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arXiv · 2610.08283

Radial Powered Mean Curvature Flow with Robin Boundary Conditions

Tianhong Pu·Xiaoqian Xin·Lixia Yuan
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Abstract

We study a positive radial graph evolving by $V=H^α+b$, with $α>0$ and $b<0$, in the unit cylinder and with $u_r(1,t)=u(1,t)$. We separate finite-time continuation estimates from the time-uniform interior estimates needed for convergence of normalized translates. On existing classical intervals we prove height and gradient bounds and an explicit conversion of velocity bounds into estimates for $H$, $u_r/r$, and $u_{rr}$. We derive the correct boundary defect for comparisons with finite-slope translators. Under independent interior velocity bounds and two-sided slope trapping, we prove compactness and identify every translated limit with the cup translator. We identify the radial zero-number comparison still required to establish that trapping. Stationary caps and power-law Robin laws are included.

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Tianhong Pu, Xiaoqian Xin, Lixia Yuan. 2026-10-06. Radial Powered Mean Curvature Flow with Robin Boundary Conditions. https://arxiv.org/abs/2610.08283

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