Heat Kernel Estimates and the Essential Spectrum on Weighted Manifolds

Charalambous, Nelia; Lu, Zhiqin

doi:10.1007/s12220-013-9438-1

Cited by 13 publications

(10 citation statements)

References 31 publications

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“…The main difficulty is the lack of effective upper bound for the f -heat kernel. In [8], by analyzing the heat kernel for a family of warped product manifolds, Charalambous and Lu also gave f -heat kernel estimates when Ric m f (m < ∞) is bounded below. In [31], the first author proved f -heat kernel estimates assuming Ric f bounded below by a negative constant and f bounded.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Heat kernel on smooth metric measure spaces with nonnegative curvature

2014

Math. Ann.

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Abstract. We derive a local Gaussian upper bound for the f -heat kernel on complete smooth metric measure space (M, g, e −f dv) with nonnegative Bakry-Émery Ricci curvature. As applications, we obtain a sharp L 1 f -Liouville theorem for f -subharmonic functions and an L 1 f -uniqueness property for nonnegative solutions of the f -heat equation, assuming f is of at most quadratic growth. In particular, any L 1 f -integrable f -subharmonic function on gradient shrinking and steady Ricci solitons must be constant. We also provide explicit f -heat kernel for Gaussian solitons.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Heat kernel on smooth metric measure spaces with nonnegative curvature

2014

Math. Ann.

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“…(1.1) Therefore when m < ∞, the Bochner formula for Ric m f can be considered as the Bochner formula for the Ricci tensor of an (n + m)-dimensional manifold, and for smooth metric measure spaces with Ric m f bounded below, one has nice f -mean curvature comparison and f -volume comparison theorems which are similar to classical ones for Riemannian manifolds, see [3,45], in particular, the comparison theorems do not depend on f ; X.-D. Li [27] derived an analogue of Li-Yau gradient estimate, using which he proved f -heat kernel estimates and several Liouville theorems; and in [9], by analyzing a family of warped product manifolds, Charalambous and Z. Lu obtained f -heat kernel estimates and essential spectrum.…”

Section: Introductionmentioning

confidence: 83%

Heat kernel on smooth metric measure spaces and applications

2015

Math. Ann.

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Abstract. We derive a Harnack inequality for positive solutions of the fheat equation and Gaussian upper and lower bounds for the f -heat kernel on complete smooth metric measure spaces (M, g, e −f dv) with Bakry-Émery Ricci curvature bounded below. The lower bound is sharp. The main argument is the De Giorgi-Nash-Moser theory. As applications, we prove an L 1 f -Liouville theorem for f -subharmonic functions and an L 1 f -uniqueness theorem for fheat equations when f has at most linear growth. We also obtain eigenvalues estimates and f -Green's function estimates for the f -Laplace operator.

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“…See [Li and Li 2014a]. and useful discussions, and Prof. Zhiqin Lu for the citation of the first version of this paper in [Charalambous and Lu 2015]. We are very grateful to the anonymous referees for their careful reading and helpful comments for the revision of the paper.…”

Section: Note Added In Proofmentioning

confidence: 97%

“…In the case where (M, g) is a complete Riemannian manifold with the bounded geometry condition, similarly to [Lott 2003;Charalambous and Lu 2015], by introducing a sequence of warped product metrics {g ε } on M = M × N defined bỹ g ε = g ⊕ ε 2 e −2φ/(m−n) g N , and using the fact that the heat kernel of the Laplace-Beltrami ( M,g ε ) on ( M,g ε ) (with renormalized volume measure) converges in the C 2,α ∩ W 2, p -topology to the heat kernel of the Witten Laplacian L = M − ∇φ • ∇ on (M, g, µ), we can use the same approach as in the compact case to give a new proof of the W-entropy formula for the heat kernel of the Witten Laplacian on complete Riemannian manifolds satisfying the bounded geometry condition in Theorem 1.1. We will study this problem in detail in the future.…”

Section: A New Proof Of Theorem 11mentioning

confidence: 99%

“…The W-entropy formula (3) only implies the monotonicity of the W-entropy for the Witten Laplacian on complete Riemannian manifolds with nonnegative mdimensional Bakry-Émery Ricci curvature, and the W-entropy formula (4) only By the same argument as in [Li and Xu 2011], or using the warped product approach as in [Charalambous and Lu 2015] and the Li-Yau-type differential Harnack inequality obtained by J. Li and X. Xu [2011], we can prove that, if Ric m,n (L) ≥ −K , then…”

Section: The W-entropy Formula For the Witten Laplacian With Negativementioning

confidence: 99%

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2015

Pacific J. Math.

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We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, nonelementary group G acting on a G-space X , we prove that if G contains a strongly contracting element and if G is not too badly distorted in X , then the action of G on X is a growth tight action. It follows that if X is a cocompact, relatively hyperbolic G-space, then the action of G on X is a growth tight action. This generalizes all previously known results for growth tightness of cocompact actions: every already known example of a group that admits a growth tight action and has some infinite, infinite index normal subgroups is relatively hyperbolic; conversely, relatively hyperbolic groups admit growth tight actions. This also allows us to prove that many CAT(0) groups, including flip-graph manifold groups and many right angled Artin groups, and snowflake groups admit cocompact, growth tight actions. These provide first examples of non relatively hyperbolic groups admitting interesting growth tight actions. Our main result applies as well to cusp uniform actions on hyperbolic spaces and to the action of the mapping class group on Teichmüller space with the Teichmüller metric. Towards the proof of our main result, we give equivalent characterizations of strongly contracting elements and produce new examples of group actions with strongly contracting elements. 0. Introduction 2 Part I. Growth tight actions 8 1. Preliminaries 8 2. Contraction and constriction 12 3. Abundance of strongly contracting elements 20 4. A minimal section 25 5. Embedding a free product set 26 6. The growth gap 27 7. The growth of conjugacy classes 29 MSC2010: 20E06,

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Heat Kernel Estimates and the Essential Spectrum on Weighted Manifolds

Cited by 13 publications

References 31 publications

Heat kernel on smooth metric measure spaces with nonnegative curvature

Heat kernel on smooth metric measure spaces with nonnegative curvature

Heat kernel on smooth metric measure spaces and applications

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