Title: Monotonicity Formulas for Capillary Surfaces Show Chinese title

Title: 单调性公式对于毛细表面

Abstract: 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}). Show Chinese abstract

Abstract: 在本文中，我们建立了半空间 $\mathbb{R}^3_+$ 和单位球 $\mathbb{B}^3$ 中的毛细表面的单调性公式，并扩展了 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}) 对具有自由边界表面的结果。 作为应用，我们得到了毛细表面的 Willmore 能量的 Li-Yau 型不等式，并将 Fraser-Schoen 在 $\mathbb{B}^3$ 中对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}) 扩展到毛细情形，这与 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}) 证明的另一种最优面积估计不同。