SQUARE-INTEGRABILITY OF SOLUTIONS OF THE YAMABE EQUATION

BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT

Abstract. We show that solutions of the Yamabe equation on certain ndimensional non-compact Riemannian manifolds which are bounded and L^{p} for p = 2n/(n−2) are also L^{2}. This L^{p}-L^{2}-implication provides explicit constants in the surgery-monotonicity formula for the smooth Yamabe invariant in our article [4]. As an application we see that the smooth Yamabe invariant of any 2connected compact 7-dimensional manifold is at least 74.5. Similar conclusions follow in dimension 8 and in dimensions ≥ 11.

Contents

1. Introduction  1
2. Preliminaries and Notation  2
3. Main Theorem  6
4. A counterexample to Theorem 3.1 for k = n − 3  11
5. Consequences for the surgery formula  12
6. Topological applications  12
Appendix A. Details of an argument  16
References  17

1. Introduction

The goal of this article is to show that bounded positive solutions of the Yamabe equation on certain n-dimensional non-compact Riemannian manifolds 𝕄_{c}^{n,k} with finite L^{p_{n}}-norm, p_{n} = 2n/(n − 2), also have finite L^{2}-norm. For c ≠ 0 these spaces 𝕄_{c}^{n,k} are products of rescaled hyperbolic space with a standard sphere, while 𝕄_{0}^{n,k} = ℝ^{k+1} × 𝕊^{n−k−1}. The integer k satisfies 0 ≤ k ≤ n − 3. The goal is achieved in some cases, we then say that the L^{p_{n}}-L^{2}-implication holds. In particular, it holds for 0 ≤ k ≤ n − 4. If k = n − 3 and if n is sufficiently large we will find counterexamples, see Section 4.

In the favourable cases, the proof of the L^{p_{n}}-L^{2}-implication is obtained by a combination of tricky integration and suitable estimates, and there is not much hope to generalize our technique to a much larger class of non-compact manifolds. The reader will thus probably ask why we develop such estimates for these special spaces.

Date: January 9, 2014.

2010 Mathematics Subject Classification. 35J60 (Primary), 35P30, 57R65, 58J50, 58C40 (Secondary).

Key words and phrases. Yamabe invariant, surgery, L^{p}-L^{2} implication.

The reason is that these spaces appear naturally as limit spaces in the surgery construction of our article [4], and this limit construction also provides L^{p_{n}}-solutions of the Yamabe equation. The main result of [4] is a surgery formula for the smooth Yamabe invariant σ(M), see Subsection 2.3 for the definition. The surgery formula states that if M is a compact manifold of dimension n and if N is obtained from M through k-dimensional surgery, then the smooth Yamabe invariants σ(M) and σ(N) satisfy

σ(N) ≥ min(σ(M), Λ_{n,k}),  (1)

provided that k ≤ n − 3. The definition of the numbers Λ_{n,k} is quite involved, see Subsection 2.6, but they are proven to be positive and to depend only on n and k. The L^{p_{n}}-L^{2}-implication helps to derive explicit lower bounds for these numbers, see Subsections 2.4 to 2.6.

Using estimates for product manifolds from the article [3] we obtain explicit positive lower bounds for Λ_{n,k} in the case 2 ≤ k ≤ n − 4, see Corollary 5.1 for details. This leads directly to a uniform positive lower bound for the smooth Yamabe invariant of 2-connected manifolds that are boundaries of compact spin manifolds, see Corollary 6.3. In dimensions n ∈ {7, 8} and n ≥ 11 we obtain an explicit positive lower bound σ(M), provided that M is 2-connected with vanishing index α(M) ∈ KO_{n}(pt), see Corollary 6.7. Note that S^{5} is the only 2-connected closed 5-manifold. Further, a 2-connected closed 6-manifold is diffeomorphic to the connected sum of a number of copies of S^{3} × S^{3} and the smooth Yamabe invariant can be estimated using results of [8] and [3]. It is also known that α(M) ≠ 0 implies σ(M) = 0.

Using results from [2], [11], and [12] an explicit positive lower bound can also be obtained in the cases (n, k) ∈ {(4, 1), (5, 1), (5, 2), (6, 3), (9, 1), (10, 1)}, see [2] for details about these dimensions.

Acknowledgements. B. Ammann was partially supported by the DFG Sachbeihilfe AM 144/2-1. M. Dahl was partially supported by the Swedish Research Council. E. Humbert was partially supported by ANR-10-BLAN 0105. We want to thank J. Petean for enlightening discussions related to this article. We thank N. Große for interesting discussions on partial differential equations on non-compact manifolds of bounded geometry and for showing us the reference [13]. Further, we thank the anonymous referee for comments that helped us improve the paper in several ways.

2. Preliminaries and Notation

2.1. Spaces of constant curvature and the model spaces 𝕄_{c}^{n,k}. Here we fix notation for the spaces of constant curvature and we define the spaces 𝕄_{c}^{n,k}.

We denote the Euclidean metric on ℝ^{n} by ξ^{n}. The sphere S^{n} ⊂ ℝ^{n+1} equipped with its standard round metric ρ^{n} is denoted by 𝕊^{n}. We set ω_{ℓ} := vol(𝕊^{ℓ}). For c ∈ ℝ let ℍ_{c}^{k+1} be the simply connected, connected, complete Riemannian manifold of constant sectional curvature −c^{2}. Its metric will be denoted as η_{c}^{k+1}. Polar coordinates around a point x_{0} gives an identification of ℍ_{c}^{k+1}\{x_{0}} with (0, ∞)×S^{k} under which

η_{c}^{k+1} = dr^{2} + sh_{c}(r)^{2}ρ^{k}

where

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

We denote the product metric on ℍ_{c}^{k+1} × 𝕊^{n−k−1} by

G_{c} := η_{c}^{k+1} + ρ^{n−k−1}

and we define the model space 𝕄_{c}^{n,k} through

𝕄_{c}^{n,k} := ℍ_{c}^{k+1} × 𝕊^{n−k−1}.

The scalar curvature of 𝕄_{c}^{n,k} is s^{G_{c}} = −c^{2}k(k + 1) + (n − k − 1)(n − k − 2). Note that 𝕄_{c}^{n,k} = 𝕄_{−c}^{n,k}. As a consequence all following infima over c ∈ [−1, 1] could be taken over c ∈ [0, 1]. From the conformal point of view the case c = ±1 is special since 𝕄_{±1}^{n,k} = ℍ_{±1}^{k+1} × 𝕊^{n−k−1} is conformal to 𝕊^{n} \ 𝕊^{k}, see [4, Proposition 3.1].

2.2. The conformal Yamabe constant. For integers n ≥ 3 we set a_{n} := \frac{4(n−1)}{n−2} and p_{n} := \frac{2n}{n−2}. For a Riemannian manifold (M, g) we denote the scalar curvature by s^{g}, the Laplace operator Δ^{g}, and the volume form dv^{g}. The conformal Laplacian is defined as L^{g} := a_{n}Δ^{g} +s^{g}. In general the dependence on the Riemannian metric is indicated by adding the metric as a superscript.

