能量凸性与 $\mathbb{R}^{3}$ 中带 Dirichlet 边界条件的 $H$-曲面流的均匀性

中文摘要 AI 辅助

我们证明了在 $\mathbb{R}^{3}$ 中，与给定平均曲率曲面相关的能量泛函，当限制在从单位 $2$-圆盘出发、具有小 Dirichlet 能量和固定边界值的映射上时，表现出凸性性质，前提是 $3$-形式 $H \cdot \text{vol}_{\mathbb{R}^{3}}$ 具有满足某些界限的原像。在关于 $H:\mathbb{R}^{3} \to \mathbb{R}$ 的更温和假设下，我们证明了当初始 Dirichlet 能量足够小时，类似的凸性估计沿泛函的热流（即 $H$-曲面流）成立。这首先对经典解证明，然后通过近似扩展到弱解，使用了从调和映射热流的先前工作改编的定量唯一性结果。作为凸性估计的推论，我们证明了在单位 $2$-圆盘上具有小能量初始映射（属于 $C^0 \cap W^{1,2}$）的 $H$-曲面流在无限时间时均匀收敛到唯一极限，该极限以相同边界值求解相应的平稳问题（即 $H$-曲面系统）。

英文摘要

We prove that the energy functional associated with surfaces of prescribed mean curvature in $\mathbb{R}^3$ exhibits a convexity property when restricted to maps from the unit $2$-disk having small Dirichlet energy and a fixed boundary value, provided that the $3$-form $H \cdot \text{vol}_{\mathbb{R}^3}$ has a primitive satisfying certain bounds. Under milder assumptions on $H:\mathbb{R}^3 \to \mathbb{R}$, we show that an analogous convexity estimate holds along the heat flow of the functional (the $H$-surface flow) when the initial Dirichlet energy is sufficiently small. This is done first for classical solutions, and then extended by approximation to weak solutions using a quantitative uniqueness result adapted from previous work on the harmonic map heat flow. As a consequence of the convexity estimate, we show that the $H$-surface flow with small-energy initial map of class $C^0 \cap W^{1, 2}$ on the unit $2$-disk converges uniformly to a unique limit at infinite time, which solves the corresponding stationary problem (the $H$-surface system) with the same boundary value.

发表机构

• University of Miami（迈阿密大学）
• University of California, Santa Cruz（加州大学圣克鲁兹分校）
• Cornell University（康奈尔大学）

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