Communications in Analysis and Geometry

Volume 24 (2016)

Number 1

A monotonicity formula for free boundary surfaces with respect to the unit ball

Pages: 195 – 221



Alexander Volkmann (Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Germany)


We prove a monotonicity identity for compact surfaces with free boundaries inside the boundary of the 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 [19, Proposition 3], and Fraser and Schoen [11, Theorem 5.4].

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 [22].

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2016-June-8: revised author contact information at end of article