Let C_{c}^{∞}(M) denote the space of compactly supported smooth functions on M. For a Riemannian manifold (M, g) of dimension n ≥ 3 we define the Yamabe functional by

F^{g}(u) := \frac{∫_{M} (a_{n}|du|_{g}^{2} + s^{g}u^{2}) dv^{g}}{(∫_{M} |u|^{p_{n}} dv^{g})^{\frac{2}{pn}}} ,

where u ∈ C_{c}^{∞}(M) does not vanish identically. The conformal Yamabe constant μ(M, g) of (M, g) is defined by

μ(M, g) := inf_{u∈C_{c}^{∞}(M),u≢0} F^{g}(u).

Here M is allowed to be compact or non-compact.

If M is compact, then the infimum is attained by a positive smooth function. It thus satisfies the associated Euler-Lagrange equation, called the Yamabe equation,

L^{g}u = μu^{p_{n}−1}  (2)

for a suitable constant μ. The invariant is actually a conformal invariant: If g_{1} is conformal to g_{2}, then μ(M, g_{1}) = μ(M, g_{2}).

2.3. The smooth Yamabe invariant. Let M be a compact manifold of dimension n ≥ 3. The smooth Yamabe invariant of M defined as

σ(M) := sup μ(M, g)

where the supremum is taken over the set of all Riemannian metrics on M. It is known that σ(S^{n}) = μ(𝕊^{n}) = n(n − 1)ω_{n}^{2/n}.

2.4. The role of the model spaces in the surgery formula. To give some background and motivation we will now briefly explain the role of the model spaces 𝕄_{c}^{n,k} in [4], and we want to give a rough idea why the invariants Λ_{n,k} defined in the following subsections appear. The discussion in the present subsection is not needed in the proofs of the following results. We thus try to avoid technical details, and do not aim for logical completeness.

Assume that (M, g) is a compact Riemannian manifold of dimension n ≥ 3 and that N is obtained by k-dimensional surgery from M, where 0 ≤ k ≤ n − 3. In [4] a sequence of metrics g_{i} is constructed on N. To prove the surgery formula (1) we show

lim sup_{i→∞} μ(N, g_{i}) ≥ min(μ(M, g), Λ_{n,k})  (3)

for a suitable positive constant Λ_{n,k}.

The solution of the Yamabe problem on (N, g_{i}) provides positive smooth functions u_{i} ∈ C^{∞}(N) satisfying the Yamabe equation

L^{g_{i}}u = μ_{i}u_{i}^{p_{n}−1}

where μ_{i} = μ(N, g_{i}) and ∥u_{i}∥_{L^{pn}} = 1.

In order to prove (3), one analyses the “limits” of the functions u_{i} in various cases. In some cases these functions u_{i} “converge” to a nontrivial solution of the Yamabe equation L^{g}u = μu^{p_{n}−1} on (M, g) with μ = lim sup μ_{i}, and it follows that lim sup_{i→∞} μ(N, g_{i}) ≥ μ(M, g). In other cases the functions concentrate in some points, and lim sup_{i→∞} μ(N, g_{i}) ≥ μ(𝕊^{n}) follows by “blowing-up” such points. This means that one suitably rescales normal coordinates centered in the blow-up point, and then the solutions of the Yamabe equation converge to a solution on ℝ^{n}, which then yields a solution of the Yamabe equation on a sphere. This phenomenon is also often described in the literature by saying that “a sphere bubbles off”.

However, it can also happen that the functions u_{i} converge to a solution of the Yamabe equation on a model space 𝕄_{c}^{n,k} with |c| ≤ 1, this corresponds to Subcases II.1.1 and II.2 in the proof of Theorem 6.1 in [4]. In this case one obtains points x_{i} ∈ N, such that the pointed Riemannian manifolds (N, g_{i}, x_{i}) “converge” to (𝕄_{c}^{n,k}, x̄). Here “convergence” means that balls of arbitrary radius R around x_{i} in (N, g_{i}) converge for fixed R and i → ∞ in the C^{∞}-sense to a ball of radius R around x̄ in 𝕄_{c}^{n,k}. The functions u_{i} will then converge to a positive solution ū of the Yamabe equation (2) on 𝕄_{c}^{n,k}. The L^{p_{n}}-norm of the limit function does not increase, that is ∥ū∥_{L^{pn}} ≤ 1.

This raises the following question.

Question 2.1. Assume that ū ∈ C^{∞}(𝕄_{c}^{n,k}) is a positive solution of the Yamabe equation L^{G_{c}} ū = λū^{p_{n}−1} with 0 < ∥ū∥_{L^{pn}} ≤ 1. Does this imply λ ≥ μ(𝕄_{c}^{n,k})?

If ū is in L^{2}, then it is already in the Sobolev space H^{1,2} and integration by parts ∫ ūΔū dv = ∫ |dū|^{2} dv is allowed on 𝕄_{c}^{n,k}, see Appendix A. It then easily follows that that the answer to the question is positive. In this case we will say that the L^{p_{n}}-L^{2}-implication holds.

In turn the L^{p_{n}}-L^{2}-implication implies that (3) holds for

Λ_{n,k} := inf_{c∈[−1,1]} μ(𝕄_{c}^{n,k}).

We will find conditions under which the L^{p_{n}}-L^{2}-implication holds. But we will also obtain examples where it is violated, see Section 4.

The fact that the L^{p_{n}}-L^{2}-implication is violated in some cases led to a technical difficulty in [4] which was solved by introducing the constant Λ_{n,k}^{(2)} into the definition of Λ_{n,k} in [4, Definition 3.2]. This is sufficient for proving the positivity of the constant Λ_{n,k}. However, in order to obtain an explicit positive lower bound for Λ_{n,k} one would like to avoid the constant Λ_{n,k}^{(2)}. The possibility to prove the L^{p_{n}}-L^{2}- implication in some cases was mentioned in [4, Remark 3.4]. As a consequence finding a positive lower bound for Λ_{n,k} reduces to finding a positive lower bound for the constants μ(𝕄_{c}^{n,k}), uniform in c ∈ [0, 1].

In the meantime new results for the explicit lower bounds for μ(𝕄_{c}^{n,k}) were obtained in [3] and [2], and a proof of [4, Remark 3.4] is needed. The goal of the present article is to provide this proof.

2.5. Modified conformal Yamabe constants. The technical difficulty described in the previous subsection required the introduction of a modified conformal Yamabe constant. In fact, two different subcases require two versions of modified constants, namely the modified conformal Yamabe constants μ^{(1)}(N, h) and μ^{(2)}(N, h), defined below. Our article aims to give some clarification of the relation between these invariants for the model spaces 𝕄_{c}^{n,k}.

Let (N, h) be a Riemannian manifold of dimension n. For i = 1, 2 we let Ω^{(i)}(N, h) be the set of non-negative C^{2} functions u on N which solve the Yamabe equation

L^{h}u = μu^{p_{n}−1}  (4)

for some μ = μ(u) ∈ ℝ. We also require that the functions u ∈ Ω^{(i)}(N, h) satisfy (a) u ≢ 0,

(b) ∥u∥_{L^{pn} (N)} ≤ 1,
(c) u ∈ L^{∞}(N),

together with

(d1) u ∈ L^{2}(N), for i = 1, or

(d2) μ(u)∥u∥_{L^{∞}(N)}^{p_{n}−2} ≥ \frac{(n−k−2)^{2}(n−1)}{8(n−2)}, for i = 2. For i = 1, 2 we set

μ^{(i)}(N, h) := inf_{u∈Ω^{(i)}(N,h)} μ(u).

