2018
DOI: 10.1007/s11425-017-9296-8
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Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds

Abstract: Let M be a complete Riemannian manifold possibly with a boundary ∂M . For any C 1 -vector field Z, by using gradient/functional inequalities of the (reflecting) diffusion process generated by L := ∆ + Z, pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of ∂M if exists. These extend and strengthen the recent results derived by A. Naber for the uniform norm Ric Z ∞ on manifolds without boundary. A key point of the present study is to apply the asymptoti… Show more

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Cited by 8 publications
(9 citation statements)
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References 13 publications
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“…
In this paper, we will present some characterizations for the upper bound of the Bakry-Emery curvature on a Riemannian manifold by using functional inequalities on path space. Moreover, some characterizations for general lower and upper bounds of Ricci curvature are also given, which extends the recent results derived by Naber [18] and Wang-Wu [26]. A crucial point of the present study is to use the symmetrical idea for the lower and upper bounds of Ricci curvature, and the localization technique.
…”
supporting
confidence: 72%
See 1 more Smart Citation
“…
In this paper, we will present some characterizations for the upper bound of the Bakry-Emery curvature on a Riemannian manifold by using functional inequalities on path space. Moreover, some characterizations for general lower and upper bounds of Ricci curvature are also given, which extends the recent results derived by Naber [18] and Wang-Wu [26]. A crucial point of the present study is to use the symmetrical idea for the lower and upper bounds of Ricci curvature, and the localization technique.
…”
supporting
confidence: 72%
“…In this paper, there are two goals: one is to present some characterizations for general lower and upper bounds of Ric Z by using the symmetrical idea; Based on the above result and the localization technique, the other one is that we will give some equivalent characterizations for the upper bound of Ric Z . The first result extends the recent results derived by Naber [18] and Wang-Wu [26]. The motivation of this work comes from the following observation: The upper and lower bounds of the Bakry-Emery curvature Ric Z at any point x ∈ M are in essence actually determined by the distribution properties of all paths near this point.…”
supporting
confidence: 68%
“…These ideas have also been implemented in the parabolic setting to characterize solutions of the Ricci flow [15]. Another interesting variant of the characterizations of bounded Ricci curvature from [19] has been obtained recently by Fang-Wu [12] and Wang-Wu [24].…”
Section: Background On Lower and Bounded Ricci Curvaturementioning
confidence: 99%
“…The upper and lower bounds for the Ricci curvature on a Riemannian manifold were well characterized in terms of the twisted Malliavin gradient-Dirichlet form E OU for the O-U process on the path space (see A in the introduction) in [46,59,60,17]. If the Malliavin gradient is replaced by the L 2 -gradient DF , then we obtain characterizations for the lower boundedness of the Ricci curvature in terms of a properly decomposition of the L 2 gradient -Dirichlet form.…”
Section: Characterization Of the Lower Bound Of The Ricci Curvaturementioning
confidence: 99%
“…Recently, Naber in [46] characterizes uniform boundedness of the Ricci curvature using the O-U process on path space. Wang-Wu [59] obtained a more general characterization of the Ricci curvature and the second fundamental form on the boundary of the Riemannian manifold using a new method. After that, this result has been extended to general uniform bounds of the Ricci curvature by Wu [60] and Cheng-Thalimaier [17].…”
Section: Introductionmentioning
confidence: 99%