surgery theorem. We know from [6, Thm. 3.3] that all these invariants are positive for 0 ≤ k ≤ m − 3. In the spinorial case we define similarly

Λ̃_{m,k}^{spin} = inf_{c∈[0,1]} Q̃_{spin}(𝕄_{c}^{m,k}) and Λ_{m,k}^{spin,∗} := inf_{c∈[0,1]} Q_{spin}^{∗}(𝕄_{c}^{m,k}).

It is known from [4, Thm. 1.1] that Λ̃_{m,k}^{spin} > 0 for all 0 ≤ k ≤ m − 2. The invariant Λ_{m,k}^{spin} in the Spinorial Surgery Theorem 2.6 can be chosen to be Λ̃_{m,k}^{spin} for all 0 ≤ k ≤ m − 2, [4, Cor. 1.4]. We introduce the notation Λ_{m,k}^{spin} := Λ̃_{m,k}^{spin} to make the presentation analogous to the non-spin case. One of the main goals of this article is to search for relations between these five possibly different Λ-invariants.

Remark 2.4. The Q-invariants for the spheres play a special role. We collect the main properties: For all manifolds (M^{m}, g) it holds Q^{∗}(M^{m}, g) ≤ Q^{∗}(𝕊^{m}). If M is spin, then Q_{spin}^{∗}(M^{m}, g) ≤ Q_{spin}^{∗}(𝕊^{m}). The invariant Q^{∗}(𝕊^{m}) is attained by a constant test function

v such that ∥v∥_{\frac{2m}{m−2}} = 1. Thus, Lv = m(m − 1)v = Q^{∗}(𝕊^{m})v^{\frac{m+2}{m−2}} and Q^{∗}(𝕊^{m}) = m(m −

1)vol(𝕊^{m})^{\frac{2}{m}} . The invariant λ_{min}^{+,∗}(𝕊^{m}) is attained by a Killing spinor φ to the Killing constant −\frac{1}{2}. Note that the normalization in (2) is chosen such that Q_{spin}^{∗}(𝕊^{m}) = Q^{∗}(𝕊^{m}). Since 𝕊^{m} is closed Q^{∗}(𝕊^{m}) = Q̃(𝕊^{m}) and Q_{spin}^{∗}(𝕊^{m}) = Q̃_{spin}(𝕊^{m}). Moreover, for (M^{m}, g) not locally conformally flat and m ≥ 6 Aubin showed, see [14, p.292], Q^{∗}(M^{m}, g) < Q^{∗}(𝕊^{m}).

2.6. Surgery-monotonicity for Yamabe type invariants below thresholds. In order to define the constant Λ_{m,k} mentioned above we set

Q^{(2)}(M, g) := inf{μ_{u} | u ∈ Ω^{(2)}(M, g)}

where Ω^{(2)}(M, g) is the set of all nonnegative functions u ∈ C^{2}(M) ∩ L^{∞}(M) satisfying L^{g}u = μ_{u}u^{p−1} for a nonnegative real number μ_{u}, ∥u∥_{L^{p}(M,g)} = 1, where p = \frac{2m}{m−2} as always,

and μ_{u}∥u∥L^{∞}^{\frac{4}{m−2}} ≥ \frac{(m−k−2)^{2}(m−1)}{8(m−2)}. Then, we set, cf. [5, Sect. 3], [6, Sect. 2.6],

Λ_{m,k} = min{Λ̃_{m,k}, inf_{c∈[0,1]} Q^{(2)}(𝕄_{c}^{m,k})}.

It follows from [6, Thm. 3.1 and below] that in case k = m − 3 ≤ 6 or k ≤ m − 4 we already have Λ_{m,k} = Λ̃_{m,k}.

Theorem 2.5 (Surgery-monotonicity for the Yamabe invariant, [5, Cor. 1.4]). Assume that N^{m} is a closed Riemannian manifold that is obtained from M^{m} by a surgery of codimension m − k ≥ 3. Then

σ^{∗}(N) ≥ min{σ^{∗}(M), Λ_{m,k}}.

Note that a surgery from M to N is called spin preserving if the spin structures on M and N extend to a spin structure on the corresponding bordism. In particular this implies that the spin structures on M and N coincide outside the region of surgery.

Theorem 2.6 (Surgery-monotonicity for the spinorial Yamabe invariant, [4, Cor. 1.4]). Assume that N^{m} is a closed Riemannian spin manifold that is obtained from M^{m} by a spin-preserving surgery of codimension m − k ≥ 2. Then

σ_{spin}^{∗}(N) ≥ min{σ_{spin}^{∗}(M), Λ_{m,k}^{spin}}.

In the case k = m − 2 the sphere 𝕊^{1} carries the bounding spin structure, as explained in the Notations 2.1.

Label  Meaning
no spin label associated to the Yamabe problem
spin  associated to the spinorial Yamabe problem for the Dirac operator
∗  defined as a variational problem
∼  defined by solutions of the associated Euler-Lagrange equation

(as a nonlinear eigenvalue problem)

no ∗ and no ∼ appears in an associated surgery result
Q  for a fixed Riemannian (spin) manifold
Λ  = inf_{c∈[0,1]} of the Q-invariants (decorated with the same labels as Λ) for

the model spaces 𝕄_{c}

σ  the supremum of the Q-invariants (with the same labels) for a

(spin) manifold over all conformal classes

Figure 2. Overview of the notations

Since there is whole zoo of different Q- and Λ-invariants, we summarize the logic of our notation in Figure 2.

3. Overview on the results

Many of the inequalities established in this article are summarized in Figure 3. For example, Λ_{m,k}^{spin,∗} = Λ̃_{m,k}^{spin} ≥ Λ_{m,k}^{∗} for all k ≤ m − 2. Other inequalities hold under additional assumptions, e.g. in the case k ≤ m − 4 and in the case k ≤ m − 3 ≤ 3 we have Λ_{m,k}^{∗} ≥ Λ̃_{m,k}. Thus, together with previously mentioned relations we obtain

Theorem 3.1. In the case k ≤ m − 4 and in the case k ≤ m − 3 ≤ 3

Λ_{m,k}^{spin,∗} = Λ̃_{m,k}^{spin} = Λ_{m,k}^{spin} ≥ Λ_{m,k}^{∗} = Λ̃_{m,k} = Λ_{m,k}.

Thus, in most of the cases the inequality Λ_{m,k}^{spin} ≥ Λ_{m,k} conjectured in the introduction holds. Together with the explicit positive lower bounds for Λ_{m,k} in [7, 6], we then obtain explicit positive lower bounds for Λ_{m,k}^{spin}. Theorem 3.1 does not provide for Λ_{m,m−3}^{spin} for m > 6. But nevertheless our techniques also allow to obtain explicit positive lower bounds for Λ_{m,m−3}^{spin} for m > 6, see Section 11. The right hand side of Figure 3 gives relations between the Q-invariants of the model spaces. Some of them require additional assumptions which are given as footnotes. The parameter c ranges in the interval [0, 1]. However, the case c = 1 is very special as then 𝕄_{1}^{m,k} is conformal to a subset of 𝕊^{m} which allows much stronger statements. This is summarized in Section 7. Another special case is k = m−1. These invariants do not have similar geometric applications. But for the sake of completeness we summarize in Section 8. As already mentioned in the introduction, the explicit positive lower bounds for Λ_{m,k}^{spin}, k ≤ m − 3 lead to bordism invariant. As we only want to give an overview here, many proofs will be given later, i.e. in section 12. Let m ≥ 5. We set

Λ_{m}^{spin} := min{Λ_{m,2}^{spin}, Λ_{m,3}^{spin}, …, Λ_{m,m−3}^{spin}}.

Λ_{m,k}^{spin,∗} Λ̃_{m,k}^{spin}

≤ (for k ≤ m − 2, Cor. 6.4)

≥ (for k ≤ m − 2, Prop. 5.3)

Λ_{m,k}^{∗} Λ̃_{m,k}

≤ (Cor. 6.2)

≥ (for k ≤ m − 4 or k ≤ m − 3 ≤ 3, Prop. 5.3)

≤ (for k ≤ m − 2, Cor. 9.8)

Q_{spin}^{∗}(𝕄_{c}^{m,k})

Q̃_{spin}(𝕄_{c}^{m,k})

≤ (for k ≤ m − 2, Cor. 6.4)

≥ (Cor. 10.6^{[1]})

Q^{∗}(𝕄_{c}^{m,k}) Q̃(𝕄_{c}^{m,k})

≤ (Cor. 6.2)

≥ (Cor. 10.2^{[2]})

≤ (for k ≤ m − 2, Cor. 9.8)

Figure 3. Summary of the results for the Q-invariants of the model spaces (right) and the corresponding Λ-invariants (left). [1]: for (m−k −1)^{2} > c^{2}k(k +1) and Q̃_{spin}(𝕄_{c}^{m,k}) < Q_{spin}^{∗}(𝕊^{m}) [2]: for (m − k − 1)(m − k − 2) > c^{2}k(k + 1), c ∈ [0, 1) or for k ≤ m − 3, c = 1

\begin{matrix}m & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ Q^{∗}(𝕊^{m}) & 80.0 & 96.3 & 113.5 & 130.7 & 147.9 & 165.0 & 182.2 & 199.3 & 216.4 \\ Λ_{m}^{spin} ≥ & 45.1 & 50.0 & 65.2 & 78.7 & 91.8 & 104.9 & 118.1 & 131.5 & 145.0 \\ Q^{∗}(ℍP^{2} × ℝ^{m−8}) ≥ & — & — & — & 121.4 & 138.5 & 97.3 & 135.9 & 158.7 & 178.0 \end{matrix}

Table 1. Some explicit lower bounds for Λ_{m}^{spin}: The values are rounded – Q^{∗}(𝕊^{m}) is rounded to the nearest multiple of 1/10 and the lower bounds for Λ_{m}^{spin} and Q^{∗}(ℍP^{2}×ℝ^{m−8}) are always rounded down. Note that Q^{∗}(ℍP^{2}) = 121.4967... is attained by the canonical metric on ℍP^{2}, due to Obata’s Theorem.

From Theorem 3.1, Section 11, and results in [6] and [7] we obtain explicit positive lower bounds for Λ_{m}^{spin}, summarized in Table 1 for low dimensions. Using standard techniques from bordism theory (see Section 12 for details) one obtains several conclusions:

Proposition 3.2. Let M be an m-dimensional closed connected, simply connected spin manifold, α(M) = 0. If 5 ≤ m ≤ 7, then

σ_{spin}^{∗}(M) ≥ Λ_{m}^{spin}.

For m ≥ 11 or m = 8 we have

σ_{spin}^{∗}(M) ≥ min{Λ_{m}^{spin}, Q^{∗}(ℍP^{2} × ℝ^{m−8})}.

For m = 9, 10 we have

σ_{spin}^{∗}(M) ≥ min{Λ_{m,1}^{spin}, Λ_{m}^{spin}, Q^{∗}(ℍP^{2} × ℝ^{m−8})}.

Note that by definition α(M) ≠ 0 implies that there are no invertible Dirac operators, thus by definition σ_{spin}^{∗}(M) = 0. We conjecture Λ_{m}^{spin} ≤ Q^{∗}(ℍP^{2}×ℝ^{m−8}) for all m ≥ 11, which would imply σ_{spin}^{∗}(M) ≥ Λ_{m}^{spin} for all closed simply connected spin manifolds M with dimension m ≥ 5, m ≠ 9, 10. A similar bound also exists for non-simply connected manifolds, namely in this case for m ≠ 9, 10

σ_{spin}^{∗}(M) ≥ min{Λ_{m}^{spin}, Λ_{m,m−2}^{spin}, Q^{∗}(ℍP^{2} × ℝ^{m−8})} > 0,

and for m = 9, 10

σ_{spin}^{∗}(M) ≥ min{Λ_{m,1}^{spin}, Λ_{m}^{spin}, Λ_{m,m−2}^{spin}, Q^{∗}(ℍP^{2} × ℝ^{m−8})} > 0,

however this positive lower bound is not explicit as no explicit lower bound for Λ_{m,m−2}^{spin} is currently available. Numerical calculations and some further assumptions indicate that Λ_{m,m−2}^{spin} < Λ_{m}^{spin}.

Proposition 3.3. Assume that M is an m-dimensional closed connected spin manifold with m ≥ 5. We consider the bordism groups Ω_{m}^{spin}(BΓ), Γ := π_{1}(M) where the boundaries and the bordisms are spin manifolds together with maps to BΓ. Let c_{M}: M → BΓ be a classifying map of the universal covering of M, i.e., the map which induces an isomorphism from π_{1}(M) to Γ = π_{1}(BΓ). Let [N, f] = [M, c_{M}] ∈ Ω_{m}^{spin}(BΓ), and let N be connected. Then

σ_{spin}^{∗}(N) ≥ min{σ_{spin}^{∗}(M), Λ_{m}^{spin}, Λ_{m,m−2}^{spin}}.

If N is connected and if f induces an isomorphism from π_{1}(N) to Γ, then

σ_{spin}^{∗}(N) ≥ min{σ_{spin}^{∗}(M), Λ_{m}^{spin}}.  (3)

Note that every class in Ω_{m}^{spin}(BΓ) → ℝ can be written as (M, c_{M}). By applying (3) twice, it follows from Proposition 3.3 that there is a well-defined map s^{spin} : Ω_{m}^{spin} → ℝ such that for all connected, simply connected spin manifolds M

s^{spin}([M]) = min{σ_{spin}^{∗}(M), Λ_{m}^{spin}}.

Thus if M is a connected spin manifold, we have

σ_{spin}^{∗}(M) ≥ min{s^{spin}([M]), Λ_{m,m−2}^{spin}}.

It follows from standard arguments of surgery theorem that

s^{spin}([M] + [N]) ≥ min{s^{spin}([M]), s^{spin}([N])}.

Thus,

Ω_{m}^{spin,>t} := {[M] ∈ Ω_{m}^{spin} | s^{spin}([M]) > t}

is a subgroup of Ω_{m}^{spin}. For example, Ω_{m}^{spin,>0} is the kernel of the index map α: Ω_{m}^{spin} → KO_{m}. Similarly as above, for an arbitrary finitely presented group Γ we obtain a well-defined map s_{Γ}^{spin} : Ω_{m}^{spin}(BΓ) → ℝ as follows: For every [M, f] ∈ Ω_{m}^{spin}(BΓ) where M is connected and f induces an isomorphism from π_{1}(M) to Γ we have

s_{Γ}^{spin}([M, f]) = min{σ_{spin}^{∗}(M), Λ_{m,1}^{spin}, Λ_{m}^{spin}}.