In particular μ^{(i)}(N, h) = ∞ if Ω^{(i)}(N, h) is empty. If N is compact then the solution of the Yamabe problem trivially implies μ(N, h) = μ^{(1)}(N, h) ≤ μ^{(2)}(N, h).

We will use this for (N, h) = 𝕄_{c}^{n,k}. In [4, Lemma 3.5] we already showed μ^{(1)}(𝕄_{c}^{n,k}) ≥ μ(𝕄_{c}^{n,k}) if 0 ≤ k ≤ n − 3. In the present article we will show that (b) implies (d1) in many cases. As a consequence we will obtain

μ^{(2)}(𝕄_{c}^{n,k}) ≥ μ^{(1)}(𝕄_{c}^{n,k}) ≥ μ(𝕄_{c}^{n,k})

in theses cases. We refer to Theorem 3.1 and Corollary 3.2 for details.

2.6. The numbers Λ_{n,k}. For integers n ≥ 3 and 0 ≤ k ≤ n − 3 set

Λ_{n,k}^{(i)} := inf_{c∈[−1,1]} μ^{(i)}(𝕄_{c}^{n,k})

and

Λ_{n,k} := min {Λ_{n,k}^{(1)}, Λ_{n,k}^{(2)}} .

It is not hard to show that Λ_{n,0} = μ(𝕊^{n}), see [4, Subsection 3.5]. The following positivity result for Λ_{n,k} is proved in [4, Theorem 3.3].

Theorem 2.2. For all n ≥ 3 and 0 ≤ k ≤ n − 3, we have Λ_{n,k} > 0.

Furthermore, the following surgery result is concluded in [4, Corollary 1.4].

Theorem 2.3. Inequality (1) holds for 0 ≤ k ≤ n − 3 and the numbers Λ_{n,k} > 0 defined above.

3. Main Theorem

Theorem 3.1. Let c ∈ [−1, 1] and let u ∈ L^{∞}(𝕄_{c}^{n,k}) ∩ L^{p_{n}}(𝕄_{c}^{n,k}) ∩ C^{2}(𝕄_{c}^{n,k}) be a uniformly continuous, positive solution of

L^{G_{c}}u ≤ μu^{p_{n}−1}.  (5)

Assume that

2k |c| < n(n − k − 2),  (6)

then u ∈ L^{2}(𝕄_{c}^{n,k}).

Inequality (6) holds when • n ≤ 5, k ∈ {0, ⋯, n − 3}, and |c| ≤ 1; or

• n ≥ 6, k ∈ {0, ⋯, n − 4}, and |c| ≤ 1; or

• n = 6, k = n − 3, and |c| < 1. This follows from the fact that 2k ≤ n(n − k − 2) is equivalent to k ≤ n − 4 + \frac{8}{n+2}.

Non-negative L^{∞}-solutions of (4) on manifolds with bounded geometry will be uniformly continuous due to standard local regularity and Sobolev embeddings (one even can show boundedness in C^{3,α} for small α > 0). Thus the conditions in Theorem 3.1 are satisfied for all L^{∞}-solutions of (4) on 𝕄_{c}^{n,k}, in particular for all u ∈ Ω^{(i)}(𝕄_{c}^{n,k}), i = 1, 2.

Also note that a non-negative solution of (4) will be either identically zero or everywhere positive, due to the maximum principle.

Remark. We expect that a similar theorem for weak solutions of (5) holds as well. Such a version for weak solutions is required in current work [5] of Nadine Große in collaboration with the first author. This project will relate the spinorial analogue of Λ_{n,k} defined in [1] to the constants Λ_{n,k} defined above.

More precisely, we expect that Theorem 3.1 also holds for uniformly continuous, non-negative functions u ∈ L^{∞}(𝕄_{c}^{n,k}) ∩ L^{p_{n}}(𝕄_{c}^{n,k}) satisfying (5) in the distributional sense, without the conditions of positivity and u ∈ C^{2}. Unfortunately the proof presented below is no longer valid in the distributional setting, as square roots of distributions and products of distributions would appear and it remains unclear how to handle these objects.

However, we worked out a distributional version of the proof below for nonnegative functions u ∈ L^{∞}(𝕄_{c}^{n,k}) ∩ L^{p_{n}}(𝕄_{c}^{n,k}) that are C^{2} on Ω for Ω := {x ∈ 𝕄_{c}^{n,k} | u(x) > 0}, that satisfy (5) on Ω, and that satisfy u^{2} ∈ C^{1}(𝕄_{c}^{n,k}). Under the above conditions on c, n and k we then get the conclusion u ∈ L^{2}(𝕄_{c}^{n,k}), as well. The non-C^{2} functions in [5] satisfy all these conditions. In the proof of the generalized version of Theorem 3.1 one has to replace integrals over F_{r} by integrals over {x ∈ F_{r} | u(x) ≥ ε}. The boundary terms thus obtained can be estimated by 0.

Corollary 3.2. We have

μ^{(2)}(𝕄_{c}^{n,k}) ≥ μ(𝕄_{c}^{n,k})

for all k ≤ n − 4 and |c| ≤ 1. The same statement holds for k = n − 3, n ∈ {4, 5} and |c| ≤ 1. And it also holds for k = 3, n = 6, |c| < 1.

Proof of Corollary 3.2. Under the conditions of the corollary Assumption (6) holds, and hence Theorem 3.1 implies

Ω^{(2)}(𝕄_{c}^{n,k}) ⊂ Ω^{(1)}(𝕄_{c}^{n,k})

and as a consequence we get

μ^{(2)}(𝕄_{c}^{n,k}) ≥ μ^{(1)}(𝕄_{c}^{n,k}).

On the other hand it is proved in [4, Lemma 3.5] that

μ^{(1)}(𝕄_{c}^{n,k}) ≥ μ(𝕄_{c}^{n,k})

for all k ∈ {0, ⋯, k − 3}. The corollary follows. □

Proof of Theorem 3.1. We will now give the proof of Theorem 3.1 in five steps. Let u be as in the statement of this theorem.

Step 1. The function u tends to 0 at infinity.

We proceed by contradiction and assume that there is an ε > 0 and a sequence of points (x_{j})_{j∈ℕ} tending to infinity (as a sequence in j) such that u(x_{j}) ≥ ε. Denote by \overline{B(x, 1)} the closed ball of radius 1 around a point x. By taking a subsequence of (x_{j}) which tends fast enough to infinity, we can assume that the balls \overline{B(x_{j}, 1)} are all disjoint. Since u is in L^{p_{n}} we have

lim_{j→∞} ∫\overline{B(x_{j},1)} u^{p_{n}} dv^{G_{c}} = 0.

