自由边界曲面关于单位球的单调性公式
A monotonicity formula for free boundary surfaces with respect to the unit ball
摘要：
证明了在$\mathbb R^n$中单位球边界内具有自由边界且具有平方可积〔的〕中曲率的紧致曲面的单调性恒等式。作为一个结果，我们得到了一个Li-Yau型不等式，从而推广了Oliveira和Soret，Fraser和Schoen的结果。在本文的最后一节中，我们得到了在任意可定向支撑曲面类$C^2$内具有自由边界的紧致曲面的一些尖锐的几何不等式。此外，我们还得到了$\mathbb R^3$中闭曲线的$L^1$ -切点能量的一个尖锐的下界，从而回答了Strzelecki，Szumaiska和von der Mosel提出的一个问题.
英文摘要：
We prove a monotonicity identity for compact surfaces with free boundaries inside the boundary of unit ball in $\mathbb R^n$ that have square integrable mean curvature. As one consequence we obtain a Li-Yau type inequality in this setting, thereby generalizing results of Oliveira and Soret, and Fraser and Schoen. In the final section of this paper we derive some sharp geometric inequalities for compact surfaces with free boundaries inside arbitrary orientable support surfaces of class $C^2$. Furthermore, we obtain a sharp lower bound for the $L^1$-tangent-point energy of closed curves in $\mathbb R^3$ thereby answering a question raised by Strzelecki, Szumańska and von der Mosel.
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