Sharp Isoperimetric Inequalities - Old and New

Isoperimetric inequalities constitute some of the most beautiful and ancient results in geometry, and play a key role in numerous facets of differential geometry, analysis, calculus of variations, geometric measure theory, minimal surfaces, probability and more.Isoperimetric minimizers have classically been determined on Euclidean, spherical, hyperbolic and Gaussian spaces.

The isoperimetric problem is well-understood on surfaces, but besides some minor variations on these examples and some three-dimensional cases, remains open on numerous fundamental spaces, like projective spaces, the flat torus or hypercube, and for symmetric sets in Gaussian space.

When partitioning the space into multiple regions of prescribed volume so that the common surface-area is minimized, the Euclidean double-bubble conjecture was established by Hutchings-Morgan-Ritoré-Ros, and the Gaussian multi-bubble conjecture was recently established in our work with Neeman, but the Euclidean and spherical multi-bubble conjectures remain wide open.

Isoperimetric comparison theorems like the Gromov-Lévy and Bakry-Ledoux theorems are well-understood under a Ricci curvature lower bound, but under an upper-bound K ≤ 0 on the sectional curvature, the Cartan-Hadamard conjecture remains open in dimension five and higher despite recent progress.

In the sub-Riemannian setting, the isoperimetric problem remains open on the simplest example of the Heisenberg group.The above long-standing problems lie at the very forefront of the theory and present some of the biggest challenges on both conceptual and technical levels.

Any progress made would be extremely important and would open the door for tackling even more general isoperimetric problems.

To address these questions, we propose adding several concrete new tools, some of which have only recently become available, to the traditional ones typically used in the study of isoperimetric problems.