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arXiv · 2611.08309

The isoperimetric problem in the Riemannian Heisenberg $3$-space

José M. Espinar·Pablo Mira
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Abstract

We prove that every closed weakly stable constant mean curvature surface immersed in the Riemannian Heisenberg space ${\rm Nil}_3$ is an embedded rotational sphere. As a consequence, we solve the isoperimetric problem in ${\rm Nil}_3$: the isoperimetric domains of ${\rm Nil}_3$ are exactly the regions enclosed by rotational constant mean curvature spheres, and for each prescribed volume there is a unique such domain, up to ambient isometry. For small volumes, the isoperimetric result was proved by Tomter in 1993.

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José M. Espinar, Pablo Mira. 2026-11-05. The isoperimetric problem in the Riemannian Heisenberg $3$-space. https://arxiv.org/abs/2611.08309

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