Tamed spaces -- Dirichlet spaces with distribution-valued Ricci bounds

Abstract

We develop the theory of tamed spaces which are Dirichlet spaces with distributionvalued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view. To this end we analyze in detail singular perturbations of Dirichlet form by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality using the perturbed energy form and generalizing the well-known Bakry-Émery curvature-dimension condition. Among other things we show the equivalence of distributional Ricci bounds to gradient estimates for the heat semigroup in terms of the Feynman-Kac semigroup induced by the taming distribution as well as consequences in terms of functional inequalities. We give many examples of tamed spaces including in particular Riemannian manifolds with either interior singularities or singular boundary behavior. MATTHIAS ERBAR, CHIARA RIGONI, KARL-THEODOR STURM, AND LUCA TAMANINI 4.2. A manifold which is tamed but not 2-tamed 36 4.3. Manifolds with boundary and potentially singular curvature 36 4.4. A tamed domain with boundary that is not semiconvex 40 4.5. A Tamed Manifold with Highly Irregular Boundary 41 5. Functional Inequalities for Tamed Spaces 45 6. Self-Improvement of the Taming Condition 47 6.1. Measure-Valued Taming Operator and Bochner Inequality 47 6.2. Self-Improvement of the L 2 -Taming Condition 51 7. Sub-tamed Spaces 55 7.1. Reflected Dirichlet Spaces and Sub-taming 56 7.2. Doubling of Dirichlet Spaces and Sub-taming 58 7.3. Doubling of Riemannian Surfaces 64 References 66

Heat kernel bounds and Ricci curvature for Lipschitz manifolds

“…Taming for LR manifolds. The notion of tamed spaces, summarized in Section 4.1, has recently been introduced in [18]. It offers a synthetic way of speaking about the Ricci curvature of a Dirichlet space being bounded from below by a distribution κ.…”

“…Since elements in K(M ) provide special cases of the relevant distributions considered in [18] and since Ricci bounds have powerful probabilistic, analytic and geometric consequences, we believe our following condition for taming of LR manifolds, cf. Theorem 4.9 below, to be of high interest.…”

“…Chapter 3 introduces the concept of control pairs for the given heat kernel and shows the general existence of such a pair. Lastly, in Chapter 4 we briefly recapitulate basic notions about tamed spaces from [18] and provide sufficient conditions for taming of LR manifolds.…”

“…In view of the above mentioned diversity of examples as well as recent breakthroughs in metric geometry for nonuniform curvature bounds [5,8,18,26,56], we predict the class of LR manifolds -regarded as prototypes of Riemannian spaces with singularities -to have great potential for near future research, which this article aims to initiate.…”

“…The prominent role of the Kato class in mathematical physics has been well established in the past decades in a series of works [2,32,49]. Recently, its interest in the context of singular Ricci bounds has grown as well [5,6,7,9,18,24,26,43] (also with applications to Ricci flow and general relativity). The study of the class of LR manifolds in this latter setting constitutes the last part of our work.…”