Rigidity phenomena on lower N-weighted Ricci curvature bounds with ε-range for nonsymmetric Laplacian

Abstract

The notion of tamed Dirichlet space was proposed by Erbar, Rigoni, Sturm and Tamanini [27] as a Dirichlet space having a weak form of Bakry-Émery curvature lower bounds in distribution sense. After their work, Braun [14] established a vector calculus for it, in particular, the space of L 2 -normed L ∞ -module describing vector fields, 1-forms, Hessian in L 2 -sense. In this framework, we establish the Hess-Schrader-Uhlenbrock inequality for 1-forms as an element of L 2 -cotangent module (an L 2 -normed L ∞ -module), which extends the Hess-Schrader-Uhlenbrock inequality by Braun [14] under an additional condition.

Quantitative estimate of diameter for weighted manifolds under integral curvature bounds and $\varepsilon$-range

“…They presented several comparison theorems with each curvature bounds. Furthermore, Kuwae-Sakurai [6] also provided several comparison theorems with the ε(n, N)-range.…”