In this case the minimum includes the constants Λ_{m,1}^{spin} since it is required to show that

s_{Γ}^{spin}([M, f] + [N, g]) ≥ min{s_{Γ}^{spin}([M, f]), s_{Γ}^{spin}([N, g])}.

Then, analogously as above,

Ω_{m}^{spin,>t}(BΓ) := {[M, f] ∈ Ω_{m}^{spin}(BΓ) | s_{Γ}^{spin}([M, f]) > t}

is a subgroup of Ω_{m}^{spin}(BΓ).

Assume that there is a closed simply connected spin manifold M of dimension m ≥ 5 with σ_{spin}^{∗}(M) < Λ_{m}^{spin}. For such manifolds one would have: If N is a simply connected closed spin manifold spin-bordant to M, then σ_{spin}^{∗}(N) = σ_{spin}^{∗}(M). An advantage of this bordism result is that we have explicit positive lower bounds for Λ_{m}^{spin}, in contrast to a similar result for the classical Yamabe invariant. As a consequence by the Hijazi inequality we have σ^{∗}(N) ≤ σ_{spin}^{∗}(M), i.e., σ_{spin}^{∗}(M) is an upper bound for the Yamabe invariant for all simply connected manifolds in [M]. This question is related to the open problem whether there is a manifold in dimension m ≥ 5 with Yamabe invariant different from 0 and σ^{∗}(𝕊^{m}). If one finds an M as above, all simply connected manifolds in the spin bordism class of M would have a Yamabe invariant in (0, σ^{∗}(𝕊^{m})).

Many of the statements on the right hand side of Figure 3 are still valid if one replaces the model spaces by arbitrary manifolds of bounded geometry, see Sections 6 and 9. The inequalities in Section 9 are noncompact versions of the Hijazi inequality which is of central importance of our article. The reader should be aware that there are different ways to generalize from the compact to the noncompact setting. We have positive and negative results for the generalization of the Hijazi inequality to the noncompact setting, see Section 9. Our investigations also need regularity statements for the Euler-Lagrange equation of the spinorial functional. For this purpose we have included Section 4 which might be of independent interest and which goes beyond the requirements of the following sections.

4. Improvements of regularity for the Dirac Euler-Lagrange equation

Let (M^{m}, g) be a Riemannian spin manifold of bounded geometry. In this section, we consider a spinor φ ∈ L^{q} and φ ∈ L_{loc}^{s} for an s > q that fulfills

Dφ = λ|φ|^{q−2}φ weakly,  (4)

i.e., in the distributional sense, where as always q = \frac{2m}{m−1}. Note that from φ ∈ L_{loc}^{s} for an s > q it follows with the methods of [2, Thm. 5.2] that φ is C^{1,α} for all α ∈ (0, 1). We omit the proof of this local statement since the proof is completely analogous as in [2]. Furthermore, we will only use the fact that φ is continuous which is part of the assumptions in the applications of this subsection. We want to further examine the regularity of φ. First, we will show that φ ∈ L^{∞}. For that we need the following auxiliary lemma.

Lemma 4.1. Fix β, R, δ > 0. Let φ ∈ Γ(S_{M}) be continuous with ∥φ∥_{L^{∞}} = ∞. Then there is a sequence (x_{i})_{i∈ℕ} in M with |φ(x_{i})| ≥ i and

|φ(x_{i})|^{−1}∥φ∥_{L^{∞}(B_{i}^{R})} ≤ 1 + δ

where B_{i}^{R} := B_{R |φ(x_{i})|^{−1/β}}(x_{i}).

Proof. Let d(., .) denote the distance in (M, g), and fix R, δ > 0. We prove the claim by contradiction: Assume that there is a constant C > 0 such that for all x ∈ M with |φ(x)| ≥ C there is y_{x} with d(x, y_{x}) < R |φ(x)|^{−1/β} and |φ(y_{x})| > (1 + δ) |φ(x)|. Then, we define a sequence x_{i} recursively by choosing x_{0} ∈ M with |φ(x_{0})| ≥ C and x_{i+1} = y_{x_{i}} for all i ≥ 0. Then, |φ(x_{i})| ≥ (1 + δ)^{i}|φ(x_{0})| ≥ (1 + δ)^{i}C → ∞ as i → ∞. But,

d(x_{i}, x_{0}) ≤ ∑_{j=0}^{i−1} d(x_{j+1}, x_{j}) ≤ ∑_{j=0}^{i−1} R |φ(x_{j})|^{− \frac{1}{β}} ≤ RC^{− \frac{1}{β}} ∑_{j=0}^{i−1}(1 + δ)^{− \frac{j}{β}} ≤ \frac{RC^{− \frac{1}{β}}}{1 − (1 + δ)^{− \frac{1}{β}}} < ∞

which then contradicts the continuity of φ. □

Lemma 4.2. Let (M^{m}, g) be of bounded geometry. Let φ ∈ L^{q} ∩ C^{0} be a weak solution of (4). Then φ ∈ L^{∞}.

Proof. We assume the contrary, i.e., ∥φ∥_{∞} = ∞. We fix β := 2(q − 2), R smaller than the injectivity radius, and some δ > 0. Then applying Lemma 4.1 there is a sequence of points (x_{i})_{i∈ℕ} in M with |φ(x_{i})| ≥ i and ∥φ∥_{L^{∞}(B_{i}^{R})} ≤ (1 + δ)|φ(x_{i})|. After passing to a subsequence, every compact subset only contains a finite number of x_{i}. We thus assume that all B_{i}^{R} are pairwise disjoint since this can always be achieved by passing to a further subsequence. We consider the charts for B_{i}^{R} given by rescaled exponential maps

u_{i}: B_{r_{i}}(0) ⊂ ℝ^{m} → B_{i}^{R}, v ↦ exp_{x_{i}}(δ_{i}v)

where m_{i} := |φ(x_{i})|, δ_{i} := m_{i}^{− \frac{1}{q−2}} and r_{i} := δ_{i}^{−1}R |φ(x_{i})|^{− \frac{1}{β}} = R |φ(x_{i})|^{\frac{1}{2(q−2)}} . Note that m_{i} = |φ(x_{i})| ≥ i → ∞ and, hence, δ_{i} → 0 and r_{i} → ∞ as i → ∞. The map u_{i} induces a map on the frame bundles which lifts to the spinor bundles, for details see [16]. For simplicity, we denote this lift also by u_{i}, and set ψ^{i} := m_{i}^{−1}u_{i}^{∗}φ. Then ψ^{i} is a spinor on B_{r_{i}}(0), |ψ^{i}(0)| = 1 and ∥ψ^{i}∥_{L^{∞}(B_{r_{i}} (0))} ≤ 1 + δ. Using the comparison of the Dirac operator with the one on the Euclidean space [8, Sect. 3 and 4], we obtain from Dφ = λ|φ|^{q−2}φ that

D^{ℝ^{m}}ψ^{i} + \frac{1}{4} ∑_{αβγ} Γ̃_{αβ}^{γ}e_{α} · e_{β} · e_{γ} · ψ^{i} + ∑_{αβ}(b_{α}^{β} − δ_{α}^{β})e_{α} · ∇_{e_{β}}ψ^{i} = λ|ψ^{i}|^{q−2}ψ^{i}

where δ_{α}^{β} denotes the Kronecker symbol, e_{α} is the standard orthonormal frame on ℝ^{m} and

b_{α}^{β} =δ_{α}^{β} − \frac{1}{6}δ_{i}^{2}R_{αλμβ}^{i}x^{λ}x^{μ} + O(δ_{i}^{3}|x|^{3}) → δ_{α}^{β}
Γ̃_{αβ}^{γ} =∂_{α}b_{β}^{γ} − \frac{1}{3}δ_{i}(R_{αγλβ}^{i} + R_{αλγβ}^{i})x^{λ} + O(δ_{i}^{2}|x|^{2}) → 0

as δ_{i} → 0, i → ∞. Here, R_{αλμβ}^{i} = g_{x_{i}}([∇_{∂_{β}}, ∇_{∂_{μ}}]∂_{α} − ∇_{[∂_{β},∂_{μ}]}∂_{α}, ∂_{λ}) is the Riemannian curvature tensor of g at x_{i}. Let K_{j}, j = 0, 1, 2, 3 be compact subsets of ℝ^{m} with K_{j+1} ⊂ interior(K_{j}) for j = 0, 1, 2, and let i_{0} be big enough such that K_{0} ⊂ B_{r_{i_{0}}} (0). Since ψ^{i} is bounded on B_{r_{i}}(0) for i ≥ i_{0}, the inner L^{s}-estimate in Theorem 2.1 shows that for each s the ψ^{i}’s are uniformly bounded in H_{1}^{s}(K_{0}). Thus, after passing to a subsequence ψ^{i} → ψ weakly in H_{1}^{s}(K_{0}). The restriction map H_{1}^{s}(K_{0}) → C^{0,γ}(K_{1}) is bounded because of Theorem 2.1, hence the ψ^{i} are uniformly bounded also in C^{0,γ}(K_{1}) for all γ ∈ (0, 1). In particular, |ψ^{i}|^{q−2}ψ^{i} are uniformly bounded in C^{0,γ}(K_{1}). Thus, by the Schauder estimate (see Theorem 2.1) we obtain ψ ∈ C^{1,γ}(K_{2}) and, thus, by Arzela-Ascoli ψ^{i} → ψ strongly in C^{1} on K_{3}, after passing to a subsequence. We apply this construction to K_{3} := B_{k}(0) and construct a diagonal subsequence for k → ∞. This subsequence converges locally in C^{1} to a spinor ψ on ℝ^{m} with |ψ(0)| = 1, ∥ψ∥_{L^{∞}} ≤ 1+δ and D^{ℝ^{m}}ψ = λ|ψ|^{q−2}ψ. We write b_{i}dvol_{ℝ^{m}} = u_{i}^{∗}dvol_{g}, where b_{i} := \sqrt{det(u_{i}^{∗}g)} → 1 as i → ∞ in C^{1} on each compact subset of ℝ^{m}. As the balls B_{i}^{R} are disjoint, φ ∈ L^{q} implies that ∫_{B_{i}^{R}} |φ|^{q} dvol_{g} → 0 as

i → ∞. Thus, we get for all compacta K̃ and sufficiently large i

∫K̃ |ψ^{i}|^{q} dvol_{ℝ^{n}} ≤ 1.001 ∫_{|x|<r_{i}} |ψ^{i}|^{q}b_{i} dvol_{ℝ^{n}} = 1.001 ∫_{B_{i}^{R}} |φ|^{q} dvol_{g} → 0 as i → ∞.

Hence, ∥ψ∥_{q} = 0 which contradicts ψ ∈ C^{1} and |ψ(0)| = 1. Thus, φ ∈ L^{∞}. □

Lemma 4.3. Let (M^{m}, g) be of bounded geometry. Let φ ∈ L^{q} ∩C^{0} fulfill weakly (4). Then, lim_{x→∞} |φ| = 0. Moreover, φ ∈ C^{1,γ} for all γ ∈ (0, 1), lim_{x→∞} ∥φ∥_{C^{1,γ}(B_{r}(x))} = 0 for all r > 0, and ∥φ∥_{C^{1,γ}} < ∞. In particular, φ is uniformly continuous.

Proof. From Lemma 4.2 we have φ ∈ L^{∞}. Fix z ∈ M and δ > 0 to be smaller than the injectivity radius. Let d(., .) denote the distance function on (M, g). We prove the first claim by contradiction: We assume that there is constant V > 0 and a sequence (x_{i})_{i∈ℕ} ⊂ M with |φ(x_{i})| ≥ V, |x_{i}| = d(x_{i}, z) → ∞ and d(x_{i}, x_{j}) > 2δ.

Let ε ∈ (0, \frac{δ}{2}). Since φ ∈ L^{∞} and (M, g) has bounded geometry, we obtain by inner L^{s}-estimates that

∥φ∥_{H_{1}^{s}(B_{ε}(x_{i}))} ≤ C_{δ}(s)(∥φ∥_{L^{s}(B_{2ε}(x_{i}))} + ∥λ|φ|^{q−2}φ∥_{L^{s}(B_{2ε}(x_{i}))})
≤ CC_{δ}(s)vol(B_{2ε}(x_{i}))^{\frac{1}{s}} ≤ C′  (5)

where C′ does not depend on i. Fixing s > m and using the Sobolev embedding H_{1}^{s}(B_{ε}(x_{i})) ↪ C^{0,γ}(B_{ε}(x_{i})) we get that ∥φ∥_{C^{0,γ}(B_{ρ}(x_{i}))} ≤ C′′ for some γ ∈ (0, 1), ρ ∈ (0, ε), and where C′′ is independent on i. With φ ∈ L^{q} we estimate

∥φ∥_{q}^{q} ≥ ∑_{i} ∥φ∥_{L^{q}(B_{ρ}(x_{i}))}^{q} ≥ K ∑_{i} inf_{x∈B_{ρ}(x_{i})} |φ(x)|

where K := inf_{i} vol(B_{ρ}(x_{i})). Note that K > 0 since (M, g) has bounded geometry. Hence, inf_{x∈B_{ρ}(x_{i})} |φ(x)| → 0 as i → ∞. But on the other hand on each ball B_{ρ}(x_{i}) we have for all x, y ∈ B_{ρ}(x_{i}) that |φ(x) − φ(y)| ≤ C′′|x − y|^{γ} ≤ C′′ρ^{γ}. Thus

V ≤ lim sup_{i→∞} |φ(x_{i})| ≤ C′′ρ^{γ}.

By choosing ρ small enough we obtain a contradiction. Inequality (5) still holds if we replace x_{i} by an arbitrary x ∈ M. Then, C′ does not depend on x. Moreover, choose s large enough we then have for any γ ∈ (0, 1) that ∥φ∥_{C^{0,γ}(B_{ρ/2}(x))} < ∞ for all x ∈ M and lim_{x→∞} ∥φ∥_{C^{0,γ}(B_{ρ/2}(x))} = 0. Thus, φ ∈ C^{0,γ} for any γ ∈ (0, 1). Then, by a further bootstrap step we obtain the same statement for C^{1,γ} instead of C^{0,γ} and for ρ/3 instead of ρ/2. Thus, for sufficiently large compact subset K̂ the norm ∥φ∥_{C^{1,γ}(M\K̂)} is arbitrarily close to zero. This implies the lemma. □

Corollary 4.4. Let (M^{m}, g) be of bounded geometry. Let φ ∈ L^{q} ∩ C^{0} be a weak solution of Dφ = λ|φ|^{\frac{2}{m−1}} φ with ∥φ∥_{q} = 1. Then, φ ∈ C^{2,γ} for all γ ∈ (0, \frac{2}{m−1}] if m ≥ 4 and all γ ∈ (0, 1) otherwise.

