#### Evolution systems of measures and semigroups properties on evolving manifolds by Li-Juan Cheng and Anton Thalmaier

Abstract
An evolving Riemannian manifold $\left(M,g$t)t∈I consists of a smooth $d$-dimensional manifold $M$, equipped with a geometric flow $g$t of complete Riemannian metrics, parametrized by $I=\left(-\infty ,T\right)$. Given an additional $C1,1$ family of vector fields $\left(Z$t)t∈I on $M$, we study the family of operators $L$tt+Zt where $\Delta$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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