Unique continuation for locally uniformly distributed measures

TL;DR

This paper proves a unique continuation property for the support of locally uniformly distributed measures in Euclidean space, supporting conjectures about their geometric structure and extending previous work in geometric measure theory.

Contribution

It establishes a new unique continuation property for locally uniform measures, advancing understanding of their support's geometric and algebraic structure.

Findings

Supports the conjecture that connected components are contained in zero sets of quadratic polynomials

Provides a new unique continuation property for locally uniform measures

Extends previous results by Kirchheim, Preiss, David, Kenig, and Toro

Abstract

In this note we show that the support of a locally $k$-uniform measure in $\mathbb R^{n+1}$ satisfies a kind of unique continuation property. As a consequence, we show that locally uniformly distributed measures satisfy a weaker unique continuation property. This continues work of Kirchheim and Preiss (Math. Scand. 2002) and David, Kenig and Toro (Comm. Pure Appl. Math. 2001) and lends additional evidence to the conjecture proposed by Kowalski and Preiss (J. Reine Angew. Math. 1987) that each connected component of the support of a locally $n$-uniform measure in $\mathbb R^{n+1}$ is contained in the zero set of a quadratic polynomial.