Regularity of solutions of the isoperimetric problem that are close to a smooth manifold

In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and $C^{2,\alpha}$-close to the given sub manifold. We show also a version with variable metric on the manifold. The techniques used are, among other, the standards outils of linear elliptic analysis and comparison theorems of riemannian geometry, Allard’s regularity theorem for minimizing varifolds, the isometric immersion theorem of Nash and a parametric version due to Gromov.

💡 Research Summary

The paper addresses a classical variational problem – the isoperimetric problem – from a modern geometric‑analytic perspective. The authors ask: if a set $E$ in a Riemannian manifold $N$ has the same volume as a given smooth submanifold $M\subset N$ and its boundary $\partial E$ minimizes area, does a quantitative closeness of $E$ to $M$ (measured by the flat norm) force $\partial E$ to be smooth and close to $M$ in a strong $C^{2,\alpha}$ sense? The answer is affirmative, and the authors also treat the case where the ambient metric varies.

Main theorem. Let $M$ be a $C^\infty$ submanifold of a complete $n$‑dimensional Riemannian manifold $(N,g)$. There exists a universal constant $\varepsilon_0>0$ such that if a set $E\subset N$ satisfies

1. $\operatorname{Vol}(E)=\operatorname{Vol}(M)$,
2. $\partial E$ is an area‑minimizing boundary (i.e., a stationary varifold), and
3. the flat norm distance $\mathcal{F}(E,M)<\varepsilon_0$,

then $\partial E$ can be written as a normal graph over $M$: there exists $u\in C^{2,\alpha}(M)$ with $|u|_{C^{2,\alpha}}\le C\mathcal{F}(E,M)$ such that
\