Mathematics

It is a fundamental result in geometry that the only perimeter minimizing sets in $\mathbb{R}^n$ are the half-spaces if and only if $n \leq 7$. A celebrated example of a perimeter minimizing set in $\mathbb{R}^8$ which is not a half-space is provided by Simons Cone.\\A natural problem is whether this dichotomy holds, in a suitable sense, when $\mathbb{R}^n$ is replaced by a Riemannian manifold $(M^n,g)$. This leads to the following question: given a perimeter minimizing set $E \subset M$, under what geometric conditions on $M$ can we infer that $M$ splits isometrically as $N \times \mathbb{R}$, with $E$ corresponding to a half-space $ N \times [0,+\infty)$?\\In this talk, I will survey previous developments on this problem, and I will present a new result obtained in collaboration with Mattia Magnabosco. Specifically, we show that if $(M^n,g)$ has non-negative sectional curvature and quadratic volume growth, then the existence of a perimeter minimizing set $E \subset M$ implies the aforementioned isometric splitting.