Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures

TL;DR

This paper extends Gromov's waist theorem to non-radial Gaussian measures, explores measures lacking the waist property, and employs simplified pancake arguments to estimate neighborhoods of volume-critical submanifolds.

Contribution

It generalizes Gromov's waist theorem to non-radial Gaussian measures and introduces simplified methods for estimating neighborhoods of submanifolds.

Findings

Extension of Gromov's waist theorem to non-radial Gaussian measures

Identification of measures without the $t$-neighborhood waist property

Development of simplified pancake argument techniques

Abstract

We study the Gromov waist in the sense of $t$-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov's original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no $t$-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromov's pancake argument to produce some estimates of $t$-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. For reader's convenience, in one appendix of this paper we provide a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian measures. In the other appendix, we provide a comparison of different variations of Gromov's pancake method.