Metrics with $λ_1(-Δ+ k R) \geq 0$ and flexibility in the Riemannian Penrose Inequality

Chao Li, Christos Mantoulidis

Published 2021-06-29Version 1

On a closed n-dimensional 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 $ in the study of scalar curvature in dimension n+1 via minimal hypersurfaces, the Yamabe problem in dimension n, and Perelman's surgery for Ricci flow in dimension n=3. We study these spaces in unison and generalize, as appropriate, positive scalar curvature results (Gromov--Lawson's codimension-3 surgery and Marques's connectedness) that we eventually apply to k = 1/2, where the space above models apparent horizons in (n+1)-dimensional time-symmetric initial data to the Einstein equations. Specifically, we compute the Bartnik mass (n=3) of apparent horizons and the Bartnik--Bray mass (n >= 2) of their outer-minimizing generalizations.