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October 8, 2025Open Access

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

Key Points

New hypercontractivity bounds are established for heat flow, revealing sharp estimates based on asymptotic volume ratio.
The optimal bound is reached if the RCD(0,N) space resembles an N-Euclidean cone with a specific Gaussian function.
The findings extend results on topological rigidities in RCD(0,N) spaces, showing significant applicability beyond Riemannian manifolds.
These results clarify properties of non-collapsed RCD(0,N) spaces, reinforcing the relationship between geometry and analysis.

Abstract

The main goal of the present paper is to provide sharp hypercontractivity bounds of the heat flow (Hₜ) ₓ ₀ on 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 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 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 RCD (0, N) spaces. Our results are new even on complete Riemannian manifolds with non-negative Ricci curvature.

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Cite This Study

Honda et al. (Wed,) studied this question.

synapsesocial.com/papers/68e5c1c76950a706b22b5f08 https://doi.org/https://doi.org/10.48550/arxiv.2503.15236

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