Evolution systems of measures and semigroup properties on evolving manifolds

2017/08/16 by Lijuan Cheng, Cheng, Li-Juan, Anton Thalmaier +1
Computer Science · Mathematics · #53C44 #58J65 #60J60 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1708.04951

openalex publication_date 2017/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An evolving Riemannian manifold (M,g_t)_{t∈ I} consists of a smooth d-dimensional manifold M, equipped with a geometric flow g_t of complete Riemannian metrics, parametrized by I=(-∞,T). Given an additional C^1,1 family of vector fields (Z_t)_{t∈ I} on M. We study the family of operators L_t=Δ_t +Z_t where Δ_t denotes the Laplacian with respect to the metric g_t. We first give sufficient conditions, in terms of space-time Lyapunov functions, for non-explosion of the diffusion generated by L_t, and for existence of evolution systems of probability measures associated to it. Coupling methods are used to establish uniqueness of the evolution systems under suitable curvature conditions. Adopting such a unique system of probability measures as reference measures, we characterize supercontractivity, hypercontractivity and ultraboundedness of the corresponding time-inhomogeneous semigroup. To this end, gradient estimates and a family of (super-)logarithmic Sobolev inequalities are established.

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