Metrics with $\lambda_1(-\Delta + k R) \geq 0$ and flexibility in the Riemannian Penrose Inequality

TL;DR

This paper explores the space of Riemannian metrics with nonnegative spectral properties related to scalar curvature, generalizes known theorems, and applies these to compute mass in general relativity and solve fill-in problems.

Contribution

It generalizes Codá Marques's path-connectedness theorem for metrics with spectral positivity and applies it to compute Bartnik and Bartnik--Bray masses, extending results in scalar curvature and general relativity.

Findings

Computed Bartnik mass of 3D apparent horizons

Extended path-connectedness to spectral positivity spaces

Provided constructions for scalar-nonnegative fill-in problem

Abstract

On a closed manifold, consider the space of all Riemannian metrics for which -Delta + kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally for different values of k in the study of scalar curvature via minimal hypersurfaces, the Yamabe problem, and Perelman's Ricci flow with surgery. When k=1/2, the space models apparent horizons in time-symmetric initial data to the Einstein equations. We study these spaces in unison and generalize Cod\'a Marques's path-connectedness theorem. Applying this with k=1/2, we compute the Bartnik mass of 3-dimensional apparent horizons and the Bartnik--Bray mass of their outer-minimizing generalizations in all dimensions. Our methods also yield efficient constructions for the scalar-nonnegative fill-in problem.