An isoperimetric characterization of a new ADM-like mass for $C^0$-asymptotically flat manifolds

Technical Summary: An Isoperimetric Characterization of a New ADM-Like Mass for $C^0$-Asymptotically Flat Manifolds

Problem Statement
The paper addresses the relationship between two distinct notions of mass in the context of general relativity on Riemannian 3-manifolds with low regularity. The classical ADM mass, while fundamental, requires strong asymptotic decay and differentiability of the metric to be well-defined and coordinate-invariant. In contrast, the isoperimetric mass, introduced by G. Huisken, is defined purely in terms of volumes and areas, making it robust under $C^0$-regularity assumptions.

Recently, Mazurowski and Yao introduced a new "ADM-like" mass ($m_{MY}$) for continuous metrics with nonnegative scalar curvature, defined via an integral involving a specific symmetric tensor and a test function $\psi$. While $m_{MY}$ is well-posed for $C^\tau$-asymptotically flat manifolds with $\tau > 1/2$, its equivalence to Huisken's isoperimetric mass ($m_{iso}$) in the smooth case remained an open question. The authors aim to prove that these two quantities coincide under optimal decay assumptions ($\tau > 1/2$) for smooth metrics.

Methodology
The proof relies on a nonlinear potential-theoretic framework centered on the $p$-inverse mean curvature flow ($p$-IMCF) and the associated $p$-Hawking masses. The methodology proceeds through several key steps:

1. $p$-IMCF and Regularization: The authors utilize the $p$-capacitary potential $u_p$ and the associated flow $w_p = -(p-1)\log u_p$. To handle the low regularity of the level sets $\Sigma^{(p)}_t$ in the limit, they employ an elliptic regularization ($w_{p,\varepsilon}$) to approximate the flow with smooth surfaces, allowing the application of standard geometric analysis tools.
2. Asymptotic Analysis: A critical technical component is establishing the asymptotic behavior of the $p$-IMCF under merely $C^0$-asymptotic flatness. The authors adapt a blow-down procedure and utilize the Kinnunen–Zhou gradient estimate for $p$-harmonic functions. This estimate, which holds under $C^0$-convergence of metrics, allows them to derive strong $W^{1,q}_{loc}$ convergence of the rescaled potentials to the Euclidean fundamental solution.
3. Monotonicity and Comparison: The authors establish monotonicity properties for a modified $p$-Hawking mass along the flow. They prove that the limit of the $p$-Hawking mass as $t \to \infty$ is bounded above by the Mazurowski–Yao mass ($m_{MY}$). This is achieved by relating the asymptotic behavior of the $p$-Hawking mass to the integral definition of $m_{MY}$ (Proposition 2.14).
4. Connecting to Isoperimetric Mass:
   • Upper Bound ($m_{iso} \leq m_{MY}$): By showing that the Hawking mass of outward minimizing domains is bounded by $m_{MY}$, and invoking the machinery of Jauregui and Lee, they deduce that the isoperimetric mass is also bounded by $m_{MY}$.
   • Lower Bound ($m_{iso} \geq m_{MY}$): They demonstrate that $m_{MY}$ can be computed as the limit at infinity of the 2-Hawking mass. Using asymptotic comparison arguments, they bound this limit from above by the isocapacitary mass (a capacitary version of the isoperimetric mass). The authors then cite their previous work (Benatti [Ben25]) which establishes the equivalence between the isocapacitary mass and the isoperimetric mass in this setting.

Key Contributions and Results

• Theorem 1.1: The primary result establishes that for a smooth, connected, complete, $C^\tau$-asymptotically flat Riemannian 3-manifold ($\tau > 1/2$) with nonnegative scalar curvature and minimal boundary, the Mazurowski–Yao mass coincides with the isoperimetric mass ($m_{MY} = m_{iso}$).
• Asymptotic Comparison: The paper refines asymptotic comparison arguments for $p$-Hawking masses, proving that the $p$-Hawking mass converges to the classical Hawking mass as $p \to 1$ and that these quantities are controlled by the isoperimetric mass under $C^0$-asymptotic flatness.
• Corollary 3.9 (Schwarzschildian Manifolds): For manifolds that are asymptotically Schwarzschildian (where the metric deviation admits a leading term $2m/|x|$), the parameter $m$ is identified with both $m_{MY}$ and $m_{iso}$.
• Corollary 3.10 (Penrose Inequality): Combining the identification of masses with the isoperimetric Penrose inequality, the authors derive a Riemannian Penrose inequality for asymptotically Schwarzschildian 3-manifolds: $\sqrt{|N|/16\pi} \leq m$, where $N$ is a component of the outermost minimal boundary. Equality holds if and only if the manifold is isometric to a spatial Schwarzschild manifold.

Significance and Claims
The paper claims to provide a unified framework for different notions of mass in the smooth 3-dimensional setting with decay $\tau > 2/3$ (and $\tau > 1/2$ for the specific equivalence proved). By proving $m_{MY} = m_{iso}$, the authors affirmatively answer a question regarding the equivalence of the mass introduced by Burkhardt-Guim and the isoperimetric mass in this regime.

The significance lies in bridging the gap between analytic definitions of mass (relying on derivatives and integrals of the metric) and geometric definitions (relying on volume and area). This equivalence validates the use of the isoperimetric mass as a robust surrogate for ADM-type masses even when the metric is only continuous, provided the scalar curvature is nonnegative in a suitable approximate sense. The results also extend the validity of the Riemannian Penrose inequality to a broader class of asymptotically flat manifolds defined by their deviation from the Schwarzschild metric.

The authors note that while the equivalence is established for smooth metrics, the extension to continuous asymptotically flat metrics remains an open question, though the current work provides the necessary tools (specifically the $W^{1,q}$ asymptotics for $p$-IMCF under $C^0$ flatness) to potentially address it in the future.