Proof. Let β := q − 2 = \frac{2}{m−1} and ψ = |φ|^{β}φ. At first we will show that

∇ψ = |φ|^{β}∇φ + β⟨∇φ, φ⟩|φ|^{β−2}φ  (6)

is in C^{γ} for γ as above: By Lemma 4.3 φ ∈ C^{1,α} for all α ∈ (0, 1). Thus, φ is locally Lipschitz and, hence, |φ|^{β} is in C^{β}. Moreover, ∇φ ∈ C^{α}, thus the first summand in (6) is C^{min{α,β}}. By [1, Lem. B.1] |φ|^{β−2}φ⊗φ ∈ C^{β}. It follows that ⟨∇φ, φ⟩|φ|^{β−2}φ is C^{γ} as well. Thus, ∇ψ ∈ C^{γ} and ψ ∈ C^{α} for all α ∈ (0, 1). Now Schauder estimates, see Theorem 2.1, imply φ ∈ C^{2,γ}. The corollary then follows. □

Example 4.5. Let us consider Euclidean ℝ^{m}, m ≥ 2 with standard basis (e_{i})_{i=1,...,m} and with a parallel spinor ψ_{0} ≠ 0. We define

φ(x^{1}, …, x^{m}) := x^{1}e_{1} · ψ_{0} − x^{2}e_{2} · ψ_{0}.

Then ∇φ = dx^{1} ⊗e_{1} ·ψ_{0} −dx^{2} ⊗e_{2} ·ψ_{0}, and thus Dφ = e_{1} ·e_{1} ·ψ_{0} −e_{2} ·e_{2} ·ψ_{0} = −ψ_{0} +ψ_{0} = 0. Thus this spinor satisfies (4) with λ = 0, but is not L^{q} and many conclusions in this section, particularly the L^{∞}-bound, do not hold. The example thus shows that the L^{q}-condition in the above lemmata is necessary.

We know that by Lemma 4.2 φ is in L^{∞}. However, the following example shows that we cannot derive an upper bound for ∥φ∥_{L^{∞}} which only depends on (M, g), ∥φ∥_{L^{q}} and λ.

Example 4.6. Consider again Euclidean ℝ^{m}. Take a Killing spinor φ on the sphere (𝕊^{m}, σ^{m}) normalized such that its L^{q= \frac{2m}{m−1}} -norm is one. Then on 𝕊^{m} we have D^{𝕊^{m}}φ = \frac{m}{2}φ = λ_{min}^{+,∗}(𝕊^{m})|φ|^{q−2}φ, cf. Remark 2.4. The stereographic projection h is a conformal map from the sphere with a point removed to the Euclidean space. Let now h^{ρ} be the composition of φ and the scaling of ℝ^{m} by ρ. Then, g_{E} = f_{ρ}^{2} h_{∗}^{ρ}(σ^{m}) where f_{ρ} = ρ^{\frac{1}{2}} f_{1} is the conformal factor. Using the identification of spinor bundles of conformal metrics, cf. [29, Sect. 4], we get a spinor φ̃ = f_{ρ}^{− \frac{m−1}{2}} φ fulfilling D^{ℝ^{m}} φ̃ = λ_{min}^{+,∗}(𝕊^{m})|φ̃|^{q−1}φ̃ and

∥φ̃∥_{L^{q}(ℝ^{m})} = 1. But ∥φ̃∥_{L^{∞}(ℝ^{m})} = ρ^{− \frac{m−1}{4}} ∥f_{1}^{− \frac{m−1}{2}} φ∥_{L^{∞}(ℝ^{m})} → ∞ as ρ → 0. We obtain an example where L^{∞}-norm of solutions cannot be controlled in terms of its L^{q}-norm, λ and (M, g).

We close this section by some lemmata on removal of singularities for our Euler-Lagrange equations.

Lemma 4.7. Let (M, g) be an m-dimensional Riemannian spin manifold, and let S ⊂ M be an embedded submanifold of dimension ℓ ≤ m − s^{∗} where s^{∗} is the conjugate exponent of s. Assume that φ is a spinor field such that ∥φ∥_{s} < ∞ for s ∈ (1, ∞) and Dφ = λ|φ|^{s−2}φ weakly on M \ S for λ ∈ ℝ. Then Dφ = λ|φ|^{s−2}φ weakly on M.

Proof. We follow the proof for the removal of singularities for weakly harmonic spinors in [3, Lem. 2.4]: Let U_{S}(ε) consist of all points of M with distance ≤ ε to S. Let η_{δ} be a cut-off function with η_{δ} = 1 on U_{S}(δ), η_{δ} = 0 on M \ U_{S}(2δ) and |∇η_{δ}| ≤ 2δ^{−1}. Then, we obtain for a smooth and compactly supported spinor ψ on M

∫_{M} ⟨φ, Dψ⟩ − λ ∫_{M} ⟨|φ|^{s−2}φ, ψ⟩ = ∫_{M} ⟨φ, D(1 − η_{δ})ψ⟩ − λ ∫_{M} ⟨|φ|^{s−2}φ, (1 − η_{δ})ψ⟩

+ ∫_{M} ⟨φ, η_{δ}Dψ⟩ + ∫_{M} ⟨φ, ∇η_{δ} · ψ⟩ − λ ∫_{M} ⟨|φ|^{s−2}φ, η_{δ}ψ⟩ .

The sum of the first two summands on the right side vanishes since Dφ = λ|φ|^{s−2}φ weakly on M \S. Moreover, |∫_{M} ⟨φ, η_{δ}Dψ⟩| ≤ ∥φ∥_{s}∥Dψ∥_{L^{s^{∗}} (U_{S}(2δ))} → 0 and |∫_{M} ⟨|φ|^{s−2}φ, η_{δ}ψ⟩| ≤

∥φ∥_{s}^{s/s^{∗}}∥ψ∥_{L^{s}(U_{S}(2δ))} → 0 as δ → 0. The remaining term can be estimates by

|∫_{M} ⟨φ, ∇η_{δ} · ψ⟩| ≤ \frac{2}{δ}∥φ∥_{L^{s}(U_{S}(2δ))}∥ψ∥_{L^{s^{∗}} (U_{S}(2δ))} ≤ \frac{C}{δ} ∥φ∥_{L^{s}(U_{S}(2δ))}vol(U_{S}(2δ) ∩ suppψ)^{\frac{1}{s^{∗}}}

≤ C′ \underbrace{∥φ∥_{L^{s}(U_{S}(2δ))}}_{→0} δ^{\frac{m−ℓ}{s^{∗}} −1} → 0.

□

Lemma 4.8. Let (M, g) be an m-dimensional Riemannian spin manifold, and let S ⊂ M be an embedded submanifold of dimension ℓ ≤ m − 2s^{∗} where s^{∗} is the conjugate exponent of s. Assume that v is a nonnegative function such that ∥v∥_{s} < ∞ for s ∈ (1, ∞) and Lv = μv^{s−1} weakly on M \ S for μ ∈ ℝ. Then Lv = μv^{s−1} weakly on M.

Proof. The proof is similar to the one of Lemma 4.7, and we use the notations therein. The cut-off function η_{δ} is chosen such it fulfills additionally |Δη_{δ}| ≤ 4δ^{−2}. Then, the estimates are done analogously. □

5. Gromov-Hausdorff convergences

Let (M_{i}, g_{i}, x_{i}), i ∈ ℕ, and (M_{∞}, g_{∞}, x_{∞}) be pointed complete connected Riemannian manifolds. We say that (M_{i}, g_{i}, x_{i}) converges to (M_{∞}, g_{∞}, x_{∞}) in the C^{k}-topology of pointed Riemannian manifolds if for every R > 0 and every i ≥ i_{0}(R) there is an injective immersion φ_{i}^{R}: B_{R+1}^{M_{∞},g_{∞}}(x_{∞}) → B_{R+1}^{M_{i},g_{i}}(x_{i}) such that (φ_{i}^{R})^{∗}g_{i} converges to g_{∞} on B_{R}^{M_{∞},g_{∞}}(x_{∞}) in the C^{k}-topology. If all manifolds above carry spin structures, then we say that they converge in the C^{k}-topology of pointed Riemannian spin manifolds if additionally the maps φ_{i}^{R} preserve the chosen spin structures.

Lemma 5.1. If (M_{i}, g_{i}, x_{i}) converges to (M_{∞}, g_{∞}, x_{∞}) in the C^{2}-topology of pointed Riemannian manifolds, then

lim sup_{i→∞} Q^{∗}(M_{i}, g_{i}) ≤ Q^{∗}(M_{∞}, g_{∞}).

If (M_{i}, g_{i}, x_{i}) converges to (M_{∞}, g_{∞}, x_{∞}) in the C^{1}-topology of pointed Riemannian spin manifolds, then

lim sup_{i→∞} Q_{spin}^{∗}(M_{i}, g_{i}) ≤ Q_{spin}^{∗}(M_{∞}, g_{∞}).

Proof. For a given ε > 0 we take v ∈ C_{c}^{∞}(M_{∞}) with F^{g_{∞}}(v) < Q^{∗}(M_{∞}, g_{∞}) + ε. Choose R > 0 such that the support of v is contained in B_{R}^{M_{∞},g_{∞}}(x_{∞}). For sufficiently large i we then have

Q^{∗}(M_{i}, g_{i}) ≤ F^{g_{i}}(v ∘ (φ_{i}^{R})^{−1}) = F^{(φ_{i}^{R})^{∗}g_{i}}(v) ≤ F^{g_{∞}}(v) + ε < Q^{∗}(M_{∞}, g_{∞}) + 2ε

where the second inequality uses that F^{g} depends only on derivatives of g up to order 2. The first part of the lemma follows in the limit ε → 0. The spinorial statement is proven completely analogously. Here, convergence in C^{1} is enough since the Dirac operator is of first order. □

In the articles [5] and [4] the following situation was considered. Assume that N^{m} is obtained from M^{m} by a surgery of dimension k. Then for any metric g on M a family of special metrics g_{ϑ}, ϑ > 0, was constructed. It was proved in [5] in combination with estimates given in [6] that for all k ≤ m − 4 and all k = m − 3 ≤ 3 we have

lim_{ϑ→0} Q^{∗}(N, g_{ϑ}) ≥ min{Q^{∗}(M, g), Λ̃_{m,k}}.

Similarly it was proven in [4] for k ≤ m − 2 that

lim_{ϑ→0} Q_{spin}^{∗}(N, g_{ϑ}) ≥ min{Q_{spin}^{∗}(M, g), Λ̃_{m,k}^{spin}}.

We apply this construction to M = 𝕊^{m} equipped with the standard metric g = σ^{m}. Then N = S^{k+1} × S^{m−k−1}. Thus we obtain a family of metrics g_{ϑ} on N = S^{k+1} × S^{m−k−1} with

lim_{ϑ→0} Q^{∗}(S^{k+1} × S^{m−k−1}, g_{ϑ}) ≥ Λ̃_{m,k} if k ≤ m − 4 or if k = m − 3 ≤ 3,

and

lim_{ϑ→0} Q_{spin}^{∗}(S^{k+1} × S^{m−k−1}, g_{ϑ}) ≥ Λ̃_{m,k}^{spin} if k ≤ m − 2.

The following lemma is proven with exactly the same methods as in Subsection 6.3 of [5].

Lemma 5.2. For any c ∈ [0, 1], there are points x_{ϑ} ∈ S^{k+1} × S^{m−k−1}, ϑ ∈ (0, 1), such that (S^{k+1} × S^{m−k−1}, g_{ϑ}, x_{ϑ}) converges in the C^{∞}-topology of pointed Riemannian manifolds to (𝕄_{c}^{m,k}, x_{0}) where x_{0} is an arbitrary base point.

Lemmata 5.1 and 5.2 imply

lim_{ϑ→0} Q^{∗}(S^{k+1} × S^{m−k−1}, g_{ϑ}) ≤ Q^{∗}(𝕄_{c}^{m,k})

and

lim_{ϑ→0} Q_{spin}^{∗}(S^{k+1} × S^{m−k−1}, g_{ϑ}) ≤ Q_{spin}^{∗}(𝕄_{c}^{m,k})

for all c ∈ [0, 1] with the same restrictions on k as above. Hence, we immediately obtain

Proposition 5.3.

Λ̃_{m,k} ≤ Λ_{m,k}^{∗} for k ≤ m − 4 and for k = m − 3 ≤ 3,
Λ̃_{m,k}^{spin} ≤ Λ_{m,k}^{spin,∗} for k ≤ m − 2.

Note that in this proposition we do not get any statement about the invariants for 𝕄_{c}^{m,k} for a fixed c, compare to Corollary 10.6.

6. Cut-off arguments

In this section we use cut-off functions to compare the ∗-invariants (which are defined as the infimum of a functional) with there ∼-counterparts (which are defined as the infimum of nonlinear eigenvalues).

Lemma 6.1. Let (M^{m}, g) be a complete connected m-dimensional Riemannian manifold. Then, Q^{∗}(M, g) ≤ Q̃(M, g).

Proof. (cp. [5, Lem. 3.5]) Let v ∈ C^{2}(M)∩L^{∞}(M)∩L^{2}(M), v ≥ 0, satisfying L^{g}v = μ_{v}v^{p−1} with μ_{v} ∈ ℝ_{≥0} and ∥v∥_{L^{p}} = 1 where p = \frac{2m}{m−2}. We fix z ∈ M. Let η_{r} be a smooth cut-off function with values in [0, 1], η_{r} = 0 on M \ B_{2r}(z), η_{r} = 1 on B_{r}(z), and |dη_{r}| ≤ 2r^{−1}. Then,

Q^{∗}(M, g) ≤\frac{∫_{M} η_{r}vL^{g}(η_{r}v)dvol_{g}}{∥η_{r}v∥_{L^{p}(g)}^{2}} = \frac{∫_{M} η_{r}^{2}vL^{g}v + a|dη_{r}|^{2}v^{2}dvol_{g}}{∥η_{r}v∥_{L^{p}(g)}^{2}}

≤\frac{∫_{M} μ_{v}η_{r}^{2}v^{p} + a4r^{−2}v^{2}dvol_{g}}{∥η_{r}v∥_{L^{p}(g)}^{2}} → μ_{v} as r → ∞.
□

Corollary 6.2. Q^{∗}(𝕄_{c}^{m,k}) ≤ Q̃(𝕄_{c}^{m,k}) and Λ_{m,k}^{∗} ≤ Λ̃_{m,k} for all m, k.