Since 𝕄_{c}^{n,k} is homogeneous, there are isometries φ_{j} : \overline{B(x_{j}, 1)} → \overline{B(O, 1)} where O is any fixed point in 𝕄_{c}^{n,k}. Obviously v_{j} := u ∘ φ_{j}^{−1} satisfy

lim_{j→∞} ∫\overline{B(O,1)} v_{j}^{p_{n}} dv^{G_{c}} = 0,  (7)

and

v_{j}(O) ≥ ε.  (8)

Since u is assumed to be uniformly continuous and bounded, the sequence v_{j} is equicontinuous and bounded. A subsequence of (v_{j}) converges uniformly, i.e. in C^{0}(\overline{B(O, 1)}, ℝ), to a continuous function v ∈ C^{0}(\overline{B(O, 1)}, ℝ).

Finally (7) and (8) tell us that

∫\overline{B(O,1)} v^{p_{n}} dv^{G_{c}} = 0

and

v(O) ≥ ε,

which gives the desired contradiction. This ends the proof of Step 1.

We now work in polar coordinates on the hyperbolic space factor of 𝕄_{c}^{n,k} = ℍ_{c}^{k+1} × 𝕊^{n−k−1} as introduced in Subsection 2.1. We thus study the metric G_{c} = dr^{2} + sh_{c}(r)^{2}ρ^{k} + ρ^{n−k−1} on the manifold (0, ∞) × S^{k} × S^{n−k−1}. Using these coordinates, we denote by F_{r} the set of constant r, that is F_{r} := S^{k} × S^{n−k−1} and we denote the restriction of g to F_{r} by g_{r} = sh_{c}(r)^{2}ρ^{k} + ρ^{n−k−1}. We define

ω(r) := (∫_{F_{r}} u^{2} dv^{g_{r}})^{\frac{1}{2}}

for r > 0. Next we prove a differential inequality for ω.

Step 2. For any γ with 0 < γ < \frac{n−k−2}{2} there is an r_{0}(γ, u) such that

ω′′(r) ≥ γ^{2}ω(r)

for all r > r_{0}(γ, u).

The argument for this step is a modification of the proof of Theorem 5.2 in [4]. The Laplacian operators of the total space Δ^{G_{c}} and of the fibers Δ^{g_{r}} are related

by

Δ^{G_{c}} = Δ^{g_{r}} − ∂_{r}^{2} + (n − 1)H_{r}∂_{r}

where H_{r} denotes the mean curvature of the fiber F_{r} in 𝕄_{c}^{n,k}. It follows that

\begin{aligned}∫_{F_{r}} uΔ^{G_{c}}u dv^{g_{r}} & = ∫_{F_{r}} (uΔ^{g_{r}}u − u∂_{r}^{2}u + (n − 1)H_{r}u∂_{r}u) dv^{g_{r}} \\ & = ∫_{F_{r}} (|d_{vert}u|^{2} − u∂_{r}^{2}u + (n − 1)H_{r}u∂_{r}u) dv^{g_{r}}. \end{aligned}

where d_{vert}u denotes the differential of u along the fiber, that is d_{vert}u = d(u_{|F_{r}}). Using Inequality (5) we get

a_{n} ∫_{F_{r}} (u∂_{r}^{2}u − (n − 1)H_{r}u∂_{r}u) dv^{g_{r}} ≥ s^{G_{c}}ω(r)^{2} − μ ∫_{F_{r}} u^{p_{n}} dv^{g_{r}}.  (9)

Computing the derivative of ω(r)^{2}/2 we get

\begin{aligned}ω′(r)ω(r) & = \frac{1}{2} \frac{d}{dr} ∫_{F_{r}} u^{2} dv^{g_{r}} \\ & = ∫_{F_{r}} u∂_{r}u dv^{g_{r}} − \frac{n − 1}{2}H_{r}ω(r)^{2}, \end{aligned}  (10)

where we used that H_{r} is constant on F_{r}. Differentiating this again and using Inequality (9) we get

\begin{aligned}ω′(r)^{2} + ω′′(r)ω(r) & = ∫_{F_{r}}(∂_{r}u)^{2} dv^{g_{r}} + ∫_{F_{r}} (u∂_{r}^{2}u − (n − 1)H_{r}u∂_{r}u) dv^{g_{r}} \\ & − \frac{n − 1}{2}(∂_{r}H_{r})ω(r)^{2} − (n − 1)H_{r}ω′(r)ω(r) \\ & ≥ ∫_{F_{r}}(∂_{r}u)^{2} dv^{g_{r}} + \frac{s^{G_{c}}}{a_{n}} ω(r)^{2} − \frac{μ}{a_{n}} ∫_{F_{r}} u^{p_{n}} dv^{g_{r}} \\ & − \frac{n − 1}{2}(∂_{r}H_{r})ω(r)^{2} − (n − 1)H_{r}ω′(r)ω(r). \end{aligned} (11)

From the Cauchy-Schwarz inequality we get

\begin{aligned}ω(r)^{2} ∫_{F_{r}}(∂_{r}u)^{2} dv^{g_{r}} & ≥ (∫_{F_{r}} u(∂_{r}u) dv^{g_{r}})^{2} \\ & = (ω′(r)ω(r) + \frac{n − 1}{2}H_{r}ω(r)^{2})^{2} , \end{aligned}

where we used Equation (10) in the second line. Thus,

∫_{F_{r}}(∂_{r}u)^{2} dv^{g_{r}} ≥ (ω′(r) + \frac{n − 1}{2}H_{r}ω(r))^{2} .  (12)

Let ε > 0 be a constant to be fixed later. By Step 1 we have

\frac{μ}{a_{n}} ∫_{F_{r}} u^{p_{n}} dv^{g_{r}} ≤ εω(r)^{2}  (13)

for all r large enough (depending on ε). Inserting (12) and (13) into (11) we obtain

\begin{aligned}ω′(r)^{2} + ω′′(r)ω(r) & ≥ (ω′(r) + \frac{n − 1}{2}H_{r}ω(r))^{2} + \frac{s^{G_{c}}}{a_{n}} ω(r)^{2} − εω(r)^{2} \\ & − \frac{n − 1}{2}(∂_{r}H_{r})ω(r)^{2} − (n − 1)H_{r}ω′(r)ω(r), \end{aligned}

or after some rearranging,

ω′′(r) ≥ \underbrace{(\frac{(n − 1)^{2}}{4} H_{r}^{2} + \frac{s^{G_{c}}}{a_{n}} − ε − \frac{n − 1}{2}(∂_{r}H_{r}))}_{=:α(r)} ω(r).  (14)

A computation tells us that

H_{r} = −\frac{k}{n − 1}∂_{r} ln sh_{c}(r) = \begin{cases}−\frac{k}{n−1}c coth(cr) & if c ≠ 0, \\ −\frac{k}{n−1} \frac{1}{r} & if c = 0, \end{cases}

so in particular,

lim_{r→∞} H_{r} = −\frac{k}{n − 1}|c|  (15)

and

lim_{r→∞} ∂_{r}H_{r} = 0.  (16)

Using (15) and (16) together with s^{G_{c}} = −c^{2}k(k + 1) + (n − k − 1)(n − k − 2) we see that the coefficient α(r) in the right hand side in (14) tends to α_{c} − ε where

α_{c} := \frac{(n − 1)^{2}}{4} \frac{k^{2}}{(n − 1)^{2}} c^{2} + \frac{n − 2}{4(n − 1)}( − c^{2}k(k + 1) + (n − k − 1)(n − k − 2))

= −(n − k − 2)\frac{k}{4(n − 1)}c^{2} + \frac{(n − 2)(n − k − 1)(n − k − 2)}{4(n − 1)}

≥ −(n − k − 2)\frac{k}{4(n − 1)} + \frac{(n − 2)(n − k − 1)(n − k − 2)}{4(n − 1)}

= \frac{(n − k − 2)^{2}}{4} .

