[Paper Review] A rigidity theorem for surfaces in Schwarzschild manifold

Po-Ning Chen, Xiangwen Zhang|arXiv (Cornell University)|Feb 3, 2018

Geometric Analysis and Curvature Flows19 references3 citations

TL;DR

This paper establishes a rigidity theorem for closed, strictly convex surfaces in the Schwarzschild manifold: if two such surfaces are isometric and have identical mean curvature, and satisfy a non-positive Ricci curvature condition along the normal, then they must be isometric via a global isometry of the Schwarzschild space. The proof leverages a variational formula for quasi-local mass and the quasi-local Penrose inequality to show the second fundamental forms must coincide, implying isometry via symmetry.

ABSTRACT

In this article, we prove a rigidity theorem for isometric embeddings into the Schwarzschild manifold, by using the variational formula of quasi-local mass.

Motivation & Objective

• To extend Cohn-Vossen's rigidity theorem from Euclidean space to the Schwarzschild manifold.
• To address the uniqueness of isometric embeddings of convex surfaces in a curved spacetime with non-trivial geometry.
• To investigate whether isometric surfaces with the same mean curvature must be globally isometric in the Schwarzschild background.
• To provide a geometric rigidity result motivated by the weighted quasi-local Penrose inequality in general relativity.
• To establish conditions under which isometric surfaces in the Schwarzschild manifold are equivalent up to isometry, despite the space's reduced symmetry.

Proposed method

• Derive a variational formula for the quasi-local mass in the Schwarzschild manifold using the static potential and induced geometry on surfaces.
• Apply the quasi-local Penrose inequality to ensure non-negativity of the mass difference under mean convexity.
• Use the isometry between surfaces to identify functions and tensors via pull-back and push-forward operations.
• Compute the first variation of the quasi-local mass under a normal deformation, leading to an expression involving the difference of second fundamental forms.
• Employ integration by parts and the Codazzi equation to relate the variation to the $ L^2 $-norm of the difference of second fundamental forms.
• Conclude that the vanishing of the first variation implies $ |h - h'|^2 = 0 $, hence $ h = h' $, under the given conditions.

Experimental results

Research questions

• RQ1 Can Cohn-Vossen-type rigidity be extended from Euclidean space to the Schwarzschild manifold?
• RQ2 What additional geometric conditions are necessary for rigidity in a spacetime with reduced isometry group, such as Schwarzschild?
• RQ3 Does the equality case of the quasi-local Penrose inequality imply isometry when mean curvatures and isometries are preserved?
• RQ4 How does the static potential influence the variation of quasi-local mass in isometric surface families?
• RQ5 To what extent does the second fundamental form determine the embedding of a surface in the Schwarzschild manifold under isometry and mean curvature constraints?

Key findings

• The second fundamental forms of isometric surfaces $ \Sigma $ and $ \Sigma' $ in the Schwarzschild manifold are identical if they have the same mean curvature and satisfy $ \mathrm{Ric}(\nu,\nu) \leq 0 $.
• The vanishing of the first variation of the quasi-local mass at $ s = 0 $, under the assumption $ H = H' $, implies $ \int_\Sigma V |h - h'|^2 \, d\sigma = 0 $, hence $ h = h' $.
• The radial function $ r $ is uniquely determined on both surfaces under the isometry, due to the monotonicity of the Ricci curvature norm in $ r $.
• The surfaces $ \Sigma $ and $ \Sigma' $ are globally isometric via an isometry of the Schwarzschild manifold, due to rotational symmetry and the equality of $ r $-restrictions.
• The result holds even if $ \Sigma' $ is not strictly convex, as long as it is mean convex and isometric to $ \Sigma $ with matching mean curvature.
• The proof relies on the positivity of the quasi-local mass and the variational formula derived from the static potential and curvature identities.