Lemma 6.3. Let (M^{m}, g) be a complete connected m-dimensional Riemannian spin manifold. Assume that D is L^{q^{∗}= \frac{2m}{m+1}} -invertible. Then Q_{spin}^{∗}(M, g) ≤ Q̃_{spin}(M, g).

Proof. Let λ = λ̃_{min}^{+}(M, g). By definition of λ̃_{min}^{+}, cf. (1), there is a φ ∈ L^{2} ∩ L^{∞} ∩ C^{1} with Dφ = λ|φ|^{q−2}φ and ∥φ∥_{q} = 1 where q = \frac{2m}{m−1}. Then, Dφ ∈ L^{q^{∗}}, and by the L^{q^{∗}}-invertibility of D we get that φ ∈ L^{q^{∗}}. Hence, λ > 0 since otherwise φ would be a nonzero L^{q^{∗}}-harmonic spinor which contradicts the L^{q^{∗}}-invertibility. We fix z ∈ M. Let η_{r} be a smooth cut-off function with values in [0, 1], η_{r} = 0 on M \B_{2r}(z), η_{r} = 1 on B_{r}(z), and |dη_{r}| ≤ 2r^{−1}. Then

\underbrace{(D(η_{r}φ), η_{r}φ)}_{∈ℝ} = (dη_{r} · φ, φ) + \underbrace{(η_{r}^{2}Dφ, φ)}_{∈ℝ} = λ ∫_{M} η_{r}^{2}|φ|^{q}dvol_{g} > 0

where we used that the summand including dη_{r} vanishes due to ⟨dη_{r} · φ, φ⟩_{x} ∈ iℝ. Thus,

λ_{min}^{+,∗}(M, g, χ) ≤ \frac{∥D(η_{r}φ)∥_{q^{∗}}^{2}}{(D(η_{r}φ), η_{r}φ)} ≤ \frac{(∥dη_{r} · φ∥_{q^{∗}} + ∥η_{r}Dφ∥_{q^{∗}})^{2}}{λ ∫_{M} η_{r}^{2}|φ|^{q}dvol_{g}}

≤ \frac{(\frac{2}{r}∥φ∥_{q^{∗}} + λ (∫_{M} η_{r}^{q^{∗}}|φ|^{q}dvol_{g})^{1/q^{∗}})^{2}}{λ ∫_{M} η_{r}^{2}|φ|^{q}dvol_{g}}

→ λ ∥φ∥_{q}^{q(2−q^{∗})/q^{∗}} = λ ∥φ∥q^{\frac{2}{m−1}} = λ

as r → ∞. Note that the summand \frac{1}{r}∥φ∥_{q^{∗}} → 0 since φ ∈ L^{q^{∗}} as shown above. Hence, Q_{spin}^{∗} ≤ Q̃_{spin}. □

Corollary 6.4. For all c ∈ [0, 1] and k ≤ m−1, we have Q_{spin}^{∗}(𝕄_{c}^{m,k}, g_{c}) ≤ Q̃_{spin}(𝕄_{c}^{m,k}, g_{c}). In particular, Λ_{m,k}^{spin,∗} ≤ Λ̃_{m,k}^{spin}.

Proof. We start with k ≤ m − 2. Lemma 6.3 and Proposition 2.2 imply Q_{spin}^{∗}(𝕄_{c}^{m,k}, g_{c}) ≤ Q̃_{spin}(𝕄_{c}^{m,k}, g_{c}) for all \frac{m−k−1}{2} > ck(\frac{m+1}{2m} − \frac{1}{2}) = \frac{ck}{2m}, i.e., for all k ≤ m − 2 and c ∈ [0, 1]. The remaining case k = m − 1 follows directly from Lemma 7.4. □

7. The model space 𝕄_{1}^{m,k}

For c = 1 the model spaces 𝕄_{c}^{m,k} is very special: The manifold

(𝕄_{1}^{m,k} = ℍ^{k+1} × 𝕊^{m−k−1}, g_{1} = g_{ℍ^{k+1}} + σ^{m−k−1} = sinh^{2} t σ^{k} + dt^{2} + σ^{m−k−1})

is conformal to (𝕊^{m} \ 𝕊^{k}, σ^{m}), [5, Prop. 3.1],

𝔲 : ℍ^{k+1} × 𝕊^{m−k−1} → 𝕊^{m} \ 𝕊^{k}, g_{1} = f^{2}𝔲^{∗}σ^{m} where f = f(t) = cosh t  (7)

where cosh t = (sin r)^{−1} with r = dist(., 𝕊^{k}). Using this conformal map, we will immediately obtain some of the Q-invariants of 𝕄_{1}^{m,k}.

Lemma 7.1. Q^{∗}(𝕄_{1}^{m,k}) = Q^{∗}(𝕊^{m}) = Q_{spin}^{∗}(𝕄_{1}^{m,k}).

Proof. By conformal invariance Q^{∗}(𝕄_{1}^{m,k}, g_{1}) = Q^{∗}(𝕊^{m} \ 𝕊^{k}, σ^{m}). Since Q^{∗} is defined over test functions, we have Q^{∗}(𝕊^{m} \ 𝕊^{k}) ≥ Q^{∗}(𝕊^{m}). On the other hand Q^{∗}(𝕊^{m}) is the highest possible value for Q^{∗}, see Remark 2.4, and thus Q^{∗}(𝕄_{1}^{m,k}) = Q^{∗}(𝕊^{m}). With analogous arguments one gets Q_{spin}^{∗}(𝕄_{1}^{m,k}) = Q_{spin}^{∗}(𝕊^{m}). Together with Q^{∗}(𝕊^{m}) = Q_{spin}^{∗}(𝕊^{m}) the lemma follows. □

In order to examine Q̃(𝕄_{1}^{m,k}) and Q̃_{spin}(𝕄_{1}^{m,k}) we will need modifications of the removal of singularities results in Lemma 4.7 and 4.8.

Lemma 7.2. Let (M, g) be an m-dimensional Riemannian spin manifold, and let S ⊂ M be an embedded submanifold of dimension ℓ ≤ m − 1. Assume that φ is a spinor field such that ∫_{U_{ε}(S)} \frac{1}{ρ}|φ|^{2} < ∞ where U_{ε}(S) consists of all points of M with distance ρ ≤ ε to S. Moreover, let Dφ = λ|φ|^{q−2}φ weakly on M \ S for λ > 0. Then Dφ = λ|φ|^{q−2}φ weakly on M.

Proof. We adapt the proof of Lemma 4.7: Let η̃_{δ} be the function on M defined by

η̃_{δ}(x) = \begin{cases}0 & for ρ := dist(x, S) ≥ ρ_{0} := δ \\ δ log(ρ_{0}/ρ) & for ρ_{0} ≥ ρ ≥ ρ_{1} := ρ_{0}e^{−1/δ} \\ 1 & for ρ_{1} ≥ ρ. \end{cases}

We smooth out η̃_{δ} in such a way that the resulting function η_{δ} still fulfills η_{δ}(x) = 1 for ρ ≥ ρ_{0}, η_{δ}(x) = 0 for ρ ≤ ρ_{1}, and |∇η_{δ}| ≤ \frac{2δ}{ρ}. Then, for a smooth and compactly supported spinor ψ on M we obtain

∫_{M} ⟨φ, Dψ⟩ − λ ∫_{M} ⟨|φ|^{q−2}φ, ψ⟩ = ∫_{M} ⟨φ, D(1 − η_{δ})ψ⟩ − λ ∫_{M} ⟨|φ|^{q−2}φ, (1 − η_{δ})ψ⟩

+ ∫_{M} ⟨φ, η_{δ}Dψ⟩ + ∫_{M} ⟨φ, ∇η_{δ} · ψ⟩ − λ ∫_{M} ⟨|φ|^{q−2}φ, η_{δ}ψ⟩ .

The sum of the first two summands on the right side vanish because the equation holds on M \ S. The terms ∫_{M} ⟨φ, η_{δ}Dψ⟩ and ∫_{M} ⟨|φ|^{q−2}φ, η_{δ}ψ⟩ vanish for the same reason as in the proof of Lemma 4.7. The remaining term is now estimated by

|∫_{M} ⟨φ, ∇η_{δ} · ψ⟩| ≤ Cδ ∫_{U_{δ}(S)∩suppψ} \frac{1}{ρ}|φ| ≤ Cδ (∫_{U_{δ}(S)} \frac{1}{ρ}|φ|^{2})^{\frac{1}{2}} (∫_{U_{δ}(S)∩suppψ} \frac{1}{ρ})^{\frac{1}{2}}

≤ C′δ \underbrace{(∫_{U_{δ}(S)} \frac{1}{ρ}|φ|^{2})^{\frac{1}{2}}}_{→0 as δ→0} \underbrace{(∫_{δe^{−1/δ}}^{δ} \frac{1}{ρ}ρ^{m−ℓ−1} dρ)^{\frac{1}{2}}}_{is δ^{−1/2} for m = ℓ + 1 and ≤Ĉδ^{(m−ℓ−1)/2} else} → 0

as δ → 0 which concludes the proof. □

Lemma 7.3. Let (M, g) be an m-dimensional Riemannian spin manifold, and let S ⊂ M be an embedded submanifold of dimension ℓ ≤ m − 2. Assume that v be a nonnegative function such that ∫_{U_{ε}(S)} \frac{1}{ρ^{2}} v^{2} < ∞ where U_{ε}(S) consists of all points of M with distance ρ ≤ ε to S. Moreover, let Lv = μv^{p−1} weakly on M \ S. Then Lv = μv^{p−1} weakly on M.

Proof. We use an analogous argumentation as in the proof above. Now, we smooth out η̃_{δ} in such a way that the resulting η_{δ} fulfills additionally |Δη_{δ}| ≤ \frac{4δ}{ρ^{2}} . Then, for h ∈ C_{c}^{∞}(M) we estimate ∫_{M} vLh − ∫_{M} v^{p−1}h in a similar way – only Δη_{δ} gives rise to a new term:

|∫_{M} vhΔη_{δ}| ≤ Cδ ∫_{U_{δ}(S)\U_{δe^{−1/δ}} (S)∩supph} \frac{1}{ρ^{2}} v

≤ Cδ (∫_{U_{δ}(S)} \frac{1}{ρ^{2}} v^{2})^{\frac{1}{2}} (∫_{U_{δ}(S)\U_{δe^{−1/δ}} (S)∩supph} \frac{1}{ρ^{2}} )^{\frac{1}{2}}

≤ C′δ \underbrace{(∫_{U_{δ}(S)} \frac{1}{ρ^{2}} v^{2})^{\frac{1}{2}}}_{→0 as δ→0} \underbrace{(∫_{δe^{−1/δ}}^{δ} \frac{1}{ρ^{2}} ρ^{m−ℓ−1} dρ)^{\frac{1}{2}}}_{is δ^{−1/2} for m = ℓ + 2 and ≤Ĉδ^{(m−ℓ−1)/2} else} → 0 as δ → 0.

□

Lemma 7.4. For m ≥ 2

Q̃_{spin}(𝕄_{1}^{m,k}) = \begin{cases}Q^{∗}(𝕊^{m}) & for k ≤ m − 2 \\ Q̃_{spin}(ℍ^{m}) = ∞ & for k = m − 1. \end{cases}

Proof. Let φ ∈ L^{∞} ∩ L^{2} ∩ C^{1} be a solution of Dφ = λ|φ|^{q−2}φ on 𝕄_{1}^{m,k} with 0 < ∥φ∥_{L^{q}} ≤ 1. Using the conformal map 𝔲 in (7) we obtain a C^{1}-solution φ̃ = f^{\frac{m−1}{2}} φ of D^{σ^{m}} φ̃ = λ|φ̃|^{q−2}φ̃ on 𝕊^{m} \ 𝕊^{k} with 0 < ∥φ̃∥_{L^{q}} ≤ 1. Moreover, since φ is L^{2} we get ∞ > ∥φ∥_{L^{2}}^{2} = ∫_{𝕊^{m}\𝕊^{k}} f|φ̃|^{2} dvol_{σ^{m}} = ∫_{𝕊^{m}\𝕊^{k}}(sin ρ)^{−1}|φ̃|^{2} dvol_{σ^{m}}. In particular, φ̃ ∈ L^{2}. Moreover, \frac{1}{ρ} − \frac{1}{sin ρ} is bounded by O(ε) for ρ ∈ (0, ε). Thus, ∫_{U_{ε}(𝕊^{k})} \frac{1}{ρ}|φ̃|^{2} dvol_{σ^{m}} < ∞ as well. Because of Lemma 7.2 φ̃ solves D^{σ^{m}} φ̃ = λ|φ̃|^{q−2}φ̃ weakly on all of 𝕊^{m}. By regularity theory on compact manifolds φ̃ ∈ L^{q} implies φ̃ ∈ H_{1}^{q} ⊂ H_{1}^{q^{∗}}. Hence, φ̃ can serve as a test function for F^{spin} on 𝕊^{m} which implies λ ≥ Q̃_{spin}(𝕊^{m}) = Q^{∗}(𝕊^{m}). Thus, Q̃_{spin}(𝕄_{1}^{m,k}) ≥ Q^{∗}(𝕊^{m}). Let now φ̃ be a Killing spinor on 𝕊^{m} with Killing constant −\frac{1}{2} and ∥φ̃∥_{L^{q}(𝕊^{m})} = 1. Then Dφ̃ = Q_{spin}^{∗}(𝕊^{m})|φ̃|^{\frac{4}{m−1}} φ̃. Then using the identification of spinor bundles to conformal metrics as in Example 4.6 the spinor φ = f^{− \frac{m−1}{2}} φ̃ fulfills the Euler-Lagrange equation for D on 𝕄_{1}^{m,k} and is in L^{∞} ∩ L^{q}. Moreover, if m − k ≥ 2, then

∥φ∥_{L^{2}}^{2} = C_{2} ∫_{0}^{∞} cosh^{1−m} t sinh^{k} t dt ≤ C_{3} + C_{4} ∫_{1}^{∞} e^{(1−m+k)t} dt < ∞.

