TY - JOUR
T1 - Metrics with λ1(- Δ + kR) ≥ 0 and Flexibility in the Riemannian Penrose Inequality
AU - Li, Chao
AU - Mantoulidis, Christos
N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2023/7
Y1 - 2023/7
N2 - 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.
AB - 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.
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U2 - 10.1007/s00220-023-04679-9
DO - 10.1007/s00220-023-04679-9
M3 - Article
AN - SCOPUS:85149263419
SN - 0010-3616
VL - 401
SP - 1831
EP - 1877
JO - Communications In Mathematical Physics
JF - Communications In Mathematical Physics
IS - 2
ER -