[Paper Review] Positive Mass Theorem on Manifolds admitting Corners along a Hypersurface

Pengzi Miao|arXiv (Cornell University)|Dec 5, 2002

Geometric Analysis and Curvature Flows5 references20 citations

TL;DR

This paper establishes a Positive Mass Theorem for asymptotically flat Riemannian manifolds with metrics that are $C^2$ up to a hypersurface $\Sigma$ but not $C^1$ across it, proving that the total mass is non-negative if the mean curvature of $\Sigma$ in the interior region is at least as large as in the exterior. The key result shows that zero mass implies the metric is flat everywhere and matches smoothly across $\Sigma$, extending Schoen and Yau's classical result to non-smooth settings with a geometric boundary condition.

ABSTRACT

We study a class of non-smooth asymptotically flat manifolds on which metrics fails to be $C^1$ across a hypersurface $Σ$. We first give an approximation scheme to mollify the metric, then we prove that the Positive Mass Theorem still holds on these manifolds if a geometric boundary condition is satisfied by metrics separated by $Σ$.

Motivation & Objective

• To extend the Positive Mass Theorem to non-smooth Riemannian manifolds where the metric fails to be $C^1$ across a hypersurface $\Sigma$.
• To identify a geometric boundary condition—specifically, $H(\Sigma, g_-) \geq H(\Sigma, g_+)$—that ensures the validity of the Positive Mass Theorem in such non-smooth settings.
• To investigate the rigidity of the mass-zero case, showing that zero mass implies flatness and smooth matching across $\Sigma$ in dimension three.
• To connect the result to Bartnik's quasi-local mass conjecture by showing that minimal mass extensions satisfy the derived boundary condition.

Proposed method

• Constructs a mollification scheme to approximate the non-smooth metric $\mathcal{G} = (g_-, g_+)$ by smooth metrics $\tilde{g}_\delta$ that converge to $\mathcal{G}$ in $C^2$ away from $\Sigma$.
• Applies the classical Positive Mass Theorem to the mollified metrics $\tilde{g}_\delta$, showing their masses converge to the mass of $\mathcal{G}$.
• Uses conformal deformation via solutions to the conformal Laplace equation to modify $g_-$ to zero scalar curvature, preserving the mean curvature inequality.
• Employs the strong maximum principle to show that conformal factors increase the mean curvature of $\Sigma$ in the interior, leading to contradiction if mass is zero.
• Analyzes the asymptotic behavior of conformal factors to relate the mass of the deformed metric to the original mass, proving non-negativity.
• Applies results from Bray and Finster on approximation by smooth metrics to show that zero mass implies flatness in both regions and matching second fundamental forms across $\Sigma$.

Experimental results

Research questions

• RQ1 Can the Positive Mass Theorem be extended to manifolds with metrics that are not $C^1$ across a hypersurface $\Sigma$?
• RQ2 What geometric condition on the mean curvatures of $\Sigma$ in the two regions ensures non-negative total mass in such non-smooth settings?
• RQ3 Does the vanishing of the total mass imply that the metric is flat and smoothly matched across $\Sigma$?
• RQ4 How does the boundary condition $H(\Sigma, g_-) \geq H(\Sigma, g_+)$ relate to the distributional non-negativity of scalar curvature across $\Sigma$?
• RQ5 Can this result be used to characterize minimal mass extensions in Bartnik's quasi-local mass conjecture?

Key findings

• The total mass of the metric $\mathcal{G} = (g_-, g_+)$ is non-negative if the mean curvature of $\Sigma$ in the interior region is at least as large as in the exterior region, i.e., $H(\Sigma, g_-) \geq H(\Sigma, g_+)$.
• If $H(\Sigma, g_-) > H(\Sigma, g_+)$ at any point on $\Sigma$, then the mass is strictly positive.
• In dimension $n=3$, if the mass is zero, then both $g_-$ and $g_+$ are flat in $\Omega$ and $M \setminus \overline{\Omega}$, respectively.
• The second fundamental forms of $\Sigma$ in both regions must be equal when the mass is zero, i.e., $A_- = A_+$.
• The entire manifold $(M, \mathcal{G})$ is isometric to $\mathbb{R}^3$ with the Euclidean metric when the mass is zero and the metrics are $C^{3,\alpha}_{\text{loc}}$, due to smooth matching and flatness.
• The rigidity result implies that the region bounded by $\Sigma$ in a manifold with non-negative scalar curvature and sufficiently large boundary mean curvature must be isometric to a Euclidean ball.