[Paper Review] Scalar curvature rigidity for locally conformally flat manifolds with boundary

Fabian-Michael Spiegel|arXiv (Cornell University)|Nov 19, 2015

Geometric Analysis and Curvature Flows15 references3 citations

TL;DR

This paper establishes a scalar curvature rigidity theorem for compact, locally conformally flat manifolds with boundary, proving that under scalar curvature $\mathrm{Scal} \geq n(n-1)$ and boundary mean curvature $H(g) \geq \cot(\rho)$ with boundary isometric to a geodesic sphere of radius $\rho$ in $S^n$, the manifold is isometric to the geodesic ball $\overline{D_\rho}$ in the standard sphere. The proof uses conformal geometry, the developing map, and maximum principle arguments via stereographic projection and subsolution-supersolution comparison.

ABSTRACT

Inspired by the work of F. Hang and X. Wang and partial results by S. Raulot, we prove a scalar curvature rigitidy result for locally conformally flat manifolds with boundary in the spirit of the well-known Min-Oo conjecture.

Motivation & Objective

• To establish a rigidity result for locally conformally flat manifolds with boundary under scalar curvature and mean curvature bounds.
• To generalize previous results by Hang and Wang and Raulot by removing dimensional and topological restrictions (e.g., $n=4,6$, $\chi(M)=1$).
• To prove that such manifolds are isometric to geodesic balls in the standard sphere $S^n$ under natural curvature and boundary conditions.
• To extend the Min-Oo conjecture to the locally conformally flat setting without assuming global conformal equivalence to the hemisphere.

Proposed method

• Use of the developing map to lift the manifold to a subset of $S^n$, leveraging local conformal flatness.
• Application of the conformal invariance of the scalar curvature and mean curvature equations under $g \mapsto u^{\frac{4}{n-2}}g$.
• Stereographic projection to transfer the problem to $\mathbb{R}^n$, where the metric becomes $v^{\frac{4}{n-2}}\sum dx^i \otimes dx^i$.
• Construction of a supersolution $v$ and comparison with the standard spherical metric $w$ via the equation $-\Delta v \geq \frac{n(n-2)}{4} v^{\frac{n+2}{n-2}}$.
• Application of the maximum principle and Hopf lemma to show $v = w$ on $\partial B_r$, implying isometry.
• Use of the fact that $v \to \infty$ near isolated singularities to rule out nontrivial conformal defects, forcing $\Lambda = \emptyset$.

Experimental results

Research questions

• RQ1 Under what conditions is a locally conformally flat manifold with boundary isometric to a geodesic ball in $S^n$?
• RQ2 Can the Min-Oo conjecture be extended to locally conformally flat manifolds without requiring global conformal equivalence to the sphere?
• RQ3 What role does the boundary mean curvature $H(g) \geq \cot(\rho)$ play in rigidity, and can this condition be weakened?
• RQ4 Is the assumption $\rho \leq \pi/2$ optimal, and why does it fail for $\rho > \pi/2$?
• RQ5 Can rigidity be established without assuming $\chi(M) = 1$ or $n=4,6$, as in prior results?

Key findings

• The manifold $(M,g)$ is isometric to the geodesic ball $\overline{D_\rho} \subset S^n$ under the conditions $\mathrm{Scal}(g) \geq n(n-1)$ and $H(g) \geq \cot(\rho)$ with $\partial M \cong \Sigma_\rho$.
• The boundary condition $H(g) \geq \cot(\rho)$ is sharp and cannot be relaxed without additional assumptions, as shown by optimality of $\rho \leq \pi/2$.
• If $M$ is simply-connected and $\rho = \pi/2$, the mean curvature condition can be dropped, recovering a result similar to Hang and Wang.
• The proof shows that any conformal defect (i.e., isolated singularities) leads to a contradiction via the maximum principle, forcing the metric to be globally conformal to the standard sphere.
• The result holds for all $n \geq 3$, removing the dimensional and topological restrictions of earlier results by Raulot and others.
• The key step is showing $v = w$ on $\partial B_r$ via the Hopf lemma, which forces the metric to be identical to the standard spherical metric on the projected ball.