DETAILS TO EXAMPLES IN “MANIFOLDS WITH SMALL DIRAC EIGENVALUES ARE NILMANIFOLDS”

BERND AMMANN, CHAD SPROUSE

1. 3-dimensional nilmanifolds

We want to make the statements in Example (1) more explicit for the 3-dimensional nilmanifolds.

Let H_{3} := {m(x, y, z) | x, y, z ∈ R} be the 3-dimensional Heisenberg group where

m(x, y, z) := (\begin{matrix}1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix}) .

Obviously the unit e is m(0, 0, 0). Let Γ := {m(x, y, z) | x, y, z ∈ Z}, and M = Γ\H_{3}. For α > 0, there is a unique left invariant metric on H_{3} which is given on T_{e}H_{3} by dx^{2} + dy^{2} + α^{2}dz^{2}. This defines a Riemannian metric on M, denoted by g_{α}. Then as α → 0 | sec_{g_{α}} | → 0, diam(M, g_{α}) < C and V ol(M, g, α) = α → 0. For the trivial spin structure on M one calculates λ_{1}(D̸^{2}) → 0 [1, Section 3]. This example shows that the lower volume bound in [2, Corollary 1.7] is necessary.

2. Details to Examples (3)

2.1. A metric on the cylinder.

Lemma 1. Let h be the standard metric on S^{3}. The ball of radius R around 0 in euclidean space ℝ^{4} will be denoted as B_{R}^{ℝ^{4}}(0). For any ϵ, R, ρ > 0, there is a > 0 and a metric g = dt^{2} + φ^{2}(t)h on S^{3} × (−(a + ρ), (a + ρ)) such that

(1) g|_{(−(a+ρ),−a]×S^{3}} and g|_{[a,(a+ρ))×S^{3}} are isometric to the B_{R+ρ}^{ℝ^{4}}(0) \ B_{R}^{ℝ^{4}}(0),

(2) scal_{g} ≥ −ϵ, and

(3) diam(M, g) ≤ 6(R + ρ)

Proof. The scalar curvature of the metric g = dt^{2} + φ^{2}(t)h on (a, b) × S^{3} is

Scal = 6 (−\frac{φ′′}{φ} + \frac{1 − (φ′)^{2}}{φ^{2}} )

(see Petersen, Riemannian geometry, Section 3.2.3). Hence, the metric g is scalar flat iff φ is a solution of

−\frac{φ′′}{φ} + \frac{1 − (φ′)^{2}}{φ^{2}} = 0.

Obviously, this equation is equivalent to

\frac{1}{2} \frac{d^{2}}{dt^{2}} (φ^{2}) = 1.

So, if we take the initial values φ(0) = c > 0 and φ′(0) = 0, then we obtain the unique solution

φ_{c}(t) = \sqrt{t^{2} + c^{2}}.

Choose any a_{0} < R, close to R. We want to modify φ_{c} such the metric g̃ = dt^{2} + φ̃^{2}(t)h is euclidean on [a_{0}, ∞) × S^{3} and (−∞, −a_{0}] × S^{3}. For a small given δ > 0 one can choose a small c such that κ := 1 − φ′_{c}(a_{0} − δ) > 0 is arbitrarily close to 0. We also can assume φ_{c}(a_{0}) < R. We define

φ̃(t) = \begin{cases}φ_{c}(t) & for |t| ≤ a_{0} − δ \\ \frac{κ}{2δ} (|t| − (a_{0} − δ))^{2} + (1 − κ)(|t| − (a_{0} − δ)) + k_{1} & for a_{0} − δ < |t| < a_{0} \\ |t| + k_{2} & for |t| ≥ a_{0} \end{cases}

where we choose k_{1} and and k_{2} such that the function φ̃ is continuous, i.e. k_{1} := φ(a_{0} − δ) and k_{2} := δ − \frac{κδ}{2} + k_{1} − a_{0}. One directly sees that φ̃ is C^{1}. It is immediately clear from the construction that the metric g̃ = dt^{2}+φ̃^{2}(t)h is euclidean on [a_{0}, ∞) × S^{3} and (−∞, −a_{0}] × S^{3}. It is also clear that if we choose δ and c sufficiently small, then φ̃(a_{0}) = δ − \frac{κδ}{2} + φ_{c}(a_{0} − δ) < R, hence (1) holds for a := a_{0} + R − φ̃(a_{0}).

The metric is obviously scalar-flat outside of [−a_{0}, −a_{0} +δ]×S^{3} ∪[a_{0} −δ, a_{0}]×S^{3}. In order to check (2) we calculate for |t| ∈ [a_{0} − δ, a_{0}]

−\frac{φ̃′′}{φ̃} + \frac{1 − (φ̃′)^{2}}{φ̃^{2}} ≥ −\frac{φ̃′′}{φ̃}
= −\frac{κ}{δφ_{c}(a_{0} − δ)}
≥ −\frac{κ}{δ(a_{0} − δ)}

So as long as κ is small enough relative to δ and a_{0} − δ, then the minimum of the scalar curvature will be arbitrarily close to 0.

