Weighted Cheeger constant and first eigenvalue lower bound estimates on smooth metric measure spaces

We establish new eigenvalue inequalities in terms of the weighted Cheeger constant for drifting p-Laplacian on smooth metric measure spaces with or without boundary. The weighted Cheeger constant is bounded from below by a geometric constant involving the divergence of suitable vector fields. On the other hand, we establish a weighted form of Escobar–Lichnerowicz–Reilly lower bound estimates on the first nonzero eigenvalue of the drifting bi-Laplacian on weighted manifolds. As an application, we prove buckling eigenvalue lower bound estimates, first, on the weighted geodesic balls and then on submanifolds having bounded weighted mean curvature.

On Scalar and Ricci Curvatures

The purpose of this report is to acknowledge the influence of M. Gromov's vision of geometry on our own works. It is two-fold: in the first part we aim at describing some results, in dimension 3, around the question: which open 3-manifolds carry a complete Riemannian metric of positive or non negative scalar curvature? In the second part we look for weak forms of the notion of ''lower bounds of the Ricci curvature'' on non necessarily smooth metric measure spaces. We describe recent results some of which are already posted in [arXiv:1712.08386] where we proposed to use the volume entropy. We also attempt to give a new synthetic version of Ricci curvature bounded below using Bishop-Gromov's inequality.

Sharp gradient estimates on weighted manifolds with compact boundary

<p style='text-indent:20px;'>In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the Dirichlet boundary condition and some sharp growth restriction near infinity. Our results can be regarded as a refinement of recent results due to Kunikawa and Sakurai.</p>