Sharp Gaussian Isoperimetry along a Ricci Flow

TL;DR

This paper establishes a sharp Gaussian isoperimetric inequality along Ricci flows, leading to various concentration, rearrangement, and stability results with broad geometric and probabilistic implications.

Contribution

It introduces a novel monotonicity formula that proves the sharp Gaussian isoperimetric inequality in the Ricci flow context, unifying multiple inequalities and estimates.

Findings

Proves the sharp Gaussian isoperimetric inequality along Ricci flows.

Derives the Gaussian enlargement theorem and concentration estimates.

Recovers and extends known inequalities like Hein--Naber's log-Sobolev and hypercontractivity.

Abstract

We prove the sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along a Ricci flow via a monotonicity formula. As consequences, we obtain the exact Gaussian enlargement theorem and a Gaussian-quantile two-set concentration estimate. In particular, this recovers the exponential concentration estimate of Hein--Naber from a sharper isoperimetric profile. We also derive Gaussian rearrangement inequalities, recover the sharp Hein--Naber log-Sobolev inequality, and identify the universal Gaussian-model constants in Bamler's \(L^p\)-Poincar\'e inequalities. Further applications include Gaussian-profile localization near Bamler's \(H_n\)-centers, convex-order and moment estimates for logarithmic derivatives of the conjugate heat kernel, reverse hypercontractivity, entropy-regular profile stability, and a path-space Bobkov inequality.