Thus, for k ≤ m − 2 we obtained Q̃_{spin}(𝕄_{1}^{m,k}) = Q^{∗}(𝕊^{m}). Let now k = m − 1. Then, 𝕄_{1}^{m,m−1} corresponds to two copies of the hyperbolic space. Thus, Q̃_{spin}(𝕄_{1}^{m,m−1}) = Q̃_{spin}(ℍ^{m}). Let φ now be a solution as above on ℍ^{m}. By a conformal map we get as above a solution φ̃ on the lower hemisphere of 𝕊^{m}. Extending φ̃ by zero to all of 𝕊^{m}, we obtain a weak solution to our nonlinear Dirac eigenvalue equation on 𝕊^{m} \ 𝕊^{m−1}. Using again Lemma 7.2 we see that φ̃ is already a nontrivial weak solution on all of 𝕊^{m}. But since φ̃ vanishes on an open subset this contradicts the unique continuation principle, [15]. Thus, such a solution φ we started with cannot exist. Hence, Q̃_{spin}(ℍ^{m}) = Q̃_{spin}(𝕄_{1}^{m,m−1}) = ∞. □

Lemma 7.5. For m ≥ 3

Q̃(𝕄_{1}^{m,k}) = \begin{cases}Q^{∗}(𝕊^{m}) & for k ≤ m − 3 \\ ∞ & for k = m − 2 \\ Q̃(ℍ^{m}) = ∞ & for k = m − 1. \end{cases}

Proof. We start analogously as in the spin case from above with a nonnegative solution v ∈ L^{∞} ∩ L^{2} ∩ C^{2} of Lv = μv^{p−1} on 𝕄_{1}^{m,k} and use the conformal map 𝔲 in (7) to obtain ṽ on 𝕊^{m} \ 𝕊^{k}. Analogous as in the proof of Lemma 7.4 we see that ∫_{U_{ε}(𝕊^{k})} \frac{1}{ρ^{2}} ṽ^{2} dvol_{σ^{m}} < ∞ which allows to use Lemma 7.3 for k ≤ m − 2. Thus, we get as in the last lemma that Q̃(𝕄_{1}^{m,k}) ≥ Q^{∗}(𝕊^{m}) for k ≤ m − 2. On the other hand, ṽ = const such that ∥ṽ∥_{L^{p}(𝕊^{m})} = 1 is a solution of the Euler-Lagrange equation on 𝕊^{m}. Set v = f^{− \frac{m−2}{2}} 𝔲^{∗}ṽ = C cosh^{− \frac{m−2}{2}} t where C is an appropriate constant. Then by conformal invariance, v fulfills the Euler-Lagrange equation on 𝕄_{1}^{m,k} and is in L^{p}. Moreover, if m − k ≥ 3, v ∈ L^{2} as can be seen by ∥v∥_{L^{2}}^{2} = C ∫_{0}^{∞} cosh^{2−m} t sinh^{k} t dt ≤ C_{1} + C_{2} ∫_{1}^{∞} e^{(2−m+k)t} dt < ∞. Hence, for k ≤ m − 3 we have Q̃(𝕄_{1}^{m,k}) = Q^{∗}(𝕊^{m}). For m−k ≤ 2 we obtained up to now that each nonnegative solution v on 𝕄_{1}^{m,k} gives rise to a nonnegative solution ṽ on 𝕊^{m}. By [35, Thm. 5] ṽ is continuous and everywhere positive. For m − k = 2 and using that ṽ is continuous and positive we can estimate

∫_{U_{ε}(𝕊^{m−2})} \frac{1}{ρ^{2}} ṽ^{2}dvol_{σ^{m}} ≥ C ∫_{0}^{ε} \frac{1}{ρ^{2}} ρ dρ.

Thus, the left integral is not finite which gives a contradiction. Thus, Q̃(𝕄_{1}^{m,m−2}) = ∞. For k = m − 1 let v ∈ L^{∞} ∩ L^{2} ∩ C^{2} be a positive solution of Lv = μv^{p−1} on 𝕄_{1}^{m,m−1}. Thus, we have two solutions of the same equation on the hyperbolic space. We will show that a nontrivial solution of Lv = μ|v|^{p−2}v on the hyperbolic space cannot exist in L^{2}. From a solution on the hyperbolic space we can use the conformal map 𝔲 to obtain a solution ṽ on the lower hemisphere 𝕊^{m}. We extend ṽ to the upper hemisphere by reflection and changing its sign on the upper hemisphere. Thus, ṽ solves Lṽ = μ|ṽ|^{p−2}ṽ on 𝕊^{m} \ 𝕊^{m−1}. Next we show that ṽ solves this equation weakly on all of 𝕊^{m}. Since ṽ is an odd function with respect to reflection at the equator, it suffices to test with odd functions h ∈ C^{∞}(𝕊^{m}). Thus, there is a constant C > 0 such that |h(x)| ≤ Cdist(x, 𝕊^{m−1}) = Cρ. Following the arguments in Lemma 7.3 the estimates are done analogously, and it remains to estimate

|∫_{M} vhΔη_{δ}| ≤ Cδ ∫_{U_{δ}(𝕊^{m−1})\U_{δe^{−1/δ}} (𝕊^{m−1})∩supph} \frac{1}{ρ^{2}} vh ≤ C′δ ∫_{U_{δ}(𝕊^{m−1})\U_{δe^{−1/δ}} (𝕊^{m−1})∩supph} \frac{1}{ρ}

≤ C′′δ \underbrace{(∫_{δe^{−1/δ}}^{δ} \frac{1}{ρ} dρ)^{\frac{1}{2}}}_{=δ^{−1/2}} → 0 as δ → 0.

Thus, ṽ solves Lṽ = μ|ṽ|^{p−2}ṽ weakly on 𝕊^{m}. Then, regularity theory implies that ṽ ∈ C^{2} and thus ṽ|_{𝕊^{m−1}} = 0. Using a conformal transformation from the lower hemisphere to the disk D in ℝ^{m}, we obtain a solution v̂ of Lv̂ = μ|v̂|^{p−2}ṽ on D which is somewhere nonzero in the interior of D and zero on the boundary. This is a contradiction to [36], [39, Thm. III.1.3]. Thus the solution we started with cannot exist, and hence Q̃(ℍ^{m}) = ∞. □

8. The invariants for k = m − 1

The constants Λ_{m,m−1}^{∗} and Λ_{m,m−1}^{spin,∗} are easy to determine.

Lemma 8.1. We have Λ_{m,m−1}^{spin,∗} = Q^{∗}(𝕊^{m}) for all m ≥ 3 and Λ_{m,m−1}^{∗} = Q^{∗}(𝕊^{m}) for all m ≥ 2.

Proof. We show

Q_{spin}^{∗}(𝕄_{c}^{m,m−1}) = Q^{∗}(𝕄_{c}^{m,m−1}) = Q^{∗}(𝕊^{m}).

For c ≠ 0, our model space 𝕄_{c}^{m,m−1} is isometric to two copies of the rescaled hyperbolic space ℍ_{c}^{m}, and for c = 0 it is isometric to two copies of the Euclidean ℝ^{m}. Thus, Q^{∗}(𝕄_{c}^{m,m−1}) = Q^{∗}(ℍ_{c}^{m}) = Q^{∗}(ℍ^{m}) = Q^{∗}(ℝ^{m}) = Q^{∗}(𝕊^{m}), cf. Remark 2.4 and the first equality follows from [32, Lem. 1.10]. Using [21, Lem. 2.0.5] the analogous equations hold for Q_{spin}^{∗} which finishes the proof. □

For the Q̃-invariants we have by scaling and Lemma 7.4 that Q̃_{spin}(ℍ_{c}^{m}) = Q̃_{spin}(ℍ^{m}) = ∞ for c ∈ (0, 1] and m ≥ 2, and Q̃(ℍ_{c}^{m}) = Q̃(ℍ^{m}) = ∞ for c ∈ (0, 1] and m ≥ 3. It remains to consider the Euclidean space.

Lemma 8.2. We have Q̃(ℝ^{m}) = ∞ for m = 3, 4, Q̃(ℝ^{m}) = Q^{∗}(𝕊^{m}) for all m ≥ 5, Q̃_{spin}(ℝ^{m}) = Q^{∗}(𝕊^{m}) for all m ≥ 3 and Q̃_{spin}(ℝ^{2}) ≥ Q^{∗}(𝕊^{2}).

Proof. We start examining Q̃_{spin}(ℝ^{m}). Let φ ∈ L^{2} ∩ L^{∞} ∩ C^{1} be a solution on ℝ^{m} of Dφ = λ|φ|^{q−2}φ with 0 < ∥φ∥_{L^{q}} ≤ 1 for some λ > 0. By stereographic projection, we have σ^{m} = \frac{4}{(1+r^{2})^{2}} g_{E} where r is the radial function in ℝ^{m}. Using the conformal invariance

of the nonlinear Dirac eigenvalue equation above, we get for φ̃ = (\frac{1+r^{2}}{2} )^{(m−1)/2} φ that

D^{σ^{m}} φ̃ = λ|φ̃|^{q−2}φ̃ and 0 ≤ ∥φ̃∥_{L^{q}} ≤ 1 on 𝕊^{m} \ \{N\}. Moreover, ∫_{𝕊^{m}\\{N\}} \frac{1+r^{2}}{2} |φ̃|^{2} dvol_{σ^{m}} = ∫_{ℝ^{m}} |φ|^{2} dvol_{E} < ∞ and 1 + r^{2} = \frac{2}{sin^{2} ρ} where ρ is the distance to the north pole N. In particular, it now follows similarly to the proof of Lemma 7.4 that ∫_{U_{ε}(N)} \frac{1}{ρ^{2}} |φ̃|^{2} dvol_{σ^{m}} is finite. In particular, ∫_{U_{ε}(N)} \frac{1}{ρ}|φ̃|^{2} dvol_{σ^{m}} is finite as well. Thus, we can apply Lemma 7.2 and see the nonlinear Dirac eigenvalue equation from above is valid on all of 𝕊^{m}. Thus, we can conclude as in the proof of Lemma 7.4 that Q̃_{spin}(ℝ^{m}) ≥ Q^{∗}(𝕊^{m}). Let φ̃ be a Killing spinor to the Killing constant −\frac{1}{2} normalized such that ∥φ̃∥_{L^{q}} = 1. Then, Dφ̃ = λ_{min}^{+,∗}(𝕊^{m})|φ̃|^{q−2}φ̃.

Using stereographic projection we obtain a smooth spinor φ = (\frac{1+r^{2}}{2} )^{\frac{−m+1}{2}} φ̃ on ℝ^{m} with

L^{q}-norm one and which satisfies Dφ = λ_{min}^{+,∗}(𝕊^{m})|φ|^{q−2}φ. Moreover,

∫_{ℝ^{m}} |φ|^{2} dvol_{E} = C ∫_{ℝ^{m}} (\frac{1 + r^{2}}{2} )^{−m+1} |φ̃|^{2} dvol_{E} ≤ C′ ∫_{0}^{∞} (\frac{1 + r^{2}}{2} )^{−m+1} r^{m−1} dr.

Thus, φ ∈ L^{2}(ℝ^{m}) for m ≥ 3. Hence, Q̃_{spin}(ℝ^{m}) = Q^{∗}(𝕊^{m}) for m ≥ 3. An analogous argumentation for nonnegative solution v ∈ C^{2} ∩ L^{∞} ∩ L^{2} on ℝ^{m} satisfying Lv = μv^{p−1} and ∥v∥_{L^{p}} = 1 for a μ > 0 gives a nonnegative solution ṽ of the corresponding nonlinear eigenvalue equation on the sphere with ∫_{U_{ε}(N)} \frac{1}{ρ^{2}} ṽ^{2} dvol_{σ^{m}} < ∞. By regularity

ṽ ∈ L^{p} ∩ H_{2}^{p^{∗}} with p^{∗} = \frac{2m}{m+2}. Thus, by the Sobolev embedding theorem ṽ ∈ H_{1}^{2}. Thus, similar as in Lemma 7.4 we see that Q̃(ℝ^{m}) ≥ Q^{∗}(𝕊^{m}). Moreover, by [35, Thm. 5] ṽ is continuous and everywhere positive. Hence, we can estimate

∫_{ℝ^{m}} v^{2} dvol_{E} = ∫_{ℝ^{m}} (\frac{1 + r^{2}}{2} )^{−m+2} ṽ^{2} dvol_{E} ≥ C ∫_{0}^{∞} (\frac{1 + r^{2}}{2} )^{−m+2} r^{m−1} dr
≥C ∫_{1}^{∞} r^{−m+3} dr + C′.

Thus, for m = 3, 4 the solution v was not in L^{2} which contradicts the assumption. Hence, Q̃(ℝ^{m}) = ∞ for m = 3, 4. For m ≥ 5, we see with an analogous calculation that taking the constant solution v of Lv = Q^{∗}(𝕊^{m})v^{p−1}, ∥v∥_{L^{p}} = 1 on the sphere, we obtain via stereographic projection a solution ṽ on ℝ^{m} which is even in L^{2}(ℝ^{m}). Thus, Q̃(ℝ^{m}) = Q^{∗}(𝕊^{m}) for m ≥ 5. □

Example 8.3. Let φ be a Killing spinor on 𝕊^{2} with L^{q=4}-norm one. Then, Dφ = Q^{∗}(𝕊^{2})|φ|^{2}φ. We consider the three-branched covering h: 𝕊^{2} → 𝕊^{2}, z ↦ z^{3}. This map preserves the spin structure. Thus, we can pullback φ via h and obtain a spinor φ̃ on 𝕊^{2}, cp. [1, Sec. 4] fulfilling Dφ̃ = Q^{∗}(𝕊^{2})|φ̃|^{2}φ̃ and ∥φ̃∥_{L^{4}}^{4} = 3. In particular, φ̃ has zeros on the north and the south pole of 𝕊^{2}. Setting φ̂ = (\frac{1}{3})^{1/4} φ̃ we obtain Dφ̂ = 3^{1/2}Q^{∗}(𝕊^{2})|φ̂|^{2}φ̂ and ∥φ̂∥_{L^{4}} = 1. Using stereographic projection we obtain a spinor ψ = (\frac{2}{1+r^{2}} )^{1/2}φ̂ on ℝ^{2} (r being the radial coordinate in ℝ^{2}) with Dψ = 3^{1/2}Q^{∗}(𝕊^{2})|ψ|^{2}ψ and ∥ψ∥_{L^{4}} = 1. Moreover, since φ̂ vanishes at the north pole N, |φ̂(x)| ≤ Cρ on U_{ε}(N) where ρ = dist(., N). by the estimate

∫_{ℝ^{2}\B_{r}(0)} |ψ|^{2}dvol_{E} = ∫_{U_{ε(r)}(N)} \frac{1 + r^{2}}{2} |φ̂|^{2}dvol_{σ^{2}} ≤ C′ ∫_{0}^{ε} ρ^{2} \frac{1 + r^{2}}{2} ρ dρ = C′ ∫_{0}^{ε} \frac{ρ^{3}}{sin^{2} ρ} dρ.

