Characterizations of the Upper Bound of Bakry–Emery Curvature

Abstract

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.

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

“…In this section, we present a last equivalent characterization of the condition Ric ≥ k on M for the class of Kato decomposable k in terms of gradient estimates for (P t ) t≥0 . A similar result can be found in [69,Corollary 2.2]. See also [68,Theorem 2.3.1] for more geometric growth conditions on k − , and [14, Theorem 1.1] for the nonsmooth case under boundedness of k − , the condition Ric ≥ k on M interpreted in a synthetic sense [60].…”

Stochastic heat equations with values in a Riemannian manifold

“…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.…”

“…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]. In addition, Wu [60] and Wang [56] gave some characterization for the upper bound of the Ricci curvature by analysis on path space and the Weitzenböck-Bochner integration formula respectively.…”

“…After that, this result has been extended to general uniform bounds of the Ricci curvature by Wu [60] and Cheng-Thalimaier [17]. In addition, Wu [60] and Wang [56] gave some characterization for the upper bound of the Ricci curvature by analysis on path space and the Weitzenböck-Bochner integration formula respectively. Similar to the above case, in Subsection 3.2 we give some (equivalent) characterizations of the lower boundedness of the Ricci curvature by using the L 2 -gradient and a properly weighted decomposition of the L 2 -Dirichlet form on path space.…”

Rigidity of cones with bounded Ricci curvature

“…Gigli [12] and Han [14] provide a definition of the full Ricci tensor on metric measure spaces, building upon a similar contruction in the context of Γ-calculus by Sturm [21]. Naber [19] characterized two-sided bounds on the Ricci curvature in terms of functional inequalities in the path space, see also recent work of Cheng-Thalmaier [8] and of Wu [24]. A drawback of the previous approaches to detailed controls on Ricci is that they do not see curvature concentrated in singular sets such as the tip of a cone.…”