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

The main goal of the present paper is to provide sharp hypercontractivity bounds of the heat flow ( Hₜ)t≥ 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.