Hypercontractivity of the heat flow on \sf RCD(0,N) spaces: sharpness and rigidities

2025/03/19 by Shouhei Honda, Alexandru Kristály, Honda, Shouhei +3
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2503.15236

openalex publication_date 2025/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The main goal of the present paper is to provide sharp hypercontractivity bounds of the heat flow (\sf H_t)_{t≥ 0} on \sf RCD(0,N) metric measure spaces. The best constant in this estimate involves the asymptotic volume ratio, and its optimality is obtained by means of the sharp L²-logarithmic Sobolev inequality on \sf RCD(0,N) spaces and a blow-down rescaling argument. Equality holds in this sharp estimate for a prescribed time t₀>0 and a non-zero extremizer f if and only if the \sf RCD(0,N) space has an N-Euclidean cone structure and f is a Gaussian whose dilation factor is reciprocal to t₀, up to a multiplicative constant. Applications include an extension of Li's rigidity result, almost rigidities, as well as topological rigidities of non-collapsed \sf RCD(0, N) spaces. Our results are new even on complete Riemannian manifolds with non-negative Ricci curvature.

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