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.