拟球面方程中的扩散与反应：填充问题的平均曲率形变

中文摘要 AI 辅助

我们研究了Bartnik拟球面方程在具有下界标量曲率的填充问题中形变边界平均曲率的扩散与反应效应。扩散效应为$\partial_tu=u^2\Delta u$导出了一个显式的$L^p$到$L^\infty$估计，从而将填充的最小边界平均曲率的已知上界推广为其调和平均的定量上界。对于自旋填充，该上界是显式的，且仅涉及粗略的内在边界数据。通过引入一个吸收反应项，我们还构造了一个形变，将任意非负初始平均曲率转化为具有一致正下界的终端平均曲率。对于Gromov关于总平均曲率的猜想，这可将$H\geq 0$的情形归结为Frenck、Hanke和Hirsch的定理A（假设$H\geq\kappa>0$）。这涵盖了非自旋边界的情形，补充了他们针对该猜想在自旋边界上的结果（定理B）。

英文摘要

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-$L^\infty$ estimate for $\partial_tu=u^2Δu$, 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 $H\geq 0$ case to Theorem A of Frenck, Hanke, and Hirsch \cite{FHH}, which assumes $H\geqκ>0$. This covers the case of non-spin boundaries, complementing their result for spin boundaries (Theorem B) for this conjecture.

发表机构

• Nankai University（南开大学）
• Sun Yat-sen University（中山大学）

机构由 AI 辅助整理，请以论文原文为准。