(3) is clear from the construction. □ Remark. The obtained metric is only piecewise smooth and C^{1} everywhere.

However, it is easy to show that after a suitable smoothing, one obtains a smooth metric for which the lemma holds.

2.2. Constructing the example.

Theorem 2. Let n ≥ 4. We set r := 2^{[n/2]} which is the rank of the spinor bundle in n dimensions. For given ϵ > 0, D > 0 there is a sequence of compact n-dimensional Riemannian spin manifolds (M_{j}, g_{j}, χ_{j})_{j∈ℕ} such that

(1) diam(M_{j}, g_{j}) ≤ D

(2) scal_{g_{j}} ≥ −ϵ

(3) The smallest r eigenvalues of the square of the Dirac operator on (M_{j}, g_{j}, χ_{j}) are smaller than ϵ.

(4) b_{1}(M_{j}) → ∞ for j → ∞.

Proof.

DETAILS TO EXAMPLES IN “MANIFOLDS WITH SMALL DIRAC EIGENVALUES ARE NILMANIFOLDS” 3

It is sufficient to prove the theorem for n = 4. As soon as we have a 4-dimensional family of manifolds (M_{j}, g_{j}, χ_{j}) with the above properties one obtains a higher dimensional family by taking the product with (S^{1})^{n−4} equipped with any flat metric and a trivial spin structure (i.e. the Spin(n − 4)-principal bundle admits a parallel section) also satisfies the above properties (with D replaced by a slightly larger number).

Let us assume n = 4. To obtain (M_{,}g_{j}) we start with a flat torus T^{4} = (S^{1})^{4} and remove 2k small disks and glue in k handles provided by the lemma above with small R, ρ and ϵ. Properties (1) and (2) are then obvious, and one calculates b_{1}(M_{k}) = 4 + k, which implies (4).

The spin structure is obtained by taking the trivial spin structure on T^{4}, which yields by restriction a spin structure on the torus with the disks removed. This spin structure extends to a spin structure on M_{k}. The extension is not unique, but our construction will be independent from the choice of this extension.

Let φ_{1}, …, φ_{r} be a basis of parallel section of ΣT^{4}.

Now we follow the lines of [Bär-Dahl, Surgery and the spectrum of the Dirac operator, First page of section 2.4]. We see that for any ϵ > 0 there is a suitable cut-off function χ that vanishes on the disks to be removed such that any parallel spinor φ on T^{4} satisfies

\frac{∫ ⟨D(χφ), D(χφ)⟩}{∫ ⟨Dφ, Dφ⟩} < ϵ.

This spinor also lives naturally on M_{k}. While φ runs over all parallel spinors, we obtain an r-dimensional space of test spinors for which the above inequality holds. Hence, by the standard Rayleigh quotient argument we obtain (3). □

3. Details to Examples (5)

We assume that we have a torus. Let sys be a shortest non-contractible loop. Assume K > −δ^{2}, δ > 0, |K| ≤ K_{max}, diam ≤ D. Then for any p > 1

∥K∥_{L^{p}} ≤ area^{1/p}K_{max}

and

∥K∥_{L^{1}} = ∫ |K| = 2 ∫ |K^{−}| ≤ δ^{2}area.

Then

area ≤ \frac{1}{δ} sinh (δdiamM) sys.

As a consequence

(∫ |K|) \frac{area}{sys^{2}} ≤ 2 (∫ |K^{−}|) \frac{area}{sys^{2}} ≤ 2δ^{2} \frac{area^{2}}{sys^{2}} ≤ 2 sinh^{2} (δdiamM) .

Similarly as in the sphere case:

area ≤ \frac{2π}{δ^{2}} [cosh (δdiamM) − 1] .

We can assume that δ^{2} ≤ K_{max}. Hence,

∥K∥_{L^{1}} ≤ 2π [cosh(\sqrt{K_{max}}D) − 1] .

and

area ≤ \frac{2π}{K_{max}} [cosh(\sqrt{K_{max}}D) − 1]

Using the estimates in [ammann:02] this implies

max u − min u ≤ S(K, D)

for a continuous function S with S(0, D) = 0. Obviously

diam ≥ 1/2 spinsys

References

[1] B. Ammann, C. Bär, The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles, Ann. Global Anal. Geom. 16 no. 3 (1998), 221–253

[2] B. Ammann, C. Sprouse, Manifolds with small Dirac eigenvalues are nilmanifolds,