  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.
