[Paper Review] A sphere theorem for Bach-flat manifolds with positive constant scalar curvature

Yi Fang, Wei Yuan|arXiv (Cornell University)|Apr 21, 2017

Geometric Analysis and Curvature Flows1 references3 citations

TL;DR

This paper establishes a sphere theorem for closed Bach-flat Riemannian manifolds with positive constant scalar curvature, proving that if the Weyl and traceless Ricci tensors are sufficiently small in $L^\infty$ or $L^{n/2}$ norm, the manifold must be locally spherical. The results generalize Singer’s rigidity theorem for Einstein manifolds and partially recover the 4D conformal sphere theorem of Chang-Gursky-Yang using a novel global estimate for symmetric 2-tensors and Sobolev inequalities under Yamabe constant control.

ABSTRACT

We show a closed Bach-flat Riemannian manifold with a fixed positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either $L^\infty$ or $L^{\frac{n}{2}}$-norm. Compared with the complete non-compact case done by Kim, we apply a different method to achieve these results. These results generalize a rigidity theorem of positive Einstein manifolds due to M.-A.Singer. As an application, we can partially recover the well-known Chang-Gursky-Yang's $4$-dimensional conformal sphere theorem.

Motivation & Objective

• To establish a rigidity result for closed Bach-flat manifolds with positive constant scalar curvature under smallness conditions on curvature tensors.
• To generalize M.-A. Singer’s rigidity theorem for positive Einstein manifolds to the broader class of Bach-flat manifolds.
• To recover a partial version of the 4-dimensional conformal sphere theorem of Chang-Gursky-Yang under curvature and Yamabe constant constraints.
• To provide a new method for global rigidity on compact manifolds, distinct from Kim’s non-compact case approach based on $L^2$-norm decay.

Proposed method

• Utilizes a global estimate for symmetric 2-tensors (Proposition 2.3) to control the traceless Ricci tensor $E$ and Weyl tensor $W$.
• Applies Sobolev inequalities with constants depending only on dimension $n$ and the lower bound $\alpha_0$ of the Yamabe constant.
• Employs Kato’s inequality and Hölder’s inequality to bound $L^{2n/(n-2)}$-norms of $E$ and $W$ in terms of their $L^{n/2}$-norms.
• Uses the conformal invariance of the $L^2$-norm of the Weyl tensor in 4D to relate the original metric $g$ to the Yamabe metric $\hat{g}$.
• Applies the Gauss-Bonnet-Chern formula in 4D to bound the $L^2$-norm of the Weyl tensor in terms of the Euler characteristic and volume.
• Establishes vanishing of $E$ and $W$ via a contradiction argument when their $L^{n/2}$-norms are below a threshold depending on $\alpha_0$.

Experimental results

Research questions

• RQ1 Under what curvature smallness conditions does a closed Bach-flat manifold with positive constant scalar curvature become locally spherical?
• RQ2 Can the rigidity result for positive Einstein manifolds be extended to Bach-flat manifolds?
• RQ3 To what extent can the 4-dimensional conformal sphere theorem of Chang-Gursky-Yang be recovered under $L^{n/2}$-norm bounds on curvature tensors?
• RQ4 How does the Yamabe constant influence the rigidity of Bach-flat manifolds with positive scalar curvature?
• RQ5 Is there a global rigidity mechanism for compact Bach-flat manifolds that avoids the $r \to \infty$ argument used in non-compact settings?

Key findings

• If $||W||_{L^\infty} + ||E||_{L^\infty} < \frac{n-1}{4}$, then the manifold is isometric to a quotient of $\mathbb{S}^n$.
• If $||W||_{L^{n/2}} + ||E||_{L^{n/2}} < \frac{3\alpha_0}{32n(n-1)}$ and the Yamabe constant $Y(M,[g]) \geq \alpha_0 > 0$, then the manifold is isometric to a quotient of $\mathbb{S}^n$.
• The manifold is Einstein if $||W||_{L^{n/2}} + ||E||_{L^{n/2}} < \frac{\alpha_0}{4n(n-1)}$, under the same Yamabe constant condition.
• In 4D, if $\int_M |W_g|^2 dv_g < \frac{32}{3}\pi^2(\chi(M^4) - 2) + \frac{\alpha_0}{192}$, then $(M^4,g)$ is conformal to $\mathbb{S}^4$ or $\mathbb{RP}^4$.
• The proof relies on a novel global estimate for symmetric 2-tensors and Sobolev inequalities with $C_S$ depending only on $n$ and $\alpha_0$, not on the metric.
• The results generalize Singer’s $L^{n/2}$-gap theorem for Einstein manifolds and partially recover the 4D conformal sphere theorem without requiring full curvature pinching.