Manifolds with Ricci Curvature in the Kato Class: Heat Kernel Bounds and Applications

Rose, Christian; Stollmann, Peter

doi:10.1017/9781108615259.006

Cited by 9 publications

(11 citation statements)

References 22 publications

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“…For a recent survey, see Sect. 2 in [26]; for an abstract point of view and a more thorough discussion of the literature on semigroup domination, see the recent [19].…”

Section: Remark 25mentioning

confidence: 99%

“…We conclude the paper with the following list of examples, stating conditions on M where quantitative heat kernel bounds can be obtained. For more details, the reader should consult [3,11,22,24,26].…”

Section: Example 35mentioning

confidence: 99%

“…A starting point for our analysis is the paper [9], where a criterion is formulated under which b 1 (M) = 0. The results of [9] were later generalized and put into a more quantitative form in [26] and [4], respectively (for some related work see also the literature cited in these two papers). The new feature of our main results can easily be explained: Roughly speaking, we employ methods from Functional Analysis and Operator Theory that originated in Mathematical Physics.…”

Section: Introductionmentioning

confidence: 99%

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Bounds on the First Betti Number: An Approach via Schatten Norm Estimates on Semigroup Differences

2022

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“…For a recent survey, see Sect. 2 in [26]; for an abstract point of view and a more thorough discussion of the literature on semigroup domination, see the recent [19].…”

Section: Remark 25mentioning

confidence: 99%

Section: Example 35mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bounds on the First Betti Number: An Approach via Schatten Norm Estimates on Semigroup Differences

2022

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“…Kato (decomposable) functions have been studied in great detail in the literature in the context of (scalar) Schrödinger operators, see [2,9,19,32,56,58] and the references therein. The survey [55] provides a concise overview over the use of Kato decomposability in the context of Riemannian manifolds and its connections to semigroup domination. A detailed study of the Kato class and the induced Schrödinger semigroups corresponding to a large class of Hunt processes can be found in [23].…”

Section: Consequences Of Variable Lower Ricci Boundsmentioning

confidence: 99%

Heat flow regularity, Bismut–Elworthy–Li’s derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature

Braun

Güneysu

2021

Electron. J. Probab.

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We prove that if the Ricci tensor Ric of a geodesically complete Riemannian manifold M , endowed with the Riemannian distance ρ and the Riemannian measure m, is bounded from below by a continuous function k : M → R whose negative part k − satisfies, for every t > 0, the exponential integrability conditionthen the lifetime ζ x of Brownian motion X x on M starting in any x ∈ M is a.s. infinite. This assumption on k holds if k − belongs to the Kato class of M . We also derive a Bismut-Elworthy-Li derivative formula for ∇Ptf for every f ∈ L ∞ (M ) and t > 0 along the heat flow (Pt) t≥0 with generator ∆/2, yielding its L ∞ -Lip-regularization as a corollary.Moreover, given the stochastic completeness of M , but without any assumption on k except continuity, we prove the equivalence of lower boundedness of Ric by k to the existence, given any x, y ∈ M , of a coupling (X x , X y ) of Brownian motions on M starting in (x, y) such that a.s.,holds for every s, t ≥ 0 with s ≤ t, involving the "average" k(u, v) := infγ 1 0 k(γr) dr of k along geodesics from u to v.Our results generalize to weighted Riemannian manifolds, where the Ricci curvature is replaced by the corresponding Bakry-Émery Ricci tensor.

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“…Of particular interest in the outlined business of singular Ricci bounds is the extended Kato class K 1− (M ) of signed measures on M , already for Riemannian manifolds without boundary [15,24,49,50,51,79,85,86] or their Ricci limits [25]. This is just the right class of measure-valued potentials for which the associated Feynman-Kac semigroup has good properties [92].…”

Section: Introductionmentioning

confidence: 99%

Vector calculus for tamed Dirichlet spaces

Braun¹

2021

Preprint

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In the language of L ∞ -modules proposed by Gigli, we introduce a first order calculus on a topological Lusin measure space (M , m) carrying a quasi-regular, strongly local Dirichlet form E. Furthermore, we develop a second order calculus if (M , E, m) is tamed by a signed measure in the extended Kato class in the sense of Erbar, Rigoni, Sturm and Tamanini. This allows us to define e.g. Hessians, covariant and exterior derivatives, Ricci curvature, and second fundamental form.

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Manifolds with Ricci Curvature in the Kato Class: Heat Kernel Bounds and Applications

Cited by 9 publications

References 22 publications

Bounds on the First Betti Number: An Approach via Schatten Norm Estimates on Semigroup Differences

Bounds on the First Betti Number: An Approach via Schatten Norm Estimates on Semigroup Differences

Heat flow regularity, Bismut–Elworthy–Li’s derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature

Vector calculus for tamed Dirichlet spaces

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