Metrics with $$\lambda _1(-\Delta + k R) \ge 0$$ and Flexibility in the Riemannian Penrose Inequality

On a closed manifold, consider the space of all Riemannian metrics for which -Delta + kR is positive (nonnegative) definite, where k > 0 and R scalar curvature. This spectral generalization curvature arises naturally different values in study via minimal hypersurfaces, Yamabe problem, Perelman's Ricci flow with surgery. When k=1/2, models apparent horizons time-symmetric initial data to Einstein equations. We these spaces unison generalize Cod\'a Marques's path-connectedness theorem. Applying this we compute Bartnik mass 3-dimensional Bartnik--Bray their outer-minimizing generalizations dimensions. Our methods also yield efficient constructions scalar-nonnegative fill-in problem.