Unique continuation for locally uniformly distributed measures

TL;DR

This work studies the geometry of supports of locally $k$-uniform measures in $\mathbb{R}^{n+1}$ and their relation to unique continuation and quadratic varieties. It proves a strong unique continuation principle for locally $k$-uniform measures and a weak version for locally uniformly distributed measures, extending prior results and providing evidence toward Kowalski–Preiss's conjecture that connected components of the support of a locally $n$-uniform measure lie in the zero set of a quadratic polynomial. The key technical innovation is a new identity linking the mean curvature vector, a conical moment $b_{z,r}$, and the height-energy, which yields flatness from vanishing curvature and propagates along connected components. These results advance endpoint density theory and support the conjectured analytic structure of supports for globally uniform measures.

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.

Theorems & Definitions (9)

• Conjecture 1: See Kowalski-Preiss-1987
• Theorem 1.1: Strong Unique Continuation
• Theorem 1.2: Weak Unique Continuation
• proof : Proof of \ref{['c:WUCP']}
• Lemma 3.1: Key equality
• proof : Proof of \ref{['t:SUCP']}
• Lemma 3.2
• proof
• proof : Proof of \ref{['lemma:key equality']}