# Convexity Properties of Harmonic Functions on Parameterized Families of Hypersurfaces

## TL;DR

This paper investigates the convexity properties of the $L^{2}$-norms of harmonic functions over various evolving hypersurfaces, extending known results from spheres to more general geometries and providing new inequalities.

## Contribution

It introduces a differential inequality for $L^{2}$-norms of harmonic functions on a broad class of hypersurfaces, generalizing convexity results beyond spheres and ellipses.

## Key findings

- Established a differential inequality for harmonic functions on evolving hypersurfaces.
- Extended convexity results to positively curved Riemannian manifolds with variable curvature.
- Provided examples with ellipses and tori illustrating the theoretical results.

## Abstract

It is known that the $L^{2}$-norms of a harmonic function over spheres satisfies some convexity inequality strongly linked to the Almgren's frequency function. We examine the $L^{2}$-norms of harmonic functions over a wide class of evolving hypersurfaces. More precisely, we consider compact level sets of smooth regular functions and obtain a differential inequality for the $L^{2}$-norms of harmonic functions over these hypersurfaces. To illustrate our result, we consider ellipses with constant eccentricity and growing tori in $\mathbf{R}^3.$ Moreover, we give a new proof of the convexity result for harmonic functions on a Riemannian manifold when integrating over spheres. The inequality we obtain for the case of positively curved Riemannian manifolds with non-constant curvature is slightly better than the one previously known.