# Characterizing Level-set Families of Harmonic Functions

## TL;DR

This paper characterizes families of level sets of harmonic functions without critical points using differential geometry and explores how the gradient evolution relates to the mean curvature of these level sets.

## Contribution

It provides a local geometric condition for level-set families of harmonic functions and constructs harmonic functions from geometric data, linking gradient flow to mean curvature.

## Key findings

- Level-set families of harmonic functions are characterized by a differential-geometric condition.
- Harmonic functions can be constructed from given geometric data.
- Gradient evolution along flow is governed by the mean curvature of level sets.

## Abstract

Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of a harmonic function along the gradient flow is determined by the mean curvature of the level sets that the flow intersects.