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Mass-Capacity Rigidity for Asymptotically Flat Manifolds with Arbitrary Ends
Abstract
We characterize equality in the mass-capacity inequality for asymptotically flat manifolds with arbitrary ends in every dimension n≥ 3. If u is the capacity minimizer, then the equality forces (M,u4/(n−2)g) to be isometric to (ℝn∖ S, gEuc) for a compact set S satisfying dimℋS≤ (n−2)/2. A key ingredient is that conformal changes by positive harmonic functions transform nonnegative Ricci curvature into nonnegative Bakry--Émery Ricci curvature of effective dimension 4−n.
Related
- Positive mass theorems for singular asymptotically hyperbolic manifolds
- Positive mass and rigidity for asymptotically flat tangent bundles
- A new boundary mass for asymptotically flat half-manifolds
- Convex hulls and area minimizing currents in asymptotically locally hyperbolic manifolds
- Isometric Embeddings of Asymptotically Hyperbolic Manifolds into Hyperbolic Space
BibTeX
@misc{bi2026masscapacity,
title = {{Mass-Capacity Rigidity for Asymptotically Flat Manifolds with Arbitrary Ends}},
author = {Yuchen Bi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08579},
eprint = {2609.08579},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08579},
}
RIS
TY - UNPB
TI - Mass-Capacity Rigidity for Asymptotically Flat Manifolds with Arbitrary Ends
AU - Yuchen Bi
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08579
UR - https://arxiv.org/abs/2609.08579
AB - We characterize equality in the mass-capacity inequality for asymptotically flat manifolds with arbitrary ends in every dimension n≥ 3. If u is the capacity minimizer, then the equality forces (M,u^4/(n−2)g) to be isometric to (ℝ^n∖ S, g_Euc) for a compact set S satisfying dim_ℋS≤ (n−2)/2. A key ingredient is that conformal changes by positive harmonic functions transform nonnegative Ricci curvature into nonnegative Bakry--Émery Ricci curvature of effective dimension 4−n.
ER -