Here the inequality comes from the fact the coefficient of c^{2} is negative so the smallest value over c ∈ [−1, 1] is attained for c = 1. Choosing ε small enough we have proved Step 2.

By assumption u and thus ω are positive. As a consequence we can define

τ(r) := (ln ω(r))′

for r > 0.

Step 3. One of the following statements is true,

lim inf_{r→∞} τ(r) ≥ (n − k − 2)/2,  (17a)
lim sup_{r→∞} τ(r) ≤ −(n − k − 2)/2.  (17b)

Assume (17b) is not true. Then there is a r̃ ≥ r_{0}(γ) with τ(r̃) > −(n − k − 2)/2.

We assume τ(r̃)^{2} ≤ \frac{(n−k−2)^{2}}{4} − 2ε where ε > 0. Choose γ := \sqrt{\frac{(n−k−2)^{2}}{4} − ε}. Using Step 2 we calculate

τ′(r) = \frac{ω′′(r)}{ω(r)} − (\frac{ω′(r)}{ω(r)})^{2} ≥ γ^{2} − τ(r)^{2}

for all r ≥ r̃, and thus τ′(r) ≥ ε as long as τ(r)^{2} ≤ \frac{(n−k−2)^{2}}{4} −2ε. An easy argument on first order ordinary differential equations yields an R > r̃ such that

τ(r) ≥ \sqrt{\frac{(n − k − 2)^{2}}{4} − 2ε}

for all r ≥ R. As ε > 0 can be chosen arbitrarily small we conclude that (17a) must hold, and Step (3) follows.

Step 4. If (17a) holds then L^{p_{n}}-boundedness of u contradicts the assumption (6).

Since 𝕄_{−c}^{n,k} = 𝕄_{c}^{n,k} we may assume that c ≥ 0. At first we consider the case c > 0. In the following argument we denote by C a positive constant that might change value from line to line. Since

vol^{g_{r}}(F_{r}) = ∫_{F_{r}} (\frac{sinh(cr)}{c})^{k} dv^{ρ^{k}+ρ^{n−k−1}} ≤ Ce^{kcr}

we get

\begin{aligned}∫_{0}^{∞} ω(r)^{p_{n}}e^{− \frac{2kc}{n−2} r} dr & = ∫_{0}^{∞} (∫_{F_{r}} u^{2} dv^{g_{r}})^{\frac{pn}{2}} e^{− \frac{2kc}{n−2} r} dr \\ & ≤ C ∫_{0}^{∞} ∫_{F_{r}} u^{p_{n}} dv^{g_{r}}dr \\ & ≤ C ∫_{𝕄_{c}^{n,k}} u^{p_{n}} dv^{G_{c}} \end{aligned}  (18)

using the Hölder inequality. This is bounded since u is assumed to be in L^{p_{n}}(𝕄_{c}^{n,k}). If (17a) holds then for any γ ∈ (0, \frac{n−k−2}{2}) there is an r_{1} = r_{1}(γ) so that

ω(r) ≥ Ce^{γr}

for all r ≥ r_{1}. Thus

∫_{0}^{∞} ω(r)^{p_{n}}e^{− \frac{2kc}{n−2} r} dr ≥ C ∫_{r_{1}}^{∞} e^{br} dr  (19)

where

b := p_{n}γ − \frac{2kc}{n − 2} = \frac{2}{n − 2}(nγ − kc).

If b ≥ 0 then the right hand side of (19) is infinite, which gives a contradiction to the boundedness of (18). Thus we have b < 0, implying γ < kc/n. Taking γ → (n − k − 2)/2 yields (n − k − 2)/2 ≤ k|c|/n, which finishes the proof of Step (4) for c > 0.

The case c = 0 can be solved with similar estimates.

Step 5. Conclusion.

It remains to show the L^{2}-boundedness of u if (17b) of Step 3 holds. We choose any γ ∈ (0, (n − k − 2)/2). If (17b) holds we have

ω(r) ≤ Ce^{−γr}

for all r ≥ r_{2}(γ), and by possibly enlarging C this holds for all r. From this estimate we have

∫_{𝕄_{c}^{n,k}} u^{2} dv^{G_{c}} = ∫_{0}^{∞} ω(r)^{2} dr < ∞,

which ends the proof of Theorem 3.1. □

4. A counterexample to Theorem 3.1 for k = n − 3

As noted after Theorem 3.1, Assumption (6) holds if n ≤ 6, k ∈ {0, …, n − 3}, |c| < 1 or n ≥ 7, k ∈ {0, …, n − 4}. It is natural to ask whether the conclusion of Theorem 3.1 holds for all k ∈ {0, …, n − 3}. The following proposition answers this in the negative.

Proposition 4.1. Let n ≥ 7. There exists a smooth positive function u ∈ L^{∞}(ℍ_{1}^{n−2}× 𝕊^{2}) ∩ L^{p_{n}}(ℍ_{1}^{n−2} × 𝕊^{2}) which satisfies

L^{G_{1}}u = 0

and is not in L^{2}(ℍ_{1}^{n−2} × 𝕊^{2}).

Note that the function u given by Proposition 4.1 satisfies Inequality (5) with μ = 0.

Proof. Consider 𝕊^{n−3} as a totally geodesic sphere in 𝕊^{n}. For y ∈ 𝕊^{n} let Γ_{y} be the Green’s function of L^{ρ^{n}} at y. That is Γ_{y} satisfies L^{ρ^{n}}Γ_{y} = δ_{y} in the sense of distributions, where δ_{y} is the Dirac distribution at y. It is well known that Γ_{y} exists and satisfies Γ_{y}(x) ∼ (4(n − 1)ω_{n−1})^{−1}r(x)^{−(n−2)} when x tends to y. Here r(x) denotes the geodesic distance from x to y. Define H on 𝕊^{n} \ 𝕊^{n−3} by

H(x) := ∫_{𝕊^{n−3}} Γ_{y}(x) dv^{ρ^{n−3}}(y).

It is straightforward to check that for x tending to 𝕊^{n−3} we have H(x) ∼ c′_{n}r′(x)^{−1} where c′_{n} depends only on n and where r′ is the geodesic distance to 𝕊^{n−3}. Hence we have H ∈ L^{p_{n}}(𝕊^{n} \ 𝕊^{n−3}) since n ≥ 7. In [4, Proposition 3.1] it was proven that 𝕊^{n} \ 𝕊^{n−3} and ℍ_{1}^{n−2} × 𝕊^{2} are conformal. Let f be the conformal factor so that G_{1} = f^{\frac{4}{n−2}} ρ^{n}. As explained in [4], f(r′) ∼ (r′)^{− \frac{2}{n−2}} when r′ tends to 0. We set u := f^{−1}H. By conformal covariance of the conformal Laplacian we have

L^{G_{1}}u = 0.

