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

Chao Li; Christos Mantoulidis

doi:10.1007/s00220-023-04679-9

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arXiv · 2106.15709

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

Chao Li·Christos Mantoulidis

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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.

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Chao Li, Christos Mantoulidis. 2021-06-29. Metrics with $\lambda_1(-\Delta + k R) \geq 0$ and flexibility in the Riemannian Penrose Inequality. https://doi.org/10.1007/s00220-023-04679-9

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