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The total geodesic curvature and the $(2+1)$-dimensional hyperbolic mass

Xiaokai He, Xiaoning Wu, Naqing Xie

TL;DR

The paper derives a boundary‑only upper bound on the total geodesic curvature of the boundary $\Sigma$ of a 2D domain with a lower bound on Gauss curvature, by relating it to a hyperbolic reference geometry via the BYST mass. The core approach combines a Bartnik–Shi–Tam gluing construction, spinorial positivity arguments for the hyperbolic Hamiltonian mass $H^0$, and a monotonicity formula comparing the actual boundary curvature to its hyperbolic counterpart. A key result is $\int_\Sigma k\,ds \le \int_\Sigma \hat{k}\,ds=2\pi\sqrt{1+r_0^2}$, with equality forcing $(\Omega,g)$ to be a geodesic disk in $(\mathbb{H}^2,g_{-1})$, and $H^0\ge0$ with rigidity when the boundary data match hyperbolic data. The paper also connects this boundary inequality to the positive mass theorem in the hyperbolic setting and analyzes the BYST mass for large ellipses in BTZ geometries, showing that the BYST limit can differ from the Hamiltonian mass for nonround ellipses and providing asymptotic expansions.

Abstract

We consider a Jordan domain diffeomorphic to a closed two-dimensional disk with a smooth boundary. Assuming the Gauss curvature of the domain has a negative lower bound, the Gauss-Bonnet formula provides an upper bound for the total geodesic curvature of the boundary curve. This bound, however, inherently depends on the interior geometry of the region. In this paper, we derive an upper bound for the total geodesic curvature expressed solely in terms of the boundary data. Notably, the proof is connected to the positivity of the hyperbolic Hamiltonian mass in the (2+1)-dimensional gravity theory.

The total geodesic curvature and the $(2+1)$-dimensional hyperbolic mass

TL;DR

The paper derives a boundary‑only upper bound on the total geodesic curvature of the boundary of a 2D domain with a lower bound on Gauss curvature, by relating it to a hyperbolic reference geometry via the BYST mass. The core approach combines a Bartnik–Shi–Tam gluing construction, spinorial positivity arguments for the hyperbolic Hamiltonian mass , and a monotonicity formula comparing the actual boundary curvature to its hyperbolic counterpart. A key result is , with equality forcing to be a geodesic disk in , and with rigidity when the boundary data match hyperbolic data. The paper also connects this boundary inequality to the positive mass theorem in the hyperbolic setting and analyzes the BYST mass for large ellipses in BTZ geometries, showing that the BYST limit can differ from the Hamiltonian mass for nonround ellipses and providing asymptotic expansions.

Abstract

We consider a Jordan domain diffeomorphic to a closed two-dimensional disk with a smooth boundary. Assuming the Gauss curvature of the domain has a negative lower bound, the Gauss-Bonnet formula provides an upper bound for the total geodesic curvature of the boundary curve. This bound, however, inherently depends on the interior geometry of the region. In this paper, we derive an upper bound for the total geodesic curvature expressed solely in terms of the boundary data. Notably, the proof is connected to the positivity of the hyperbolic Hamiltonian mass in the (2+1)-dimensional gravity theory.
Paper Structure (6 sections, 2 theorems, 64 equations, 1 figure)

This paper contains 6 sections, 2 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Prescribed scalar curvature
  3. Positivity of the mass expression
  4. A monotonicity formula
  5. An equivalent statement of the positive mass theorem
  6. Limits of the BYST mass for large ellipses

Key Result

Theorem 1

Let $(\Omega,g)$ be a Jordan domain with boundary $\partial \Omega =\Sigma$. We choose the orthornomal frame $\{e_1,e_2\}$ so that $e_1$ is tangent to $\Sigma$ and $e_2$ is normal to $\Sigma$. The boundary curve $\Sigma$ is parameterised as Denote by $L=2\pi r_0$ and $k=g(\nabla^g_{e_1}e_2,e_1)$ the length and the geodesic curvature of $\Sigma$ with respect to the outward normal $e_2$ of $\Sigma$

Figures (1)

  • Figure 1: Graph of $\mathfrak{m}_{\infty}(m,\epsilon)$ with respect to $\epsilon$ for $m=1,0,-1$.

Theorems & Definitions (2)

  • Theorem 1
  • Proposition 1