毛细管表面的单调性公式

arXiv - MATH - Differential Geometry Pub Date : 2024-09-05 DOI:arxiv-2409.03314

Guofang Wang, Chao Xia, Xuwen Zhang

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引用次数: 0

摘要

本文建立了半空间 $\mathbb{R}^3_+$ 和单位球 $\mathbb{B}^3$ 中毛细管表面的单调性公式，并扩展了 Volkmann 的结果（Comm.Anal.Geom.24(2016), no.1, 195~221.\href{https://doi.org/10.4310/CAG.2016.v24.n1.a7}{https://doi.org/10.4310/CAG.2016.v24.n1.a7})for surfaces with free boundary.作为应用，我们得到了毛细管表面的 Willmore 能量的 Li-Yau-typeinequalities，并将 Fraser-Schoen 对$\mathbb{B}^3$中最小自由边界表面的最优面积估计（Adv. Math.226(2011), no.5, 4011~4030.\href{https://doi.org/10.1016/j.aim.2010.11.007}{https://doi.org/10.1016/j.aim.2010.11.007} ）扩展到毛细管环境，这与 Brendle 所证明的另一个最优面积估计不同（Ann.Fac.(6)32(2023), no.1, 179~201.\href{https://doi.org/10.5802/afst.1734}{https://doi.org/10.5802/afst.1734}).

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Monotonicity Formulas for Capillary Surfaces

In this paper, we establish monotonicity formulas for capillary surfaces in the half-space $\mathbb{R}^3_+$ and in the unit ball $\mathbb{B}^3$ and extend the result of Volkmann (Comm. Anal. Geom.24(2016), no.1, 195~221. \href{https://doi.org/10.4310/CAG.2016.v24.n1.a7}{https://doi.org/10.4310/CAG.2016.v24.n1.a7}) for surfaces with free boundary. As applications, we obtain Li-Yau-type inequalities for the Willmore energy of capillary surfaces, and extend Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in $\mathbb{B}^3$ (Adv. Math.226(2011), no.5, 4011~4030. \href{https://doi.org/10.1016/j.aim.2010.11.007}{https://doi.org/10.1016/j.aim.2010.11.007}) to the capillary setting, which is different to another optimal area estimate proved by Brendle (Ann. Fac. Sci. Toulouse Math. (6)32(2023), no.1, 179~201. \href{https://doi.org/10.5802/afst.1734}{https://doi.org/10.5802/afst.1734}).

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arXiv - MATH - Differential Geometry

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