Abstract

A classical inequality due to Li and Yau states that for a closed immersed surface the Willmore energy can be bounded from below by $4 \pi$ times the maximum multiplicity of the surface. Subsequently, Leon Simon proved a monotonicity identity for closed immersed surfaces, which as a corollary lead to a new proof of the Li-Yau inequality. In this talk we consider compact free boundary surfaces with respect to the unit ball in $\mathbb Rⁿ$, i.e. compact surfaces in $\mathbb R^n$, the boundaries of which meet the boundary of the unit ball orthogonally. Inspired by Simon’s idea we prove a monotonicity identity in this setting. As a corollary we obtain a Li-Yau type inequality, which can be seen as a generalization of an inequality due to Fraser and Schoen to not necessarily minimal surfaces. Using a similar idea Simon Brendle had already extended Fraser-Schoen’s inequality to higher dimensional minimal surfaces in all codimensions.