3 problems

Let a weighted manifold be diffeomorphic to $\mathbb{R}^n$ and satisfy the separate curvature bounds … Here $\operatorname{Ric}_g$ is the Riemannian Ricci tensor and $\nabla_g^2V$…

Let $(S^n,g,\operatorname{vol}_g)$ satisfy $\operatorname{Ric}_g\geq\rho g$ with $\rho>0$, and let $g^\rho_{\mathrm{can}}$ be the canonical metric on $S^n$ rescaled so that…

Let $(\mathbb{R}^n,g,\mu)$ satisfy $CD(\rho,\infty)$ with $\rho>0$, and let $\gamma^n_\rho$ be the $n$-dimensional Gaussian measure. A map pushes forward $\gamma^n_\rho$ onto $\mu$…