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

Abstract

We prove a monotonicity identity for compact surfaces with free boundaries inside the boundary of unit ball in 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 R 3 thereby answering a question raised by Strzelecki, Szumańska and von der Mosel [23].

Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball

“…Note that in [45] Volkmann obtained an inequality similar to (1.2) (with equality on caps and disks) in case n = 2 for a much broader class of hypersurfaces using methods from geometric measure theory. In arbitrary higher dimensions, a Willmore type estimate for the convex case was deduced in [28], however with equality only on flat disks.…”

Local solutions to a free boundary problem for the Willmore functional

“…For Ω convex the length bound could in fact be dropped using the Gauß Bonnet theorem. Global bounds for the Willmore energy of surfaces with free boundary are proved in recent work by Volkmann [21].…”

Capacity inequalities and rigidity of cornered/conical manifolds

“…Although the result originally stated in Lemma 5.1 of [22] does not includes any information about the regularity of the boundary, it can be extended up to the boundary applying Proposition 3.2 of [35] and the Grüter-Jost's free boundary regularity [11]. Hence for those above values of t, Σ t is orthogonal to S in the classical sense and any neighborhood of points x ∈ S ∩∂ * M Ω t 2 Definition 2.…”