Yau's Gradient Estimates on Alexandrov Spaces

Zhang, Hui-Chun; Zhu, Xi-Ping

doi:10.4310/jdg/1349292672

Cited by 44 publications

(75 citation statements)

References 41 publications

(162 reference statements)

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“…See Remarks 4.23 and 4.30. It is worth pointing out that recently Zhang-Zhu proved a similar result on an Alexandrov space in [62].…”

Section: Introductionmentioning

confidence: 79%

Ricci curvature and L^p -convergence

Honda

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We give the definition of L p -convergence of tensor fields with respect to the Gromov-Hausdorff topology and several fundamental properties of the convergence. We apply this to establish a Bochner-type inequality which keeps the term of Hessian on the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a lower Ricci curvature bound and to give a geometric explicit formula for the Dirichlet Laplacian on a limit space defined by Cheeger-Colding. We also prove the continuity of the first eigenvalues of the p-Laplacian with respect to the Gromov-Hausdorff topology.

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“…See Remarks 4.23 and 4.30. It is worth pointing out that recently Zhang-Zhu proved a similar result on an Alexandrov space in [62].…”

Section: Introductionmentioning

confidence: 79%

Ricci curvature and L^p -convergence

Honda

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…These two gradient estimates are fundamental tools in geometric analysis and related fields, and there have been many efforts afterwards to generalise them to different settings, see for instance [34,38,39,40,44,52,67,73,82,90,91,95,113,114,115]. Let us review some of these generalisations.…”

Section: Background and Main Resultsmentioning

confidence: 99%

“…We shall see in Lemma 2.3 below that, under (D) and (P 2 ), (RH ∞ ) is equivalent to Yau's gradient estimate (Y ∞ ) with K = 0. See [30,72,112,114] for more about (Y ∞ ) and (RH ∞ ). Actually, a more natural formulation for the reverse L p -Hölder inequality for gradients of harmonic functions is…”

Section: Background and Main Resultsmentioning

confidence: 99%

Gradient estimates for heat kernels and harmonic functions

Coulhon

Jiang

Koskela

et al. 2020

Journal of Functional Analysis

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Let (X, d, µ) be a doubling metric measure space endowed with a Dirichlet form E deriving from a "carré du champ". Assume that (X, d, µ, E ) supports a scale-invariant L 2 -Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for p ∈ (2, ∞]: (i) (G p ): L p -estimate for the gradient of the associated heat semigroup; (ii) (RH p ): L p -reverse Hölder inequality for the gradients of harmonic functions; (iii) (R p ): L p -boundedness of the Riesz transform (p < ∞); (iv) (GBE): a generalised Bakry-Émery condition. We show that, for p ∈ (2, ∞), (i), (ii) (iii) are equivalent, while for p = ∞, (i), (ii), (iv) are equivalent. Moreover, some of these equivalences still hold under weaker conditions than the L 2 -Poincaré inequality. Our result gives a characterisation of Li-Yau's gradient estimate of heat kernels for p = ∞, while for p ∈ (2, ∞) it is a substantial improvement as well as a generalisation of earlier results by Auscher-Coulhon-Duong-Hofmann [7] and Auscher-Coulhon [6]. Applications to isoperimetric inequalities and Sobolev inequalities are given. Our results apply to Riemannian and sub-Riemannian manifolds as well as to non-smooth spaces, and to degenerate elliptic/parabolic equations in these settings.

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“…Even more, they prove the full equivalence between the reduced Riemannian curvature-dimension condition (RCD * (κ, N )) and (1) for any parameters κ ∈ R and N ∈ [1, ∞). There is also a related result in the context of Alexandrov spaces by Zhang and Zhu [51]. Bochner's inequality captures the Eulerian picture of curvature-dimension bounds.…”

Section: Introductionmentioning

confidence: 94%

“…where we used (51) in the first and Jensen's inequality in the second inequality. Finally, we integrate h from 0 to t and the rest of the proof is exactly the same as in Proposition 4.9 in [21].…”

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confidence: 99%

Cones over metric measure spaces and the maximal diameter theorem

Ketterer¹

2015

Journal de Mathématiques Pures et Appliquées

113

112

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The main result of this article states that the (K, N )-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD * (KN, N +1) if and only if the underlying space satisfies RCD * (N −1, N ). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner's inequality that was established for general metric measure spaces by Erbar, Kuwada and Sturm in [21] (independently, the same result has been announced by Ambrosio, Mondino and Savaré). As a corollary of our result and the Gigli-Cheeger-Gromoll splitting theorem [25] we obtain a maximal diameter theorem in the context of metric measure spaces that satisfy the condition RCD * .

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Yau's Gradient Estimates on Alexandrov Spaces

Abstract: In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yau's gradient estimate for harmonic functions is also obtained on Alexandrov spaces.

Cited by 44 publications

References 41 publications

Ricci curvature and L^p -convergence

Ricci curvature and L^p -convergence

Gradient estimates for heat kernels and harmonic functions

Cones over metric measure spaces and the maximal diameter theorem

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Yau's Gradient Estimates on Alexandrov Spaces

Abstract: In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yau's gradient estimate for harmonic functions is also obtained on Alexandrov spaces.

Cited by 44 publications

References 41 publications

Ricci curvature and Lp -convergence

Ricci curvature and Lp -convergence

Gradient estimates for heat kernels and harmonic functions

Cones over metric measure spaces and the maximal diameter theorem

Contact Info

Product

Resources

About

Ricci curvature and L^p -convergence

Ricci curvature and L^p -convergence