Moreover,

∫_{ℍ_{1}^{n−2}×𝕊^{2}} u^{p_{n}} dv^{G_{1}} = ∫_{𝕊^{n}\𝕊^{n−3}} H^{p_{n}} dv^{ρ^{n}} < ∞.

In addition, using the asymptotics of f given above, it is easy to check that u is not in L^{2} and hence provides the desired counterexample. □

5. Consequences for the surgery formula

The goal of this paper is to find explicit lower bounds for Λ_{n,k}. We find the following.

Corollary 5.1. Assume that k ∈ {2, ⋯, n − 4}, then

Λ_{n,k} ≥ \underline{Λ}_{n,k}

where

\underline{Λ}_{n,k} := \frac{na_{n}}{((k + 1)a_{k+1})^{\frac{k+1}{n}} ((n − k − 1)a_{n−k−1})^{\frac{n−k−1}{n}}} μ(𝕊^{k+1})^{\frac{k+1}{n}} μ(𝕊^{n−k−1})^{\frac{n−k−1}{n}} .

Note that we have Λ_{n,0} = μ(𝕊^{n}) from [4, Section 3.5], and hence the only cases not covered by this corollary are k = 1 and k = n − 3. Further,

\underline{Λ}_{n,2} = \underline{Λ}_{n,n−4} = na_{n} (\frac{μ(𝕊^{3})}{3a_{3}})^{\frac{3}{n}} (\frac{μ(𝕊^{n−3})}{(n − 3)a_{n−3}} )^{\frac{n−3}{n}} = na_{n} (\frac{π^{4}}{16} )^{\frac{1}{n}} ν_{n−3}^{1/n},

where we defined

ν_{ℓ} := (\frac{μ(𝕊^{ℓ})}{ℓa_{ℓ}})^{ℓ} = ω_{ℓ}^{2} (\frac{ℓ − 2}{4})^{ℓ}

and it holds that

ω_{ℓ} = vol(𝕊^{ℓ}) = \frac{2π^{(ℓ+1)/2}}{Γ (\frac{ℓ+1}{2})} .

We define

\underline{Λ}_{n,2+} := min\{\underline{Λ}_{n,2}, …, \underline{Λ}_{n,n−4}\}.

Some values for \underline{Λ}_{n,2+} are listed in Figure 1. Numerically we calculated \underline{Λ}_{n,2+} = \underline{Λ}_{n,2} for n ≤ 3000, and it seems reasonable to conjecture this for all n, but we do not have a proof.

Proof of Corollary 5.1. The conformal Yamabe invariant μ(𝕄_{c}^{n,k}) as defined in Section 2 for non-compact manifolds is, by virtue of [3, Theorem 2.3], bounded from below by \underline{Λ}_{n,k}. To see this we just have to notice that μ(ℍ_{c}^{k+1}) = μ(𝕊^{k+1}), which holds since ℍ_{c}^{k+1} is conformal to a subset of 𝕊^{k+1}. Thus Corollary 5.1 is a direct consequence of Corollary 3.2. □

6. Topological applications

In this section we derive some topological consequences of our main theorem. Recall that by definition a manifold M is k-connected, k ≥ 1, if it is connected and if π_{1}(M) = π_{2}(M) = ⋯ = π_{k}(M) = 0.

Proposition 6.1. Let M_{0} and M_{1} be non-empty compact 2-connected manifolds of dimension n ≥ 7, and assume that M_{0} is spin bordant to M_{1}. Then M_{1} can be obtained from M_{0} by a sequence of surgeries of dimensions ℓ, 3 ≤ ℓ ≤ n − 4.

Note that 2-connected manifolds are orientable and spin, and they carry a unique spin structure.

The proposition is well-known, but for the sake of being self-contained we include a proof following the lines of [10, Lemma 4.2]). As a first step we prove a lemma.

Lemma 6.2. Let M_{0} and M_{1} compact spin manifolds of dimension n ≥ 7 and assume that M_{0} is spin bordant to M_{1}. Then there is a 3-connected spin bordism W from M_{0} to M_{1}.

Proof of the Lemma. We start with a given spin bordism W_{0} from M_{0} to M_{1}. From this bordism we construct a bordism W which is 3-connected.

By performing 0-dimensional surgeries at the bordism, one can modify the original bordism W_{0} to be connected. This can be done such that the bordism W_{1} thus obtained is again orientable, and W_{1} then carries a spin structure.

We now perform 1-dimensional surgeries to reduce the fundamental group to the trivial group. Assume that [γ] ∈ π_{1}(W_{1}). We can assume that γ : S^{1} → W_{1} is an embedding. Its normal bundle is trivial as W_{1} is orientable. Performing a 1-dimensional surgery along γ using a trivialization ν of this normal bundle yields a new bordism W_{1}^{γ,ν} which depends both on γ and ν. This bordism is orientable. For any γ one can choose a trivialization ν such that the bordism W_{1}^{γ,ν} carries a spin structure that coincides with the spin structure of W_{1} outside a tubular neighborhood of the image of γ. The van Kampen Theorem gives a surjective homomorphism π_{1}(W_{1}) → π_{1}(W_{1}^{γ,ν}) such that [γ] is in the kernel. The fundamental group π_{1}(W) is finitely generated, let γ_{i} be disjoint embedded circles such that [γ_{1}], …, [γ_{ℓ}] is a set of generators π_{1}(W). Performing 1-dimensional surgeries along the γ_{i} with suitable trivializations of their normal bundles then yields a simplyconnected spin bordism W_{2} from M_{0} to M_{1}.

Next we perform 2-dimensional surgeries to remove π_{2}(W_{2}). Assume that [σ] ∈ π_{2}(W_{2}) is given and assume that σ : S^{2} → W_{2} is an embedding. Since W_{2} is spin the normal bundle of the image of σ is trivial. Performing a 2-dimensional surgery along σ yields a new spin bordism W_{2}^{σ} which depends on the choice of σ. However, it is independent of the choice of trivialization as different trivializations are homotopic. After a finite number of 2-dimensional surgeries we obtain a 2connected spin bordism W_{3} from M_{0} to M_{1}.

