Convergence of Riemannian 4-manifolds with L2-curvature bounds

Norman Zergänge

doi:10.1515/acv-2017-0058

Published by De Gruyter January 20, 2019

Convergence of Riemannian 4-manifolds with L2-curvature bounds

Norman Zergänge

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2017-0058

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Abstract

In this work we prove convergence results of sequences of Riemannian 4-manifolds with almost vanishing L2-norm of a curvature tensor and a non-collapsing bound on the volume of small balls. In Theorem 1.1 we consider a sequence of closed Riemannian 4-manifolds, whose L2-norm of the Riemannian curvature tensor tends to zero. Under the assumption of a uniform non-collapsing bound and a uniform diameter bound, we prove that there exists a subsequence that converges with respect to the Gromov–Hausdorff topology to a flat manifold. In Theorem 1.2 we consider a sequence of closed Riemannian 4-manifolds, whose L2-norm of the Riemannian curvature tensor is uniformly bounded from above, and whose L2-norm of the traceless Ricci-tensor tends to zero. Here, under the assumption of a uniform non-collapsing bound, which is very close to the Euclidean situation, and a uniform diameter bound, we show that there exists a subsequence which converges in the Gromov–Hausdorff sense to an Einstein manifold. In order to prove Theorem 1.1 and Theorem 1.2, we use a smoothing technique, which is called L2-curvature flow. This method was introduced by Jeffrey Streets. In particular, we use his “tubular averaging technique” in order to prove distance estimates of the L2-curvature flow, which only depend on significant geometric bounds. This is the content of Theorem 1.3.

Keywords: Compactness theorems in Riemannian geometry; integral curvature bounds,fourth-order curvature flow; gradient flow; critical exponent; Einstein manifold

MSC 2010: 53C25; 53C44

Communicated by Frank Duzaar

A Auxiliary Results

Here we present some results which we have used in this work.

Lemma A.1.

Let (Mn,g⁢(t))t∈[t1,t2] be a smooth family of Riemannian manifolds and let γ:[0,L]→M be a smooth curve. Then we have the estimates:

(A.1)|dd⁢tL(γ,t)|≤∫γ|g′(t)|g⁢(t)dσt,

(A.2)|log(|v|g⁢(t2)2|v|g⁢(t1)2)|≤∫t1t2∥g′(t)∥L∞⁢(M,g⁢(t))dt for all v∈TM,

(A.3)|∂∂⁡t|∇γ˙γ˙|g⁢(t)2|≤|g′|g⁢(t)|∇γ˙γ˙|g⁢(t)2+C(n)|γ˙|g⁢(t)2|∇γ˙γ˙|g⁢(t)|∇g′|g⁢(t)

on M×(t1,t2).

Proof.

Using a unit-speed-parametrization of γ, we infer (A.1). Estimate (A.2) is proved in [9, Lemma 14.2, p. 279]. Estimate (A.3) is stated in [23, p. 271]. ∎

Lemma A.2.

Let M4 be a closed Riemannian manifold and let (M,g⁢(t))t∈[0,T] be a solution to the L2-flow. We have the following estimates:

(A.4)Volg⁢(t)⁡(M)=Volg⁢(0)⁡(M) for all ⁢t∈(0,T]

and

(A.5)Volg⁢(t)(U)12≥Volg⁢(0)(U)12-Ct12(∫0t∫U|gradℱg⁢(s)|g⁢(s)2dVg⁢(s)ds)12 for all t∈(0,T] and U⊆M open.

Proof.

Equation (A.4) is a special case of the first equation in [22, p. 44]. Estimate (A.5) follows from (1.1). ∎

Lemma A.3.

Let (Mn,g) be a complete n-dimensional Ricci-flat Riemannian manifold having bounded curvature and injectivity radius bounded from below by ι>0. Then there exists a constant C⁢(n,ι)≥0 such that

∥R⁢mg∥L∞⁢(Mn,g)≤C⁢(n,ι).

Proof.

We argue by contradiction. Suppose this statement is not true. Then we could find a sequence of complete n-dimensional Ricci-flat manifolds (Mi,gi)i∈ℕ such that injgi⁢(Mi)≥ι and ∥R⁢mgi∥L∞⁢(Mi,gi)=Ci, where limi→∞⁡Ci=∞. We construct a blow-up sequence as follows: for each i∈ℕ, let hi:=Ci⋅gi be such that injhi⁢(Mi)≥Ci⁢ι and ∥R⁢mhi∥L∞⁢(Mi,hi)=1. For each i∈ℕ, we choose a fixed point pi∈Mi such that |R⁢mhi⁢(pi)|hi≥12. Using R⁢chi≡0, the first equation on [1, p. 461] or [9, Theorem 7.1, p. 274] implies Δhi⁢R⁢mhi=R⁢mhi∗R⁢mhi and, consequently, ∥Δhi⁢R⁢mhi∥L∞⁢(Mi,hi)≤K⁢(n). Furthermore, from [10, Lemma 1], we obtain uniform C0-bounds on the metrics (hi)i∈ℕ in normal coordinates. Hence, an iterative application of the theory of linear elliptic equations of second order, following the arguments of [1, p. 478, second paragraph], we obtain uniform higher order estimates, i.e., ∥∇hik⁡R⁢mhi∥L∞⁢(Mi,hi)≤K⁢(n,k) for all i,k∈ℕ. Hence, [1, Theorem 2.2, pp. 464–466] implies that there exists a subsequence (Mi,gi,pi)i∈ℕ that converges in the pointed Ck,α-sense, where k∈ℕ is arbitrary, to a smooth manifold (X,h,p) satisfying |R⁢mh⁢(p)|h≥12 and, using [16], injh⁢(X,p)=∞. An iterative application of [6, Theorem 2] implies that (X,h,p)=(ℝn,geuc,0) which yields a contradiction. ∎

Acknowledgements

This work is a part of the author’s doctoral thesis [28], written under the supervision of Miles Simon at the Otto-von-Guericke-Universität Magdeburg.

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Received: 2017-12-02

Revised: 2018-12-11

Accepted: 2018-12-17

Published Online: 2019-01-20

Published in Print: 2021-01-01

Convergence of Riemannian 4-manifolds with L2-curvature bounds

Abstract

A Auxiliary Results

Lemma A.1.

Proof.

Lemma A.2.

Proof.

Lemma A.3.

Proof.

Acknowledgements

References

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