Diffusion and reaction in the quasi-spherical equation: mean curvature deformations for fill-Ins
Abstract
We study the diffusion and reaction effects of Bartnik's quasi-spherical equation to deform boundary mean curvature in fill-in problems with scalar curvature bounded below. The diffusion effect yields an explicit Lp-to-L∞ estimate for ∂tu=u2Δu, thereby extending the known upper bound for the minimum boundary mean curvature of fill-ins to a quantitative upper bound for its harmonic mean. For spin fill-ins, this bound is explicit and involves only coarse intrinsic boundary data. By introducing an absorbing reaction term, we also construct a deformation that transforms any nonnegative initial mean curvature into a terminal mean curvature with a uniform positive lower bound. For Gromov's conjecture on total mean curvature, this reduces the H≥ 0 case to Theorem A of Frenck, Hanke, and Hirsch \cite{FHH}, which assumes H≥κ>0. This covers the case of non-spin boundaries, complementing their result for spin boundaries (Theorem B) for this conjecture.
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BibTeX
@misc{wang2026diffusion,
title = {{Diffusion and reaction in the quasi-spherical equation: mean curvature deformations for fill-Ins}},
author = {Wenlong Wang and Guodong Wei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13999},
eprint = {2609.13999},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13999},
}
RIS
TY - UNPB
TI - Diffusion and reaction in the quasi-spherical equation: mean curvature deformations for fill-Ins
AU - Wenlong Wang
AU - Guodong Wei
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13999
UR - https://arxiv.org/abs/2609.13999
AB - We study the diffusion and reaction effects of Bartnik's quasi-spherical equation to deform boundary mean curvature in fill-in problems with scalar curvature bounded below. The diffusion effect yields an explicit L^p-to-L^∞ estimate for ∂_tu=u^2Δu, thereby extending the known upper bound for the minimum boundary mean curvature of fill-ins to a quantitative upper bound for its harmonic mean. For spin fill-ins, this bound is explicit and involves only coarse intrinsic boundary data. By introducing an absorbing reaction term, we also construct a deformation that transforms any nonnegative initial mean curvature into a terminal mean curvature with a uniform positive lower bound. For Gromov's conjecture on total mean curvature, this reduces the H≥ 0 case to Theorem A of Frenck, Hanke, and Hirsch \cite{FHH}, which assumes H≥κ>0. This covers the case of non-spin boundaries, complementing their result for spin boundaries (Theorem B) for this conjecture.
ER -