In a similar way one can also remove π_{3}(W_{3}). The Whitney embedding theorem implies that any class in π_{3}(W_{3}) can be represented by an embedding τ : S^{3} → W_{3}, as n ≥ 6. The normal bundle of the image of τ is trivial, as π_{2}(O(n − 3)) = 0. A surgery along τ with any trivialization ν will then kill [τ], and since n ≥ 7 the spin bordism W_{3}^{τ,ν} thus obtained is again 2-connected and will have π_{3}(W_{3}^{τ,ν}) ≅ π_{3}(W_{3})/[τ]. After finitely many surgery steps we obtain a 3-connected bordism W as claimed in the lemma. □

Proof of the Proposition. Assume that W is 3-connected spin bordism from M_{0} to M_{1}. Then H_{i}(W, M_{j}) = 0 for i = 0, 1, 2, 3, in particular b_{i}(W, M_{0}) = 0 for these numbers i. We can apply [9, VIII Theorem 4.1] for k = 4 and m = n + 1. One obtains that there is a presentation of the bordism such that for any i < 4 and any i > n − 3 the number of i-handles is given by b_{i}(W, M_{0}). Any i-handle corresponds to a surgery of dimension i − 1. It remains to show that b_{i}(W, M_{0}) = 0 for i ∈ {0, 1, 2, 3, n + 1, n, n − 1, n − 2}. For i ∈ {0, 1, 2, 3} this was discussed above. By Poincaré duality H^{n+1−i}(W, M_{0}) is dual to H_{i}(W, M_{1}) which vanishes for i = 0, 1, 2, 3. One the other hand the universal coefficient theorem tells us that the free parts of H^{i}(W, M_{0}) and H_{i}(W, M_{0}) are isomorphic. Thus b_{i}(W, M_{0}) which is by definition the rank of (the free part of) H_{i}(W, M_{0}) vanishes for i ∈ {n + 1, n, n − 1, n − 2}. □

Corollary 6.3. Let M be a 2-connected compact manifold of dimension n ≥ 7 which is a spin boundary. Then

σ(M) ≥ \underline{Λ}_{n,2+}

where \underline{Λ}_{n,2+} is defined in Section 5.

Proof. Assume that M is the boundary of a compact spin manifold W of dimension n + 1. By removing a ball we obtain a spin-bordism from S^{n} to M. The preceding proposition tells us that M can be obtained by surgeries of dimensions ℓ ∈ {3, …, n − 4} from S^{n}. By applying the surgery formula (1) and Corollary 5.1 we get the stated lower bound for σ(M). □

Theorem 6.4 (Stolz [14, Theorem B]). Let M be a compact spin manifold of dimension n ≥ 5. Assume that the index α(M) ∈ KO_{n}(pt) vanishes. Then M is spin-bordant to the total space of an ℍP^{2}-bundle over a base Q for which the structure group is PSp(3).

The base Q has to be understood as a spin manifold, so that it admits a spin structure and the choice of spin structure matters. The theorem includes the fact the every spin manifold of dimension 5, 6, or 7 is a spin boundary, in these cases Q = ∅.

Proposition 6.5 (Extended Stolz theorem). In the case n ≥ 9 one can assume that the base Q is connected, and in the case n ≥ 11 one can assume that it is simply connected.

Note that M := ℍP^{2} ⨿ ℍP^{2} is an example of an 8-dimensional manifold where Q cannot be chosen to be connected. This follows from the fact that ℍP^{2} has non-vanishing signature and thus [ℍP^{2}] is an element of infinite order in Ω_{8}^{spin}. If S^{1} carries the spin structure that does not bound a disc, then ℍP^{2} × S^{1} and ℍP^{2} × S^{1} × S^{1} are examples of dimension 9 and 10 where Q cannot be chosen to be simply connected. This is a consequence of the fact that [ℍP^{2} × S^{1}] ∈ Ω_{9}^{spin} and [ℍP^{2} × S^{1} × S^{1}] ∈ Ω_{10}^{spin} are non-zero elements (of order 2), see [6, Cor. 1.9] or [7, Cor. 2.6].

Proof. Assume that M is spin bordant to a spin manifold N_{0} which is the total space of a fiber bundle with fiber ℍP^{2} and structure group PSp(3) over a base Q_{0} of dimension n − 8 ≥ 1. By performing 0-dimensional surgery on Q_{0} we obtain a connected space Q_{1}. The spin bordism from Q_{0} to Q_{1} which is associated to the 0-dimensional surgeries yields a spin bordism from N_{0} to a total space of a fiber bundle with fiber ℍP^{2} and structure group PSp(3) over Q. This Q is connected, but not necessarily simply-connected.

Now assume n ≥ 11. Any path γ : S^{1} → Q_{1} is homotopic to an embedding and has a trivial normal bundle as Q_{1} is orientable. A tubular neighborhood of the image of γ in Q_{1} is diffeomorphic to S^{1} × B^{n−9}. Any trivialization of this normal bundle yields the germ of such a diffeomorphism up to isotopy. Because of our condition n ≥ 11 we can choose the trivialization of the normal bundle such that the induced spin structure on S^{1} × B^{n−9} is the bounding spin structure of B^{2} × B^{n−9}. Doing a surgery along γ with respect to such a trivialization we obtain a spin manifold Q_{2}, and the associated bordism from Q_{1} to Q_{2} is a spin bordism. As PSp(3) is connected the ℍP^{2}-bundle with structure group PSp(3) extends to a bundle of the same type over this bordism.

We now perform a sequence of such 1-dimensional surgeries, where γ runs through a generating set of π_{1}(Q_{1}). The space thus obtained is then simply-connected. □

Combining the previous results we obtain the following.

Corollary 6.6. Let M be a 2-connected compact manifold of dimension n = 7. Then

σ(M) ≥ \underline{Λ}_{7,2+} > 74.5.

To derive a similar result for n = 8, we remark that the conformal Yamabe constant of ℍP^{2}, equipped with the standard metric, is 128π/120^{1/4} = 121.4967... > \underline{Λ}_{8,2+} = 92.24278....

Corollary 6.7. Let M be a 2-connected compact manifold of dimension n = 8. Then σ(M) = 0 if α(M) ≠ 0, and

σ(M) ≥ \underline{Λ}_{8,2+} > 92.2

if α(M) = 0.

Proposition 6.8. Let M_{0} be the total space of a bundle with fiber ℍP^{2} and structure group PSp(3) over a base B of dimension n − 8. Then, if n ≥ 11

σ(M_{0}) ≥ \underline{λ}_{n} := na_{n} (\frac{3^{6}2^{18}}{7^{8}5^{2}} π^{8})^{1/n} ν_{n−8}^{1/n}

Proof. M. Streil has shown in [15] that σ(M_{0}) ≥ μ(ℍP^{2}×ℝ^{n−8}), where ℍP^{2}×ℝ^{n−8} carries the product metric of the standard metrics on ℍP^{2} and ℝ^{n−8}. On the other hand it follows from [3, Theorem 2.3] that

μ(ℍP^{2} × ℝ^{n−8}) ≥ \frac{na_{n}}{(8a_{8})^{8/n}((n − 8)a_{n−8})^{(n−8)/n}} μ(ℍP^{2})^{8/n}μ(𝕊^{n−8})^{(n−8)/n}.

We also have

(\frac{μ(ℍP^{2})}{8a_{8}})^{8} = \frac{3^{6}2^{18}}{7^{8}5^{2}} π^{8} = 1.32599...π^{8} = 12581.78...

This clearly implies the proposition. □

As an example we study n = 11. Then ν_{3} = π^{6}/8 and thus \underline{λ}_{11} = 178.23277. Some further values for \underline{λ}_{n} are listed in Figure 1.

Proposition 6.9. Let M be a 2-connected compact manifold of dimension n ≥ 11. Then σ(M) = 0 if α(M) ≠ 0. If α(M) = 0, then

σ(M) ≥ \begin{cases}\underline{λ}_{11} > 135.90 & if n = 11, \\ \underline{λ}_{12} > 158.72 & if n = 12, \\ \underline{Λ}_{n,2+} & if n ≥ 13. \end{cases}

Proof. If α(M) = 0, then we have seen

σ(M) ≥ min(\underline{Λ}_{n,2+}, \underline{λ}_{n}).