Thus, ψ ∈ L^{2}(ℝ^{2}) and Q̃_{spin}(ℝ^{2}) ≤ 3^{1/2}Q^{∗}(𝕊^{2}).

Summarizing we obtained for the spinorial invariants

Corollary 8.4. We have

Λ̃_{m,m−1}^{spin} = Λ_{m,m−1}^{spin} = Q^{∗}(𝕊^{m}) for all m ≥ 3

and 3^{1/2}Q^{∗}(𝕊^{2}) ≥ Λ̃_{2,1}^{spin} = Q̃_{spin}(ℝ^{2}) ≥ Q^{∗}(𝕊^{2}) = Λ_{2,1}^{spin}. Moreover,

Λ̃_{m,m−1} =Λ_{m,m−1} = Q^{∗}(𝕊^{m}) for all m ≥ 5,
∞ = Λ̃_{m,m−1} >Λ_{m,m−1} = Q^{∗}(𝕊^{m}) for m = 3, 4.

9. Hijazi inequalities

On a closed spin manifold (M^{m}, g), the Hijazi inequality provides a lower bound of the lowest eigenvalue λ_{0}^{2}(g) of the square of the Dirac operator by the lowest eigenvalue of the conformal Laplacian μ(g), [29, Thm. A],

λ_{0}^{2}(g) ≥ \frac{m}{4(m − 1)}μ(g).  (8)

Taking the infimum over all metrics conformal to g with constant volume, one obtains the conformal Hijazi inequality [22]

Q_{spin}^{∗}(M) ≥ Q^{∗}(M).  (9)

We call (8) the metric Hijazi inequality and (9) the conformal Hijazi inequality. In this section, we want to discuss whether similar inequalities also hold on noncompact manifolds. In this context one should replace the lowest eigenvalues in (8) by the infimum of the corresponding spectra whereas (9) remains unchanged. In [22, Thm. 1.1 and 1.2] the metric Hijazi inequality was shown by the second author for complete spin manifold of finite volume fulfilling one of the following conditions:

(1) The infimum of the spectrum of the squared Dirac operator is an eigenvalue.

(2) The infimum of the spectrum of the squared Dirac operator is in the essential spec-

trum, m ≥ 5 and the scalar curvature is bounded from below. In particular, this already implies the conformal Hijazi inequality for manifolds which admit a conformal metric ḡ that is complete and of finite volume and where zero is not in the essential spectrum of the Dirac operator for ḡ or where the second condition from above is fulfilled, cf. [22, Thm. 1.3]. There are also examples of manifolds of bounded geometry where the metric Hijazi inequality does not hold. The simplest example is the hyperbolic space ℍ^{m} where 0 is in the spectrum of the Dirac operator and the spectrum of the conformal Laplacian is [μ, ∞) with μ = 4\frac{m−1}{m−2} \frac{(m−1)^{2}}{4} − m(m − 1) = \frac{m−1}{m−2} > 0.

On the other hand, the hyperbolic space is conformal to a subset of the standard sphere. Thus, Q_{spin}^{∗}(ℍ^{m}) = Q_{spin}^{∗}(ℍ^{m}) = Q_{spin}^{∗}(𝕊^{m}) = Q^{∗}(𝕊^{m}), see Lemma 7.1 for details. Unfortunately it is still unclear whether the conformal Hijazi inequality (9) holds for all complete Riemannian spin manifolds. In this section we prove slightly modified conformal Hijazi inequalities. Some inequalities are proven only for the model spaces, some on more general manifolds, e.g. for manifolds of bounded geometry with uniformly positive scalar curvature.

Proposition 9.1. Let (M^{m}, g) be of bounded geometry with m ≥ 3. Let φ ∈ L^{q} ∩ C^{0} and λ ∈ ℝ with Dφ = λ|φ|^{\frac{2}{m−1}} φ weakly and ∥φ∥_{q} = 1 where q = \frac{2m}{m−1}. Then u := |φ|^{\frac{m−2}{m−1}} satisfies

Lu ≤ 4\frac{m − 1}{m}λ^{2}u^{\frac{m+2}{m−2}}  (10)

in the sense of distributions. Moreover, the equation holds classically outside the zero-set of u.

Proof. By Corollary 4.4 and Lemma 4.2 φ ∈ L^{∞} ∩ C^{2}. We use the idea of Christian Bär and Andrei Moroianu written down in [18, Prop. 3.4]. Let α := \frac{m−2}{m−1} and f := λ|φ|^{q−2}. We define the Friedrich connection ∇^{f} as ∇_{X}^{f}ψ = ∇_{X}ψ + \frac{f}{m}X · ψ. Then for all points where φ ≠ 0 we estimate

d^{∗}d|φ|^{α} = \frac{α}{2}|φ|^{α−2}d^{∗}d|φ|^{2} − α(α − 2)|φ|^{α−2}|d|φ||^{2}
=α|φ|^{α−2} (⟨Δφ, φ⟩ − |∇φ|^{2} − (α − 2)|d|φ||^{2})
=α|φ|^{α−2} (⟨Δφ, φ⟩ − |∇^{f}φ|^{2} − 2\frac{f}{m}⟨Dφ, φ⟩ + \frac{f^{2}}{m} |φ|^{2} − (α − 2)|d|φ||^{2})
=α|φ|^{α−2} (⟨D^{2}φ, φ⟩ − \frac{scal_{M}}{4} |φ|^{2} − |∇^{f}φ|^{2} − 2\frac{f}{m}⟨Dφ, φ⟩ + \frac{f^{2}}{m} |φ|^{2} − (α − 2)|d|φ||^{2})
=α|φ|^{α−2} (⟨(D − f)^{2}φ, φ⟩ − \frac{scal_{M}}{4} |φ|^{2} − |∇^{f}φ|^{2} + 2\frac{m − 1}{m}f⟨(D − f)φ, φ⟩ + \frac{m − 1}{m}f^{2}|φ|^{2}

+\frac{m}{m − 1}|d|φ||^{2})

where we used in the last step that ⟨[D, f]φ, φ⟩ = ⟨df ·φ, φ⟩ has to vanish since ⟨df ·φ, φ⟩_{x} ∈ iℝ but all the other terms are real. By Corollary 4.4, φ is in C^{2} and all equations above hold in the classical sense. In particular (D − f)φ = 0. As the spinor φ is in the kernel of the operator D − f we can use the refined Kato inequality |∇^{f}φ|^{2} ≥ \frac{m}{m−1}|d|φ||^{2}, see [18, (3.9)]. Thus, we get for the u defined in the proposition

d^{∗}du ≤ −α\frac{scal_{M}}{4} u + α\frac{m − 1}{m}f^{2}u.

Thus, using a = \frac{4}{α} this means that

Lu = aΔu + scal_{M}u ≤ 4\frac{m − 1}{m}f^{2}u = 4\frac{m − 1}{m}λ^{2}u^{\frac{m+2}{m−2}} .

As remarked above this all holds outside of the zero set of u = |φ|^{\frac{m−2}{m−1}} . From Corollary A.2 we see that inequality (10) holds distributionally since u is a nonnegative function. □

Proposition 9.2. We assume the conditions of Proposition 9.1. Additionally we assume that scal_{M} ≥ s_{0} > 0 or φ ∈ L^{2 \frac{m−2}{m−1}} , then

Q^{∗}(M, g) ≤ 4\frac{m − 1}{m}λ^{2}.  (11)

Proof. Let u be defined as in the previous proposition. For any regular value ε > 0 of u, we consider V_{ε} := {u ≥ ε}. Note that by Lemma 4.3 lim_{x→∞} u(x) = 0, and hence (V_{ε})_{ε} exhausts M as ε → 0. Let ν be the exterior unit normal field of the boundary of V_{ε}. Then, ∂_{ν}u ≤ 0. Thus, integration over V_{ε} of Inequality (10) multiplied by u gives

4\frac{m − 1}{m}λ^{2} \underbrace{∫_{V_{ε}} u^{\frac{2m}{m−2}} dvol_{g}}_{→1 as ε→0} ≥ ∫_{V_{ε}} (au d^{∗}du + scal_{M}u^{2}) dvol_{g}

= ∫_{V_{ε}} (a|du|^{2} + scal_{M}u^{2}) dvol_{g} − ∫_{∂V_{ε}} au∂_{ν}u dvol_{g}
≥ ∫_{V_{ε}} (a|du|^{2} + scal_{M}u^{2}) dvol_{g}.  (12)

In the case that scal_{M} ≥ s_{0} > 0 this implies u ∈ H_{1}^{2}(M). For the remaining case that φ ∈ L^{2 \frac{m−2}{m−1}} we have u ∈ L^{2}. Thus, we obtain

lim_{ε→0} ∫_{V_{ε}} |du|^{2}dvol_{g} ≤ 4\frac{m − 1}{ma}λ^{2} + \frac{sup_{M} |scal_{M}|}{a}∥u∥_{L^{2}}^{2},

and this as well implies u ∈ H_{1}^{2}. Sard’s theorem tells us that the set of regular ε is dense. In the limit ε → 0 we then get (11) from (12). □

Example 9.3 (Spherical cap solution on hyperbolic space). Let B_{r} ⊂ 𝕊^{m} be a ball in the standard sphere of radius r. Let φ be a Killing spinor on the sphere with Killing constant −\frac{1}{2} normalized as |φ| = vol(B_{r})^{− \frac{1}{q}} for q = \frac{2m}{m−1}. Then ∥φ∥_{L^{p}(B_{r})} = 1 and

D^{𝕊^{m}}φ = \frac{m}{2}φ = \frac{m}{2}vol(B_{r})^{\frac{q−2}{q}} |φ|^{q−2}φ. Let u: ℍ^{m} → B_{r} be a conformal map from the hyperbolic space to the spherical cap such that g_{ℍ^{m}} = f^{2}u^{∗}σ^{m}. Then, using identification of the spinor bundles, as in Example 4.6 and setting φ̃ := f^{− \frac{m−1}{2}} φ we get by conformal invariance that

D^{ℍ^{m}} φ̃ = \underbrace{\frac{m}{2}vol(B_{r})^{\frac{q−2}{q}}}_{=:λ_{r}} |φ̃|^{q−2}φ̃ and ∥φ̃∥_{L^{q}(ℍ^{m})} = 1 and ∥φ̃∥_{L^{∞}(ℍ^{m})} < ∞.

Then λ_{r} → 0 as r → 0. Nevertheless, Q^{∗}(ℍ^{m}) = Q^{∗}(𝕊^{m}). Thus, Proposition 9.2 does not hold without the assumption scal_{M} ≥ s_{0} > 0 or φ ∈ L^{2 \frac{m−2}{m−1}} . Hence, the conformal Hijazi inequality Q̃_{spin}(ℍ^{m}) ≥ Q^{∗}(ℍ^{m}) which trivially follows from Lemma 7.4 is no longer true if we remove the L^{2}-condition in the definition of Q̃_{spin}.

We now want to use these inequalities to prove Hijazi inequalities for the model spaces 𝕄_{c}^{m,k}. In this goal we will examine whether for certain m and k there is a spinor φ ∈ L^{2 \frac{m−2}{m−1}} satisfying the assumptions of Proposition 9.1.

Proposition 9.4. Let m ≥ 3. Let 0 ≤ k < m − 2, c ∈ [0, 1] or k = m − 2 and c ∈ [0, 1). On the manifold 𝕄_{c}^{m,k}, we consider a spinor field φ ∈ L^{q} ∩ C^{0} solving

Dφ = λ|φ|^{\frac{2}{m−1}} φ, ∥φ∥_{q} = 1

for λ ∈ ℝ and q = \frac{2m}{m−1}. Then, φ ∈ L^{2 \frac{m−2}{m−1}} .

We also now that φ ∈ L^{∞} by Lemma 4.2.

Lemma 9.5. Under the assumptions of the proposition we have (m − 2)(m − k − 1) > ck, unless k = m − 2 and c = 1.

Proof of the lemma. The condition (m − 2)(m − k − 1) > ck is equivalent to (m − 1)(m − k − 2) > −(1 − c)k. □

Proof of Proposition 9.4. By Proposition 2.2 D is L^{r}-invertible if

\frac{m − k − 1}{2} > ck |\frac{1}{r} − \frac{1}{2}| .  (13)

By assumption we have Dφ = λ|φ|^{2/(m−1)}φ ∈ L^{\frac{2m}{m+1}} . The condition (13) for r := 2m/(m+1) is equivalent to m(m − k − 1) > ck which is fulfilled by assumption. Thus, we obtain φ ∈ L^{\frac{2m}{m+1}} . Hence,

Dφ = λ|φ|^{q−2}φ ∈ L^{\frac{2m}{(m+1)(q−1)} = \frac{2m(m−1)}{(m+1)^{2}}} .

Note that for m ≥ 3 we have \frac{2m(m−1)}{(m+1)^{2}} ≤ 2\frac{m−2}{m−1} =: s. Hence, using φ in L^{∞} we get Dφ ∈ L^{s}. Moreover, D is L^{s}-invertible as condition (13) for r := s is equivalent to (m−2)(m−k−1) > ck which is provided by assumption and Lemma 9.5. Thus φ ∈ L^{s}. □

Example 9.6. In the exceptional case k = m−2 and c = 1 the conclusion of Proposition 9.4 is not correct. To see this, we construct the following example. We consider a Killing spinor on 𝕊^{m} and transport it conformally to 𝕄_{1}^{m,m−2}, similar to the proof of Lemma 7.4. The spinor falls off as e^{−(m−1)r/2} where r is the distance to a fixed point on ℍ^{m−1} as introduced in Section 2.2. Then the L^{2 \frac{m−2}{m−1}} -norm of φ is infinite. A similar example is provided by Example 9.3 in the case k = m − 1 and c > 0.

