Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

TL;DR

This paper explores isoperimetric inequalities and the geometry of level curves of harmonic functions on various surfaces, including singular and non-constant curvature surfaces, providing new inequalities, principles, and growth estimates.

Contribution

It generalizes existing results to singular Alexandrov surfaces and surfaces with variable curvature, introducing new inequalities and principles for harmonic functions' level curves.

Findings

Logarithmic convexity of level curve length on diverse surfaces

Laplace-type equations for geodesic curvature functions

Maximum and minimum principles for curvature-related functions

Abstract

We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature $K$. Such geodesic curvature functions turn out to satisfy certain Laplace-type equations and inequalities, from which we infer various maximum and minimum principles. The results are complemented by a number of growth estimates for the derivatives $L'$ and $L''$ of the length of the level curve function $L$, as well as by examples illustrating the presentation. Our work generalizes some results due to Alessandrini, Longinetti, Talenti, Ma--Zhang and Wang--Wang.