It remains to compare \underline{Λ}_{n,2+} and \underline{λ}_{n}. Numerically we calculated \underline{λ}_{11} ≤ \underline{Λ}_{11,2+}, \underline{λ}_{12} ≤ \underline{Λ}_{12,2+}, and \underline{λ}_{n} ≥ \underline{Λ}_{n,2+} for 13 ≤ n ≤ 5000.

For n ≥ 1100 we found

\underline{λ}_{n}^{n} ≥ 1.43 \underline{Λ}_{n,2}^{n} ≥ 1.43 \underline{Λ}_{n,2+}^{n}

by studying the Γ-function and by using Γ(n)/Γ(n − 1/2) ≥ \sqrt{n − 1}. This yields the required result. □

n  Y (𝕊^{n})  \underline{Λ}_{n,2+}  \underline{λ}_{n}
7 113.5272754 74.50435
8 130.7157953 92.24278367
9 147.8778709 109.4260421
10 165.0220642 126.4134026
11 182.1536061 143.3280094 135.9033973
12 199.2758713 160.2189094 158.7256737
13 216.3911332 177.1071517 178.0562033
14 233.5009793 194.0019409 196.2714765
15 250.6065514 210.9071013 213.9967504
16 267.7086915 227.8239126 231.4689436
17 284.8080344 244.7524346 248.7967717
18 301.9050675 261.6921542 266.0365304

Figure 1. Some values for \underline{Λ}_{n,2+} and \underline{λ}_{n}.

Appendix A. Details of an argument

We here give some details on an argument sketched directly after Question 2.1.

Proposition A.1. Let (N, h) be a complete n-dimensional Riemannian manifold of bounded geometry and let u ∈ L^{2}(N) ∩ L^{p_{n}}(N) be a positive solution of the Yamabe equation (4). Then ∫_{N} uΔu dv^{h} and ∫_{N} |du|^{2} dv^{h} exist and satisfy

∫_{N} uΔu dv^{h} = ∫_{N} |du|^{2} dv^{h}.

It also follows that u ∈ H_{1}^{2}(N) and μ ≥ μ(N, h).

Proof. Since uL^{h}u = μu^{p_{n}} it is clear that ∫ uL^{h}u dv^{h} exists and as s^{h} is bounded and because of u ∈ L^{2} this also implies the existence of the integral ∫_{N} uΔu dv^{h} ∈ ℝ.

Let η_{i} : N → [0, 1] be a sequence of smooth compactly supported cut-off functions such that |dη_{i}| ≤ 1 and such that the η_{i} ≡ 1 on K_{i} where K_{i} is a compact exhaustion of N, that is N = ⋃ K_{i}. In particular ∫_{supp dη_{i}} u^{2} dv^{h} → 0 for i → ∞. We can also assume η_{i+1} ≥ η_{i}.

In the case (N, h) = 𝕄_{c}^{n,k} such functions can be obtained as η_{i}(x) := χ(r(x)−2i) where r is the radial function on ℍ_{c}^{k+1} and where χ : ℝ → [0, 1] is a smooth function with 0 ≤ χ′ ≤ 1, χ|_{(−∞,0]} ≡ 0, χ|_{[2,∞)} ≡ 1. For general (N, h) of bounded geometry this existence follows similarly using [13, Appendix A1.2., Lemma 2.1., page 70].

We calculate

\begin{aligned}∫_{N} |d(η_{i}u)|^{2} dv^{h} & = ∫_{N} (u^{2}|dη_{i}|^{2} + η_{i}u⟨dη_{i}, du⟩ + ⟨du, η_{i}d(η_{i}u)⟩) dv^{h} \\ & = ∫_{N} (u^{2}|dη_{i}|^{2} + η_{i}u⟨dη_{i}, du⟩ + ⟨du, d(η_{i}^{2}u) − η_{i}udη_{i}⟩) dv^{h} \\ & = ∫_{N} (u^{2}|dη_{i}|^{2} + ⟨du, d(η_{i}^{2}u)⟩) dv^{h} \\ & = ∫_{N} u^{2}|dη_{i}|^{2} dv^{h} + ∫_{N} η_{i}^{2}uΔ^{h}u dv^{h}. \end{aligned} (20)

As u is in L^{p_{n}} ∩ L^{2} and as s^{h} is bounded, we see that

(1 − η_{i}^{2})uΔ^{h}u = (1 − η_{i}^{2}) \underbrace{(\frac{μ}{a_{n}} u^{p_{n}} − \frac{s^{h}}{a_{n}} u^{2})}_{∈L^{1}}

converges to 0 in L^{1}. Thus ∫_{N} η_{i}^{2}uΔ^{h}u dv^{h} exists and converges to ∫_{N} uΔ^{h}u dv^{h}. We also see that

∫_{N} u^{2}|dη_{i}|^{2} dv^{h} ≤ ∫_{supp dη_{i}} u^{2} dv^{h} → 0.

Thus

∫_{N} |d(η_{i}u)|^{2} dv^{h} → ∫_{N} uΔ^{h}u dv^{h}  (21)

and together with ∥η_{i}u∥_{L^{pn}} → ∥u∥_{L^{pn}} = 1 this implies

F^{g}(η_{i}u) → μ

from which we directly see that μ ≥ μ(N, h).

From (21) we also see ∫_{N} uΔ^{h}u dv^{h} ≥ 0 and (21) can be rewritten as

∥d(η_{i}u)∥_{L^{2}} → \sqrt{∫_{N} uΔ^{h}u dv^{h}}.

As a consequence

\begin{aligned}∥η_{i}du∥_{L^{2}(N)} & ≤ ∥d(η_{i}u)∥_{L^{2}(N)} + ∥udη_{i}∥_{L^{2}(N)} \\ & ≤ ∥d(η_{i}u)∥_{L^{2}(N)} + ∥u∥_{L^{2}(supp dη_{i})} \\ & → \sqrt{∫_{N} uΔ^{h}u dv^{h}} \end{aligned}  (22)

is bounded. As η_{i}|du| is increasing with limit |du|, we get |du| ∈ L^{2} with ∥du∥_{L^{2}} ≤

\sqrt{∫_{N} uΔ^{h}u dv^{h}}, and thus u ∈ H_{1}^{2}.

On the other hand

∥d(η_{i}u)∥_{L^{2}(N)} ≤ \underbrace{∥u∥_{L^{2}(supp dη_{i})}}_{→0} +∥du∥_{L^{2}(N)},

so

∫_{N} |du|^{2} dv^{h} = ∥u∥_{L^{2}(N)}^{2} = ∫_{N} uΔ^{h}u dv^{h}.
□

References

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[15] M. Streil, PhD thesis in preparation, Univ. Regensburg.

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

   E-mail address: bernd.ammann@mathematik.uni-regensburg.de

Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden

   E-mail address: dahl@math.kth.se

Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37200 Tours - France,

   E-mail address: Emmanuel.Humbert@lmpt.univ-tours.fr