A monotonicity formula for free boundary surfaces with respect to the unit ball
A monotonicity formula for free boundary surfaces with respect to the unit ball
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DOI：
10.4310/cag.2016.v24.n1.a7
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发表时间：
2014-02
期刊：
影响因子：
--
通讯作者：
Alexander Volkmann
中科院分区：
文献类型：
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作者：
Alexander Volkmann
我们证明了在$\mathbb R^n$中具有平方可积平均曲率的单位球边界内具有自由边界的紧曲面的单调恒等式。作为一个结果，我们在这种情况下得到了Li-Yau型不等式，从而推广了Oliveira和Soret以及Fraser和Schoen的结果。最后，在C^2类任意可定向支撑面内，导出了具有自由边界的紧曲面的尖锐几何不等式。此外，我们得到了$ mathbb R^3$中闭曲线$L^1$切点能量的一个明显的下界，从而回答了Strzelecki、Szuma\ nska和von der Mosel提出的一个问题。
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, Szuma\'nska and von der Mosel.