The aim of this paper is to provide new stability results for sequences of metric measure spaces $(X_i,d_i,m_i)$ convergent in the measured Gromov-Hausdorff sense. By adopting the so-called extrinsic approach of embedding all metric spaces into a common one $(X,d)$, we extend the results of Gigli-Mondino-Savar\'e by providing Mosco convergence of Cheeger's energies and compactness theorems in the whole range of Sobolev spaces $H^{1,p}$, including the space $BV$, and even with a variable exponent $p_i\in [1,\infty]$. In addition, building on the results of Ambrosio-Stra-Trevisan, we provide local convergence results for gradient derivations. We use these tools to improve the spectral stability results, previously known for $p>1$ and for Ricci limit spaces, getting continuity of Cheeger's constant. In the dimensional case $N<\infty$, we improve some rigidity and almost rigidity results by Ketterer and Cavaletti-Mondino. On the basis of the second-order calculus by Gigli, in the class of $RCD(K,\infty)$ spaces we provide stability results for Hessians and $W^{2,2}$ functions and we treat the stability of the Bakry-\'Emery condition $BE(K,N)$ and of ${\bf Ric}\geq KI$, with $K$ and $N$ not necessarily constant.

/pdf/new-stability-results-for-sequences-of-metric-measure-spaces-42w4vgpz6l.pdf

New stability results for sequences of metric measure spaces with uniform Ricci bounds from below

The aim of this paper is to prove the sharp estimate on the first nontrivial eigenvalue of the p -Laplacian on a compact Riemannian manifold with nonnegative Ricci curvature and to characterize the equality case. The estimate applies to manifolds with empty or convex boundary, and in this latter case we also assume Neumann boundary conditions for the p -Laplacian. The main tool used for the proof is a gradient comparison based on a generalized p -Bochner formula.

Sharp estimate on the first eigenvalue of the p-Laplacian

For metric measure spaces satisfying the reduced curvature–dimension condition CD∗(K,N) we prove a series of sharp functional inequalities under the additional “essentially nonbranching” assumption. Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfying lower curvature bounds and, more generally, RCD∗(K,N) spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower Ricci curvature bounds. In particular we prove the Brunn–Minkowski inequality, the p–spectral gap (or equivalently the p–Poincare inequality) for any p ∈ [1,∞), the log-Sobolev inequality, the Talagrand inequality and finally the Sobolev inequality. All the results are proved in a sharp form involving an upper bound on the diameter of the space; all our inequalities for essentially nonbranching CD∗(K,N) spaces take the same form as the corresponding sharp ones known for a weighted Riemannian manifold satisfying the curvature–dimension condition CD(K,N) in the sense of Bakry and Emery. In this sense our inequalities are sharp. We also discuss the rigidity and almost rigidity statements associated to the p–spectral gap. In particular, we have also shown that the sharp Brunn–Minkowski inequality in the global form can be deduced from the local curvature–dimension condition, providing a step towards (the long-standing problem of) globalization for the curvature–dimension condition CD(K,N). To conclude, some of the results can be seen as answers to open problems proposed in Villani’s book Optimal transport.

/pdf/sharp-geometric-and-functional-inequalities-in-metric-18ak64w6xz.pdf

Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds

We obtain two elliptic gradient estimates for positive solutions to the $$f$$
 -heat equation on a complete smooth metric measure space with only Bakry–Emery Ricci tensor bounded below. One is a local sharp Souplet–Zhang’s type and the other is a global Hamilton’s type. As applications, we prove parabolic Liouville theorems for ancient solutions satisfying some growth restriction near infinity. In particular the Liouville results are suitable for the gradient shrinking or steady Ricci solitons. The estimates of derivation of the $$f$$
 -heat kernel are also obtained.

Elliptic gradient estimates for a weighted heat equation and applications

Sharp estimate on the first eigenvalue of the p-laplacian

Li–Yau type estimates for a nonlinear parabolic equation on complete manifolds

Upper bounds on the first eigenvalue for a diffusion operator via Bakry–Émery Ricci curvature☆

In this paper, we study elliptic gradient estimates for a nonlinear f -heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton’s and Souplet–Zhang’s gradient estimates for positive solutions to the nonlinear f -heat equation only assuming the ∞ -Bakry–Emery Ricci tensor is bounded below. As applications, we prove parabolic Liouville properties for some kind of ancient solutions to the nonlinear f -heat equation. Some special cases are also discussed.

Elliptic gradient estimates for a nonlinear heat equation and applications

We derive a local Gaussian upper bound for the $f$-heat kernel on complete smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry–Emery Ricci curvature. As applications, we obtain a sharp $L_f^1$-Liouville theorem for $f$-subharmonic functions and an $L_f^1$-uniqueness property for nonnegative solutions of the $f$-heat equation, assuming $f$ is of at most quadratic growth. In particular, any $L_f^1$-integrable $f$-subharmonic function on gradient shrinking and steady Ricci solitons must be constant. We also provide explicit $f$-heat kernel for Gaussian solitons.

Jia-Yong Wu

Papers

Elliptic gradient estimates for a weighted heat equation and applications

Li–Yau type estimates for a nonlinear parabolic equation on complete manifolds

Upper bounds on the first eigenvalue for a diffusion operator via Bakry–Émery Ricci curvature☆

Elliptic gradient estimates for a nonlinear heat equation and applications

Heat kernel on smooth metric measure spaces with nonnegative curvature