Nonexistence of isoperimetric sets in spaces of positive curvature (Q6566116)

The paper studies the non-existence of isoperimetric sets of small volumes in the setting of non-collapsed non-negatively curved spaces.\N\NMore precisely, Theorem 1.1 states that for any \(d\geq 3\) there exists a smooth complete \(d\)-dimensional Riemannian manifold \((M,g)\) such that:\N\begin{itemize}\N\item The sectional curvature of \((M,g)\) is strictly positive at each point of \(M\).\N\item \((M,g)\) has nondegenerate asymptotic cone, meaning that there exists \(c>0\) such that \(r^{-d}\mathrm{vol}_g(B(p,r))\geq c\) for every \(p\in M\) and \(r>0\); in particular, \(M\) is non-compact and non-collapsed, the latter meaning that \(\inf_{p\in M}\mathrm{vol}_g(B(p,1))>0\).\N\item If \(v<1\), then no isoperimetric set of volume \(v\) in \(M\) exists.\N\item If \(v>1\), then there exists an isoperimetric set of volume \(v\) in \(M\).\N\end{itemize}\NTheorem 1.2 gives a similar (non-)existence statement for relative isoperimetric sets in some closed strictly convex set \(C\subseteq\mathbb R^d\) (with \(d\geq 3\)) having non-empty interior, smooth boundary and non-degenerate asymptotic cone. In fact, the authors prove Theorem 1.2 first, then they obtain Theorem 1.1 by considering the boundary of \(C\).\N\NThe above results are sharp in the following respects: in non-negatively curved \(2\)-dimensional Alexandrov spaces, isoperimetric sets exist for every volume (see [\textit{M. Ritoré}, J. Geom. Anal. 11, No. 3, 509--517 (2001; Zbl 1035.53085)] and [\textit{G. Antonelli} and \textit{M. Pozzetta}, ``Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature'', Preprint, \url{arXiv:2302.10091}]); in non-negatively curved \(d\)-dimensional Alexandrov spaces (with \(d\geq 3\)) having non-degenerate asymptotic cone, isoperimetric sets of large volume always exist (see [\textit{G. P. Leonardi} et al., Isoperimetric inequalities in unbounded convex bodies. Providence, RI: American Mathematical Society (AMS) (2022; Zbl 1503.52002); \textit{G. Antonelli} et al., Math. Ann. 389, No. 2, 1677--1730 (2024; Zbl 1546.53039); \textit{G. Antonelli} et al., Calc. Var. Partial Differ. Equ. 61, No. 2, Paper No. 77, 40 p. (2022; Zbl 1494.53049)]).