Corollary 9.7. Consider 𝕄_{c}^{m,k} with m ≥ 3. Let 0 ≤ k < m − 2, c ∈ [0, 1] or k = m − 2 and c ∈ [0, 1). Let φ ∈ L^{q} ∩ C^{0} be a solution of Dφ = λ|φ|^{\frac{2}{m−1}} φ on 𝕄_{c}^{m,k} with λ ∈ ℝ and ∥φ∥_{q} = 1 where q = \frac{2m}{m−1}. Then,

λ^{2} ≥ \frac{m}{4(m − 1)}Q^{∗}(𝕄_{c}^{m,k}).

Proof. We set u = |φ|^{\frac{m−2}{m−1}} . By Proposition 9.4 u ∈ L^{2}. Then, Proposition 9.2 gives the corollary. □

Corollary 9.8 (Conformal Hijazi inequality for the model spaces). For 0 ≤ k ≤ m − 2 and c ∈ [0, 1] we have

Q̃_{spin}(𝕄_{c}^{m,k}) ≥ Q^{∗}(𝕄_{c}^{m,k}).

In particular, Λ̃_{m,k}^{spin} ≥ Λ_{m,k}^{∗} for 0 ≤ k ≤ m − 2.

Proof. For k < m − 2 or k ≤ m − 2 and c < 1 this follows immediately from Corollary 9.7 and the definition of Q̃_{spin}. The remaining case, k = m − 2 and c = 1, was treated in Lemma 7.4. □

Remark 9.9. In the case k = m−2 we obtain together with [5, Lem. 3.8] that Q̃_{spin}(𝕄_{c}^{m,m−2}) ≥ Q^{∗}(𝕄_{c}^{m,m−2}) ≥ c^{\frac{2}{m}} Q^{∗}(𝕊^{m}). For a test function v ∈ C^{∞}(𝕄_{c}^{m,m−2}) that is constant along the S^{1} one can calculate (since the scalar curvature of S^{1} is zero) that

F^{𝕄_{c}^{m,m−2}}(v) = c^{\frac{2}{m}} F^{𝕄_{1}^{m,m−2}}(v).

Since Q^{∗}(𝕄_{1}^{m,m−2}) = Q^{∗}(𝕊^{m}) is minimized by a v that is constant along S^{1}, we have Q^{∗}(𝕄_{c}^{m,m−2}) = c^{\frac{2}{m}} Q^{∗}(𝕊^{m}). Thus, together we obtain

Q̃_{spin}(𝕄_{c}^{m,m−2}) ≥ Q^{∗}(𝕄_{c}^{m,m−2}) = c^{\frac{2}{m}} Q^{∗}(𝕊^{m}).

In particular, Λ_{m,m−2}^{∗} = 0.

10. Minimizer of the variational problems

The Euler-Lagrange equations of the constants Q^{∗} and λ_{min}^{+,∗} defined via functionals read as

Lu = Q^{∗}u^{p−1} with ∥u∥_{p= \frac{2m}{m−2}} = 1

and

Dφ = λ_{min}^{+,∗}|φ|^{q−2}φ with ∥φ∥_{q= \frac{2m}{m−1}} = 1.

Assume now such minimizing solutions u ∈ H_{1}^{2} ∩ L^{∞} and φ ∈ H1^{\frac{2m}{m+1}} ∩ L^{∞} exist. Then, we also have u ∈ C^{2} and φ ∈ C^{1}. Then, Q̃ ≤ Q^{∗}. Moreover, by interpolation φ ∈ L^{2} and, thus, Q̃_{spin} ≤ Q_{spin}^{∗}.

We now recall some theorems for the existence of such solutions on almost homogeneous manifolds (M, g), i.e., on Riemannian manifolds on which there is a relatively compact set U ⊂⊂ M such that for all x ∈ M there is an isometry f: M → M with f(x) ∈ U. Note that a manifold is almost homogeneous if and only if the isometry group G = Isom(M, g) acts cocompactly on M. This follows since the distance between the orbits on M defines a metric on M/G, and the induced topology is the quotient topology of π: M → M/G.

Theorem 10.1. [24, Thm. 13] Let (M^{m}, g) be a Riemannian manifold of bounded geometry with scal_{M} ≥ const > 0 for a constant c and Q^{∗}(M, g) < Q^{∗}(𝕊^{m}). Let (M, g) be almost homogeneous. Then, there is a positive smooth solution u ∈ H_{1}^{2} ∩ L^{∞} ∩ C^{2} of the Euler-Lagrange equation Lu = Q^{∗}(M)u^{\frac{m+2}{m−2}} and ∥u∥_{\frac{2m}{m−2}} = 1.

In the reference of this reference, u ∈ C^{2} was not explicitly stated, but follows from standard elliptic regularity theory.

Corollary 10.2. Let (m − k − 1)(m − k − 2) > c^{2}(k + 1)k, 0 ≤ c < 1 or let k ≤ m − 3 and c = 1. Then,

Q̃(𝕄_{c}^{m,k}) ≤ Q^{∗}(𝕄_{c}^{m,k}).

Proof. For, k ≤ m − 3 and c = 1 we even have equality by Lemmata 7.1 and 7.5. For k = 0, 𝕄_{c}^{m,0} is diffeomorphic to 𝕄_{1}^{m,0}. Thus, by Lemma 7.5 we even have equality. Let now k > 0. Then the condition c < 1 implies that 𝕄_{c}^{m,k} is not conformally flat. In the case m ≥ 6 Aubin’s inequality, cp. Remark 2.4, provides Q^{∗}(𝕄_{c}^{m,k}) < Q^{∗}(𝕊^{m}). For m < 6 we have Q^{∗}(𝕄_{c}^{m,k}) < Q^{∗}(𝕊^{m}) by the following theorem. Thus Theorem 10.1 implies the existence of a solution u as above, which directly implies the corollary. □

Theorem 10.3. [9, Thm. 1.1] Let m = n + k + 1, m ≥ 3, k > 0, and c ∈ [0, 1). Then

Q^{∗}(𝕊^{n} × ℍ_{c}^{k+1}, σ^{n} + g_{c}) < Q^{∗}(𝕊^{m}, σ^{m}).

Theorem 10.4. [23, Thm. 16] Let (M^{m}, g) be a Riemannian spin manifold of bounded geometry with scal_{M} ≥ C > 0 for a constant C and λ_{min}^{+,∗}(M, g) < λ_{min}^{+,∗}(𝕊^{m}). Let (M, g) be almost homogeneous. Moreover, assume that the Dirac operator D on M is invertible as an operator from L^{s} to L^{s} for s = \frac{2m}{m+1}. Then, there is a positive smooth solution

φ ∈ H1^{\frac{2m}{m+1}} ∩ L^{∞} ∩ C^{1} of the Euler-Lagrange equation Dφ = λ_{min}^{+,∗}|φ|^{\frac{2}{m−1}} φ and ∥φ∥_{\frac{2m}{m−1}} = 1.

In the reference of this theorems φ ∈ L^{∞} was not stated explicitly, but can be seen directly from the proof in [23] or alternatively by Lemma 4.2. Moreover, in the original version of Theorem 10.4 it was requested that D is invertible for all s ∈ [\frac{2m}{m+1}, \frac{2m}{m+1} +ε] for some ε > 0. But if D is invertible for s = \frac{2m}{m+1}, then it is also invertible for the conjugate exponent \frac{2m}{m−1} and by interpolation for all s in between, cp. [10, App. B].

For the special case of manifolds that are product spaces M = N_{1} × N_{2} of an almost homogeneous manifold N_{1} and a closed manifold N_{2} one can relax the assumption on the positive scalar curvature in Theorem 10.4:

Theorem 10.5. Let (M^{m} = N_{1} × N_{2}, g) be a Riemannian spin manifold that is a product manifold of an almost homogeneous manifold N_{1} and a closed manifold N_{2}. Let λ_{N_{2}}^{2} be the lowest eigenvalue of the square (D^{N_{2}})^{2} of the Dirac operator on N_{2}, and let λ_{N_{2}}^{2} + \frac{1}{4} inf scal_{N_{1}} =: c > 0. Moreover, let λ_{min}^{+,∗}(M, g) < λ_{min}^{+,∗}(𝕊^{m}), and assume that the Dirac operator D on M is invertible as an operator from L^{q^{∗}} to L^{q^{∗}} for q^{∗} = \frac{2m}{m+1}. Then,

there is a positive smooth solution φ ∈ H1^{\frac{2m}{m+1}} ∩ L^{∞} ∩ C^{1} of the Euler-Lagrange equation Dφ = λ_{min}^{+,∗}|φ|^{\frac{4}{m−1}} and ∥φ∥_{\frac{2m}{m−1}} = 1.

Proof. We start as in the proof of the general result in [23] - here we shortly recall the steps that remain the same: For that let ρ be a radial admissible weight, see [23, Sect. A.1], ρ ≤ 1. Then, by [23, Lem. 13] we obtain for each s ∈ [2, q = \frac{2m}{m−1}) a sequence φ_{s} ∈ H_{1}^{s^{∗}} ∩ C^{1} with Dφ_{s} = λ_{s}^{α}ρ^{αs}|φ_{s}|^{s−2}φ_{s} and ∥ρ^{α}φ_{s}∥_{L^{s}} = 1 where s and s^{∗} are conjugate, α = α(s) → 0 as s → q and μ := lim sup_{s→q} λ_{s}^{α} ≤ λ_{min}^{+,∗}. Then by [23, Lemmata 14 and 15] a subsequence φ_{s} = φ_{α(s),s} converges to a function φ ∈ H_{1}^{s^{∗}} in C^{1}-topology on each compact subset, and we have Dφ = μ|φ|^{q−2}φ. It remains to show that ∥φ∥_{L^{q}} = 1, i.e., in particular that φ is nonzero. Then the arguments that φ is a solution as desired just follow again the lines of [23, Lemma 15]. We prove the remaining point by contradiction, i.e., we assume that φ = 0: Note that by [23, Lem. 33] for each s we have lim_{|x|→∞} |φ_{s}| = 0. Thus, we can fix x_{s} ∈ N_{1} such that ∫_{{x_{s}}×N_{2}} |φ_{s}|^{2} attains its maximum. By the almost-homogeneity of N_{1} we can assume that all x_{s} are contained in a compact subset K of M. Moreover, then

0 ≤Δ_{N_{1}} ∫_{{x_{s}}×N_{2}} |φ_{s}|^{2} = ∫_{{x_{s}}×N_{2}} Δ_{N_{1}}|φ_{s}|^{2}
=2Re ∫_{{x_{s}}×N_{2}}⟨∇_{N_{1}}^{∗}∇_{N_{1}}φ_{s}, φ_{s}⟩ − 2 ∫_{{x_{s}}×N_{2}} |∇^{N_{1}}φ_{s}|^{2} ≤ 2Re ∫_{{x_{s}}×N_{2}}⟨∇_{N_{1}}^{∗}∇_{N_{1}}φ_{s}, φ_{s}⟩.

Using ρ ≤ 1, Dφ_{s} = λ_{s}^{α}ρ^{αs}|φ_{s}|^{s−2}φ_{s} and ⟨d(function) · ψ, ψ⟩_{x} ∈ iℝ we get

Re ∫_{{x_{s}}×N_{2}}⟨D^{2}φ_{s}, φ_{s}⟩ ≤ (λ_{s}^{α})^{2} ∫_{{x_{s}}×N_{2}} |φ_{s}|^{2(s−1)}.

The square of the Dirac operator decomposes as D^{2} = (D^{N_{1}})^{2} + (D̃^{N_{2}})^{2} where D̃^{N_{2}} = diag(D^{N_{2}}, −D^{N_{2}}) in case that both manifolds are odd dimensional and D̃^{N_{2}} = D^{N_{2}} else, see e.g. [10, Sect. 2.5]. The operators (D̃^{N_{2}})^{2} and (D^{N_{2}})^{2} have the same spectrum. Then, together with the Schrödinger-Lichnerowicz formula we obtain

∫_{\{x_{s}\}×N_{2}}|D̃^{N_{2}}φ_{s}|^{2} + Re ∫_{\{x_{s}\}×N_{2}}⟨∇_{N_{1}}^{∗}∇_{N_{1}}φ_{s}, φ_{s}⟩ + \frac{scal_{N_{1}}}{4} ∫_{\{x_{s}\}×N_{2}}|φ_{s}|^{2}

≤ (λ_{s}^{α})^{2} ∫_{{x_{s}}×N_{2}}|φ_{s}|^{2(s−1)}

and, thus,

(λ_{N_{2}}^{2} + \frac{scal_{N_{1}}}{4} ) ∫_{\{x_{s}\}×N_{2}} |φ_{s}|^{2} ≤ (λ_{s}^{α})^{2} max_{\{x_{s}\}×N_{2}} |φ_{s}|^{2(s−2)} ∫_{\{x_{s}\}×N_{2}} |φ_{s}|^{2}.

Hence,

(λ_{N_{2}}^{2} + \frac{scal_{N_{1}}}{4} ) (λ_{s}^{α})^{−2} ≤ max_{\{x_{s}\}×N_{2}} |φ_{s}|^{2(s−2)}
c(λ_{min}^{+,∗})^{−2} ≤ lim inf_{s→q} (λ_{N_{2}}^{2} + \frac{scal_{N_{1}}}{4} ) (λ_{s}^{α})^{−2} ≤ lim inf_{s→q} max_{\{x_{s}\}×N_{2}} |φ_{s}|^{2(s−2)}.

\begin{matrix}m & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ Q^{∗}(𝕊^{m}) & 113.5 & 130.7 & 147.88 & 165.0 & 182.2 & 199.3 & 216.4 & 233.5 & 250.6 \\ L_{m,m−3} & 65.2 & 78.7 & 91.8 & 104.9 & 118.1 & 131.5 & 145.0 & 158.6 & 172.4 \end{matrix}

Table 2. Some explicit values for L_{m,m−3} := inf_{c∈[0,1]} L_{m}(c^{2}) – a lower bound for Λ_{m,m−3}^{∗}, Λ̃_{m,m−3} and Λ_{m,m−3}^{spin}. The values are rounded – Q^{∗}(𝕊^{m}) is rounded to the nearest multiple of 1/10 and L_{m,m−3} is always rounded down.

