4 problems

Let $\mathbb{R}^{n}$ be equipped with a smooth, radial, log-convex density, and consider regions of prescribed volume. Brakke's conjecture. Balls centered at the origin are isoperi…

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 $(\mathbb{R}^n,g,\mu)$ 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 $\mu$…

Let $(\mathbb{R}^n,g,\mu)$ be a connected weighted manifold satisfying $CD(\rho,\infty)$ with $\rho>0$, and let $\lambda_k(\mathbb{R}^n,g,\mu)$ denote its ordered eigenvalues. Spec…