Existence of exhaustions with controlled sign of mean curvature in noncompact manifolds
Abstract
Given a complete and non-compact Riemannian manifold of dimension 3≤ n≤ 7, we find conditions for the existence of compact exhaustions {Ui}i=1∞ such that all boundaries ∂ Ui have the same sign of mean curvature. In particular, we show that in these dimensions, a manifold with one end satisfying a no tentacle condition always has a weakly mean convex, weakly mean concave, or minimal exhaustion, according to the behaviour of the area profile of separating hypersurfaces. This is a version of a result sketched by Gromov, under a condition which is weaker than his thickness at infinity. In more general situations, we study the existence of exhaustions which are weakly mean concave or weakly mean convex apart from an arbitrarily small L1 error. That is, we find sufficient conditions for the existence of exhaustions {Ui}i=1∞ which satisfy ∫∂ UiH+→0, ∫∂ UiH-→0, or ∫∂ Ui|H|→0, where H-, H+, and |H| are the negative part, positive part, and the absolute value of the mean curvature respectively. As an application of our methods, we study the existence of almost c-convex exhaustions, that is, exhaustions whose boundaries satisfy ∫∂ Ui(H−c)-→0 for some c>0.
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BibTeX
@misc{das2026existence,
title = {{Existence of exhaustions with controlled sign of mean curvature in noncompact manifolds}},
author = {Anushree Das},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19241},
eprint = {2610.19241},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19241},
}
RIS
TY - UNPB
TI - Existence of exhaustions with controlled sign of mean curvature in noncompact manifolds
AU - Anushree Das
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19241
UR - https://arxiv.org/abs/2610.19241
AB - Given a complete and non-compact Riemannian manifold of dimension 3≤ n≤ 7, we find conditions for the existence of compact exhaustions {U_i}_i=1^∞ such that all boundaries ∂ U_i have the same sign of mean curvature. In particular, we show that in these dimensions, a manifold with one end satisfying a no tentacle condition always has a weakly mean convex, weakly mean concave, or minimal exhaustion, according to the behaviour of the area profile of separating hypersurfaces. This is a version of a result sketched by Gromov, under a condition which is weaker than his thickness at infinity. In more general situations, we study the existence of exhaustions which are weakly mean concave or weakly mean convex apart from an arbitrarily small L^1 error. That is, we find sufficient conditions for the existence of exhaustions {U_i}_i=1^∞ which satisfy ∫_∂ U_iH^+→0, ∫_∂ U_iH^-→0, or ∫_∂ U_i|H|→0, where H^-, H^+, and |H| are the negative part, positive part, and the absolute value of the mean curvature respectively. As an application of our methods, we study the existence of almost c-convex exhaustions, that is, exhaustions whose boundaries satisfy ∫_∂ U_i(H−c)^-→0 for some c>0.
ER -