Conformal Transformation on Metric Measure Spaces

Han, Bang-Xian

doi:10.1007/s11118-018-9705-7

Cited by 7 publications

(7 citation statements)

References 29 publications

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“…We also would like to mention the follow-up work by Bang-Xian Han and Anna Mkrtchyan [10] which extend several results of this paper to the setting of metric measure spaces without 'smoothness' assumptions.…”

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confidence: 63%

Ricci tensor for diffusion operators and curvature-dimension inequalities under conformal transformations and time changes

Sturm

2018

Journal of Functional Analysis

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Within the Γ2-calculus of Bakry and Ledoux, we define the Ricci tensor induced by a diffusion operator, we deduce precise formulas for its behavior under drift transformation, time change and conformal transformation, and we derive new transformation results for the curvature-dimension conditions of Bakry-Émery as well as for those of Lott-Sturm-Villani. Our results are based on new identities and sharp estimates for the N -Ricci tensor and for the Hessian. In particular, we obtain Bochner's formula in the general setting.

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confidence: 63%

Ricci tensor for diffusion operators and curvature-dimension inequalities under conformal transformations and time changes

Sturm

2018

Journal of Functional Analysis

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“…Combining the techniques and results in [14,20], the first author [15,16] proved the transformation formula for the Lagrangian curvature-dimension condition RCD(K, N) under conformal transformation when the reference function w is bounded and smooth…”

Section: Remarkmentioning

confidence: 99%

“…Combining the techniques and results in [ 14 , 20 ], the first author [ 15 , 16 ] proved the transformation formula for the Lagrangian curvature-dimension condition under conformal transformation when the reference function w is bounded and smooth enough. Together with the well-known transformation formula for under drift transformations, this result also provides a transformation formula for under time change.…”

Section: Introductionmentioning

confidence: 99%

“…C. One of the important applications of time-change is the “convexification” of non-convex subsets of an RCD( K , N )-space as introduced by the second author and Lierl [ 18 ]. For sublevel sets of regular semi-convex functions V , they proved convexity after suitable conformal transformations, while control of the curvature bound under these transformations follows from the work [ 15 ] of the first author. Unfortunately, these previous results do not apply to the most natural potential, the signed distance function due to lack of regularity.…”

Section: Introductionmentioning

confidence: 99%

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Curvature-dimension conditions under time change

Han

Sturm

2021

Annali di Matematica

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We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry–Émery will transform under time change. Similarly, for metric measure spaces we study how the curvature-dimension condition in the sense of Lott–Sturm–Villani will transform under time change.

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“…One of the outcomes of the tensor calculus on RCD spaces built in [4] is the existence of a measure-valued Ricci tensor. An example of application of this object to the study of the geometry of such spaces is given by the paper [7] where, inspired by some more formal computations due to Sturm [15], it is shown that transformations of the metric-measure structure (like, e.g., conformal ones) alter lower Ricci curvature bounds as in the smooth context.…”

Section: Introductionmentioning

confidence: 99%

Riemann curvature tensor on RCD spaces and possible applications

Gigli

2019

Comptes Rendus Mathematique

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We show that on every RCD spaces it is possible to introduce, by a distributional-like approach, a Riemann curvature tensor.Since after the works of Petrunin and Zhang-Zhu we know that finite dimensional Alexandrov spaces are RCD spaces, our construction applies in particular to the Alexandrov setting. We conjecture that an RCD space is Alexandrov if and only if the sectional curvature -defined in terms of such abstract Riemann tensor -is bounded from below. RésuméNous montrons que sur chaque espace RCD il est possible d'introduire, par une approche distributionnelle, un tenseur de courbure de Riemann.Puisque après les travaux de Petrunin et de Zhang-Zhu nous savons que les espaces d'Alexandrov de dimension finie sont des espaces RCD, notre construction s'applique en particulier au cadre d'Alexandrov. Nous conjecturons qu'un espace RCD est Alexandrov si et seulement si la courbure sectionnelle -définie en termes de ce tenseur de Riemann abstrait -est bornée par dessous.

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Conformal Transformation on Metric Measure Spaces

Cited by 7 publications

References 29 publications

Ricci tensor for diffusion operators and curvature-dimension inequalities under conformal transformations and time changes

Ricci tensor for diffusion operators and curvature-dimension inequalities under conformal transformations and time changes

Curvature-dimension conditions under time change

Riemann curvature tensor on RCD spaces and possible applications

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