Note that all x_{s} are contained in a compact set. Thus, a subsequence of x_{s} converges to some x ∈ M. But, φ = 0 means that then the right hand-side is zero which gives the desired contradiction. □

Corollary 10.6. Let (m − k − 1)^{2} > c^{2}(k + 1)k and Q_{spin}^{∗}(𝕄_{c}^{m,k}) < Q_{spin}^{∗}(𝕊^{m}). Then, Q̃_{spin}(𝕄_{c}^{m,k}) ≤ Q_{spin}^{∗}(𝕄_{c}^{m,k}).

Proof. This corollary follows from Theorem 10.5. Set N_{1} = ℍ_{c}^{k+1} and N_{2} = 𝕊^{m−k−1}. Then λ_{N_{2}}^{2} = \frac{(m−k−1)^{2}}{4} and scal_{N_{1}} = −c^{2}k(k + 1). □

Remark 10.7. Note that using Theorem 10.4 would lead to the condition (m − k − 1)(m − k − 2) > c^{2}k(k + 1). Thus, here Theorem 10.5 gives a better result.

11. Λ-invariants for k = m − 3

For applying the surgery monotonicity formulas, cf. Theorems 2.5 and 2.6, one needs explicit positive lower bounds for Λ_{m,k} in dimension k ≤ m−3 and for Λ_{m,k}^{spin} in dimension k ≤ m−2. In the case k ≤ m − 4 and in the case k + 3 = m ≤ 6 explicit positive lower bounds for Λ_{m,k} were provided in [7]. Using Λ_{m,k}^{spin} ≥ Λ_{m,k}, cp. Corollary 6.4, Proposition 5.3 and [6, Thm. 3.1 and Cor. 3.2], this yields explicit positive lower bounds for Λ_{m,k}^{spin} in these cases. However the techniques we have developed in the previous sections provide also explicit positive lower bounds for Λ_{m,m−3}^{∗} for any m > 6 which is the subject of the present section. Note that by Corollaries 9.8 and 6.4 and by Proposition 6.2 we have Λ̃_{m,m−3}^{spin} = Λ_{m,m−3}^{spin,∗} ≥ Λ_{m,m−3}^{∗} and Λ̃_{m,m−3} ≥ Λ_{m,m−3}^{∗}. Thus, any lower bound for Λ_{m,m−3}^{∗} is also a lower bound for the other three invariants. In Section 2.6 we pointed out that Λ_{m,m−3}^{spin} = Λ_{m,m−3}^{spin,∗}. As a first step we estimate Q_{c}: It was derived in [7, Thm. 4.1] for all c ∈ (0, 1) that

Q_{c} ≥ (\frac{Q_{0}}{Q_{1}} − \frac{c^{2}(k + 1)k}{(1 − c^{2})(m − k − 1)(m − k − 2) + c^{2}(k + 1)k} (\frac{Q_{0}}{Q_{1}} − c^{\frac{2(m−k−1)}{m}} )) Q_{1} (14)

where we defined Q_{c} := Q^{∗}(𝕄_{c}^{m,k}). On the other hand it follows for m ≥ 6 and k = m − 3 from [7, Sec. 4.1 (iv)] that

Q_{0} ≥ \frac{ma}{24^{\frac{3}{m}} ((m − 3)a_{m−3})^{\frac{m−3}{m}}} Q^{∗}(𝕊^{m−3})^{\frac{m−3}{m}} Q^{∗}(𝕊^{3})^{\frac{3}{m}} =: Q̂_{0}.

Inequality (14) reads for k = m − 3 ≥ 3 as

Q_{c} ≥ (\frac{Q_{0}}{Q_{1}} − \frac{c^{2}(m − 2)(m − 3)}{(1 − c^{2})2 + c^{2}(m − 2)(m − 3)} (\frac{Q_{0}}{Q_{1}} − c^{4/m})) Q_{1}

≥\frac{(1 − c^{2})2Q̂_{0} + c^{2+4/m}(m − 2)(m − 3)Q_{1}}{(1 − c^{2})2 + c^{2}(m − 2)(m − 3)} =: L_{m}(c^{2}).

Since Q̂_{0} and Q_{1} = Q^{∗}(𝕊^{m}) are known explicitly one can compute the infimum of L_{m}(c^{2}) numerically for fixed m, see Table 2 for some explicit values. For general m we can still estimate the infimum of L_{m}(c^{2}):

\frac{d}{ds}L_{m}(s) =\frac{−2Q̂_{0} + (1 + 2/m)s^{2/m}(m − 2)(m − 3)Q_{1}}{2 + s((m − 2)(m − 3) − 2)}

− \frac{((1 − s)2Q̂_{0} + s^{1+2/m}(m − 2)(m − 3)Q_{1}) [(m − 2)(m − 3) − 2]}{(2 + s((m − 2)(m − 3) − 2))^{2}}.

The condition \frac{d}{ds}L(s) = 0 is equivalent to

f(s) := \frac{(2 + s((m − 2)(m − 3) − 2))^{2}}{(m − 2)(m − 3)} \frac{d}{ds}L(s) = s^{\frac{2}{m} +1}A_{0} + s^{\frac{2}{m}} A_{1} + A_{2} = 0

where A_{0} = \frac{2}{m}Q_{1}((m − 2)(m − 3) − 2), A_{1} = 2(\frac{2}{m} + 1)Q_{1}, and A_{2} = −2Q̂_{0}. There is at least one zero in the interval (0, 1) since f(0) < 0 and f(1) > 0 (since A_{0} > 0 and Q_{1} ≥ Q̂_{0}). Let F (u) = f(u^{m}) = u^{m+2}A_{0} + u^{2}A_{1} + A_{2}, and let u_{0} ∈ (0, 1) be a zero of F. Then

\frac{F (u)}{u − u_{0}} = u^{m+1}A_{0} + u^{m}A_{0}u_{0} + … + uA_{0}u_{0}^{m} + uA_{1} + A_{0}u_{0}^{m+1} + A_{1}u_{0}.

But for positive u the last polynomial is always positive. Thus, there is only one zero of f in the interval (0, 1).

Moreover, f(s) > s^{\frac{2}{m}} A_{1} + A_{2}. Thus, f(c_{2}^{2}) > 0 where c_{2} := (\frac{−A_{2}}{A_{1}} )^{\frac{m}{2}} = (\frac{mQ̂_{0}}{(m+2)Q_{1}} )^{\frac{m}{2}} . Hence,

Λ_{m,m−3}^{∗} = inf_{c∈[0,1]} Q_{c} ≥ inf_{c∈[0,c_{2}]} L_{m}(c^{2})

≥ inf_{c∈[0,c_{2}]} \frac{(1 − c^{2})2Q̂_{0} + c^{4/m+2}(m − 2)(m − 3)Q_{1}}{2 + c_{2}^{2}((m − 2)(m − 3) − 2)} .

Since (1−c^{2})2Q̂_{0}+c^{4/m+2}(m−2)(m−3)Q_{1} attains its minimum for c3^{\frac{4}{m}} = \frac{2mQ̂_{0}}{(m+2)(m−2)(m−3)Q_{1}}

we obtain

Λ_{m,m−3}^{∗} ≥\frac{(1 − c_{3}^{2})2Q̂_{0} + c_{3}^{4/m+2}(m − 2)(m − 3)Q_{1}}{2 + c_{2}^{2}((m − 2)(m − 3) − 2)} .

Together with the explicit positive lower bounds Λ_{3,0}^{∗} = Q^{∗}(𝕊^{3}) = 6·2^{2/3}π^{4/3} = 43, 823233..., Λ_{4,1}^{∗} > 38.9 [6, Example 4.7], and Λ_{5,2}^{∗} > 45.1 [6, Example 4.10] we obtain explicit positive lower bounds for all Λ_{m,m−3}^{∗}. These methods obviously also yield explicit positive lower bounds for all Q̃_{spin}(ℍ^{m−2} ×c𝕊^{2}).

12. Bordism arguments

Proof of Proposition 3.2. By a theorem of Stolz, [37, Thm. B], M is spin-bordant to the total space M_{0} of an ℍP^{2}-bundle over a base Q for which the structure group is P Sp(3). In particular, in dimension m = 5, 6, 7 this implies that Q = ∅. Each manifold M in dimension m = 5, 6, 7 is a spin boundary. Moreover, by the extended Stolz theorem [6, Prop. 6.5] we can assume that Q is connected if m ≥ 9 and that Q is simply connected if m ≥ 11. Thus, in dimension m ≥ 11 also M_{0} can be chosen to be connected and simply connected. In case that Q = ∅, set M_{1} := 𝕊^{m} otherwise M_{1} := M_{0}. First, let m = 5, 6, 7 or m ≥ 11. As M_{1} is simply connected M can be obtained from M_{1} by a sequence of surgeries of dimensions ℓ where 2 ≤ ℓ ≤ m − 3, see [7, Prop. 5.1]. Using Theorem 2.5 this implies that σ_{spin}^{∗}(M) ≥ min{σ_{spin}^{∗}(M_{1}), Λ_{m}^{spin}}. In the case m = 5, 6, 7 and in the case m ≥ 8 and Q = ∅ we use Λ_{m,k}^{spin} = Λ_{m,k}^{spin,∗} ≤ σ_{spin}^{∗}(𝕊^{m}) for k ≤ m − 2, see Theorem 3.1, and obtain σ_{spin}^{∗}(M) ≥ Λ_{m}^{spin}. For m ≥ 11 and Q ≠ ∅, i.e.

M_{0} = M_{1}, we use the conformal Hijazi inequality for M_{1} and σ^{∗}(M_{1}) ≥ Q^{∗}(ℍP^{2} × ℝ^{m−8}) for m ≥ 8, see [38], where ℍP^{2} × ℝ^{m−8} carries the product metric of the standard metrics of both factors. Then we obtain

σ_{spin}^{∗}(M) ≥ min{Λ_{m}^{spin}, Q^{∗}(ℍP^{2} × ℝ^{m−8})}.

In dimension m = 8 and Q ≠ ∅, M_{1} is a disjoint sum of copies of ℍP^{2}, possibly with reversed orientation. Using 0-dimensional surgeries M_{1} is spin bordant to the connected and simply connected manifold M_{2} := ℍP^{2}# … #ℍP^{2}, possibly with reversed orientation. Thus, σ_{spin}^{∗}(M_{2}) ≥ σ_{spin}^{∗}(M_{1}) = σ_{spin}^{∗}(ℍP^{2}) ≥ Q^{∗}(ℍP^{2}). As above M can then be obtained from M_{2} by surgeries of dimensions ℓ where 2 ≤ ℓ ≤ m − 3. Using again Theorem 2.5 this implies that σ_{spin}^{∗}(M) ≥ min{Λ_{m}^{spin}, σ_{spin}^{∗}(M_{2})} ≥ min{Λ_{m}^{spin}, Q^{∗}(ℍP^{2})}. Let now m = 9, 10 and Q ≠ ∅. Then, M can be obtained from M_{1} by surgeries of dimensions ℓ where 1 ≤ ℓ ≤ m − 3. Using Theorem 2.5 this implies that σ_{spin}^{∗}(M) ≥ min{Λ_{m,1}^{spin}, Λ_{m}^{spin}, σ_{spin}^{∗}(M_{1})}. Similar to above we get

σ_{spin}^{∗}(M) ≥ min{Λ_{m,1}^{spin}, Λ_{m}^{spin}, Q^{∗}(ℍP^{2} × ℝ^{m−8})}.
□

The analogous statement for non-simply connected manifolds mentioned after Proposition 3.2 is proven analogously.

Proof of Proposition 3.3. Let (W, F) be the spin bordism from (M, c_{M}) to (N, f). First, we use 0-dimensional surgery in order to make W connected. We will abuse the notation and also denote the bordism after surgery (W, F). Note that Γ is always finitely presented. Then, again by 0-dimensional surgery, we change W and F such that F induces a surjection on π_{1}. Next, we use 1-dimensional surgery such that the resulting F induces an injection on π_{1}. As a consequence, the resulting map F induces a bijection on π_{1}. Next, we use 2-dimensional surgeries to kill π_{2}(W, M), cp. [27, Proof of Prop. 2.1.1]. This can always be achieved as every element of π_{2}(W, M) then comes from an element in π_{2}(W) and thus can be represented by an embedded S^{2}. The condition that W is spin implies that this embedded S^{2} has trivial normal bundle. Then, the embedding M ↪ W is 2-connected and N ↪ W is 1-connected. Thus, we can obtain N from M by attaching handles of dimensions ℓ with 2 ≤ ℓ ≤ m − 2. Together with Theorem 2.5 we then obtain the claim. □

Appendix A. Weak partial differential inequalities

In this appendix, we recall the connection between viscosity solutions and distributional solutions of weak partial differential inequalities. All functions in this appendix are realvalued. Let Δ = d^{∗}d be the geometric Laplacian on functions on a Riemannian manifold (M, g). Assume that P is an operator of the form P = aΔ+V where a is a positive smooth function and V is a continuous function. Let f be a continuous function. We say that

P u ≤ f

holds in the distributional sense if for all compactly supported smooth nonnegative functions v on M

∫_{M} uP v dvol_{g} ≤ ∫_{M} fv dvol_{g}.

We say that

P u ≤ f

holds in the viscosity sense if u is continuous and for every p ∈ M and ε > 0 there is a neighborhood U_{ε} of p and a C^{2}-function h_{ε} : U_{ε} → ℝ such that h_{ε}(p) = u(p), h_{ε} ≤ u in U_{ε} and P h_{ε}(p) ≤ f(p).

Theorem A.1. [30, Thm. 1 and 2] A continuous function u fulfills P u ≤ f in the distributional sense if and only if it also fulfills the inequality in the viscosity sense.

Actually, in the reference [30, Thm. 1 and 2] the statement is proven for a wide class of second-order operators on ℝ^{n}. But since this class includes the representation of aΔ_{M} in a chart of geodesic coordinates the above theorem follows.

Corollary A.2. Let f be a nonnegative continuous function. Let u ≥ 0 be a continuous function such that P u ≤ f in the classical sense whenever u is positive. Then, u fulfills P u ≤ f in the distributional sense.

Proof. We see that P u ≤ f in the viscosity sense by taking h_{ε} = u whenever u is positive and h_{ε} = 0 otherwise. Then, Theorem A.1 implies the corollary. □

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Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Email address: bernd.ammann@mathematik.uni-regensburg.de

Institut für Mathematik, Universität Leipzig, 04109 Leipzig, Germany

Email address: grosse@math.uni-leipzig.de