Metrics with ${\lambda }_1(-\Delta +kR)\ge 0$ and flexibility in the Riemannian Penrose Inequality

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 Codá 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.

To appear in Comm. Math. Phys. There are several refinements and expository improvements relative to previously uploaded version thanks to comments from the anonymous referees

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