Brown-York mass and positive scalar curvature I - First eigenvalue problem and its applications

TL;DR

This paper explores the relationship between scalar curvature and the first eigenvalue on manifolds with boundary, establishing sharp inequalities and applications to black hole horizon area estimates in static spaces.

Contribution

It introduces a new inequality linking scalar curvature and eigenvalues, with applications to geometric analysis and general relativity.

Findings

Proves a sharp inequality for first eigenfunctions under scalar curvature conditions.

Shows the first eigenvalue is at least n for manifolds with scalar curvature ≥ n(n-1).

Provides an area estimate for event horizons in vacuum static spaces with positive cosmological constant.

Abstract

In this article, we investigate the connection between scalar curvature and first eigenfunctions via positive mass theorem for Brown-York mass. For compact manifolds with nice boundary, we show that a sharp inequality holds for first eigenfunctions when posing appropriate assumptions on scalar curvature and first eigenvalue. This inequality implies that for a compact n-dimensional manifold with boundary, its first eigenvalue is no less that n, if its scalar curvature is at least n(n-1) with appropriate boundary conditions posed, where equality holds if and only the manifold is isometric to the canonical upper hemisphere. As an application, we derive an estimate for the area of event horizon in a vacuum static space with positive cosmological constant, which reveals an interesting connection between the area of event horizon and Brown-York mass. This estimate generalizes a similar result of Shen for three dimensional vacuum static spaces and also improves the uniqueness result of de Sitter space-time due to Hizagi-Montiel-Raulot.