Metrics with λ1(- Δ + kR) ≥ 0 and Flexibility in the Riemannian Penrose Inequality

Chao Li, Christos Mantoulidis

Research output: Contribution to journalArticlepeer-review

Abstract

On a closed manifold, consider the space of all Riemannian metrics for which - Δ + 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.

Original languageEnglish (US)
Pages (from-to)1831-1877
Number of pages47
JournalCommunications In Mathematical Physics
Volume401
Issue number2
DOIs
StatePublished - Jul 2023

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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