Static solutions from the point of view of comparison geometry

TL;DR

This paper uses comparison geometry to analyze static solutions of Einstein's equations in three dimensions, deriving monotonic quantities that lead to proofs of the Penrose inequality and Schwarzschild uniqueness without conformal flatness assumptions.

Contribution

It introduces a novel comparison geometry approach to static Einstein solutions, providing new monotonic quantities and simplified proofs of key inequalities and uniqueness results.

Findings

Recovered the Penrose inequality for asymptotically flat solutions.

Proved the uniqueness of the Schwarzschild solution.

Established sharp estimates for the Laplacian of the distance function.

Abstract

We analyze (the harmonic map representation of) static solutions of the Einstein Equations in dimension three from the point of view of comparison geometry. We find simple monotonic quantities capturing sharply the influence of the Lapse function on the focussing of geodesics. This allows, in particular, a sharp estimation of the Laplacian of the distance function to a given (hyper)-surface. We apply the technique to asymptotically flat solutions with regular and connected horizons and, after a detailed analysis of the distance function to the horizon, we recover the Penrose inequality and the uniqueness of the Schwarzschild solution. The proof of this last result does not require proving conformal flatness at any intermediate step.