For $n\ge2$, we prove that the $C^2$ boundary of a bounded nonempty open set in $\mathbb H^n$ is volume-preserving stable if and only if it is a Pansu sphere up to left translation and Heisenberg dilation. The proof combines centre reconstruction with a volume-constrained test field derived from Pansu dilation. Consequ...
We prove the sharp horizontal isoperimetric inequality in the first Heisenberg group for all sets of finite horizontal perimeter and positive finite volume, with equality precisely for left translates and dilates of the Pansu ball.
We prove the sharp horizontal isoperimetric inequality in $\mathbb H^n$, $n\ge2$, with equality precisely for left translates and Heisenberg dilations of Pansu balls, up to Haar-null sets.
For $n\ge2$, we prove a quantitative centre-rigidity estimate for bounded sets of finite Riemannian perimeter that are stationary and volume-constrained stable for normalized Riemannian perimeter with prescribed curvature $h_j$. Under confinement to a fixed compact set and cutoffs of vanishing full surface Dirichlet en...
