A fractional Willmore-type energy functional -- subcritical observations

TL;DR

This paper studies surfaces with bounded fractional mean curvature, demonstrating how the fractional Willmore-type functional influences local surface properties, leading to regularity results, inequalities, and stability insights under convexity assumptions.

Contribution

It introduces and analyzes a fractional Willmore-type energy functional, establishing control over surface regularity and stability in the subcritical convex setting.

Findings

Control of local parametrization by fractional Willmore functional

Lower Ahlfors-regularity established for surfaces

Weak Michael-Simon type inequality derived

Abstract

We investigate surfaces with bounded L^p-norm of the fractional mean curvature, a quantity we shall refer to as fractional Willmore-type functional. In the subcritical case and under convexity assumptions we show how this Willmore-functional controls local parametrization, and conclude as consequences lower Ahlfors-regularity, a weak Michael-Simon type inequality, and an application to stability.