Periodic Reporting for period 2 - ISOPERIMETRY (Sharp Isoperimetric Inequalities - Old and New)
Berichtszeitraum: 2024-04-01 bis 2025-09-30
Isoperimetric inequalities constitute some of the most beautiful and classical results in geometry, going back to the story of the founding of Carthage by Queen Dido. These inequalities play a key role in numerous facets of differential geometry, analysis, PDE, calculus of variations, geometric measure theory, probability theory and more.
The isoperimetric variational problem is simple to state but often very hard to resolve: among all sets of prescribed volume in a given space, characterize those sets whose surface-area is minimal. For example, in Euclidean space, the Euclidean ball minimizes the surface-area among all sets of a given volume. Similarly, the isoperimetric minimizers on spherical, hyperbolic and Gaussian spaces have been determined. The isoperimetric problem is well-understood on two-dimensional 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 Euclidean space into two bounded regions of enclosed prescribed volumes so that the total common surface-area is minimized, it was proved by Hutchings-Morgan-Ritoré-Ros that the solution is a double-bubble, such as the one we see when two soap-bubbles attach. In Gaussian space, a complete characterization of how to optimally partition the space into multiple regions was established in our prior work with Neeman, but the multi-bubble conjectures in Euclidean, spherical and hyperbolic spaces have remained widely open. Isoperimetric comparison theorems like the Gromov-Lévy and Bakry-Ledoux theorems are well-understood under a Ricci curvature lower bound, but under a upper-bound K ≤ 0 on the sectional curvature, the Cartan-Hadamard conjecture remains open in dimension higher than four 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.
The isoperimetric variational problem is simple to state but often very hard to resolve: among all sets of prescribed volume in a given space, characterize those sets whose surface-area is minimal. For example, in Euclidean space, the Euclidean ball minimizes the surface-area among all sets of a given volume. Similarly, the isoperimetric minimizers on spherical, hyperbolic and Gaussian spaces have been determined. The isoperimetric problem is well-understood on two-dimensional 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 Euclidean space into two bounded regions of enclosed prescribed volumes so that the total common surface-area is minimized, it was proved by Hutchings-Morgan-Ritoré-Ros that the solution is a double-bubble, such as the one we see when two soap-bubbles attach. In Gaussian space, a complete characterization of how to optimally partition the space into multiple regions was established in our prior work with Neeman, but the multi-bubble conjectures in Euclidean, spherical and hyperbolic spaces have remained widely open. Isoperimetric comparison theorems like the Gromov-Lévy and Bakry-Ledoux theorems are well-understood under a Ricci curvature lower bound, but under a upper-bound K ≤ 0 on the sectional curvature, the Cartan-Hadamard conjecture remains open in dimension higher than four 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.
In two joint works with Neeman, we have resolved one of the most important aims of the project, pertaining to the multi-bubble isoperimetric conjectures on Euclidean space R^n and spherical space S^n. It was conjectured by J. Sullivan in the 1990's that standard bubbles uniquely minimize total perimeter among all k-bubbles enclosing prescribed volume, for any k≤n+1. These conjectures are known to hold when n=2, but the only other previously known case was the groundbreaking confirmation of the double-bubble conjecture (case k=2) in R^3 (and later R^n) circa 2000 by Hutchings-Morgan-Ritoré-Ros. We confirmed the conjectures on R^n and S^n in the following range: the double-bubble conjectures for n≥2, the triple-bubble conjectures (k=3) for n≥3, the quadruple-bubble conjectures (k=4) for n≥4 and the quintuple-bubble conjectures (k=5) for n≥5 (but without uniqueness on R^n). In fact, we showed that for all k≤n, a minimizing cluster necessarily has spherical interfaces, and that its cells (including the unbounded one) are necessarily connected, resolving a conjecture of Heppes. Furthermore, we showed that the corresponding Jacobi operator has exactly k negative eigenvalues, thereby establishing the concavity of the isoperimetric profile.
In a joint work with our post-doc Xu, we established that all standard bubbles on R^n, S^n and H^n (hyperbolic space) are stable, an infinitesimal local form of minimizing perimeter. Moreover, our results apply to Mobius-flat non-standard bubbles and even more general partitions as well.
We also studied the isoperimetric problem on the n-dimensional cube. The isoperimetric conjecture on the 3D cube predicts that minimizers are enclosed by spheres about a corner, cylinders about an edge and coordinate planes. This has only been previously established for relative volumes close to 0, 1/2 and 1 by compactness arguments. After decades without progress, our analysis confirms the conjecture on a cube with side lengths (β,1,1) in a new range of relatives volumes v∈[0,1/2]. In particular, we confirm the conjecture for the standard cube (β=1) for all v≤0.120582 when β≤0.919431 for the entire range where spheres are conjectured to be minimizing, and also for all v∈[0,1/2]∖(1/π−β/4,1/π+β/4). As for higher dimensions, we showed that, surprisingly, the analogous conjecture is false when n≥10.
In two joint works with Ivaki, we investigated refinements of the Brunn-Minkowski inequality, a key tool for understanding isoperimetric properties of convex bodies K in R^n. The log-Brunn-Minkowski conjecture predicts that when K is origin-symmetric, it enjoys improved volumetric properties, as captured by the cone-volume measure V_K. It was shown by Brendle-Choi-Daskalopoulos that if V_K is a multiple of the Lebesgue measure, then K is necessarily a centered Euclidean ball, thereby resolving a longstanding conjecture of Firey. Using a local version of the Brunn–Minkowski inequality and centro-affine differential geometry, we obtained a new proof of Firey’s conjecture (which generalizes to a multitude of additional nonlinearities), and moreover, we confirmed the log-Minkowski conjecture whenever K enjoys very liberal curvature pinching estimates, improving upon all previously known results.
In a joint work with Shabelman and Yehudayoff, we resolved a longstanding conjecture from the 1990s of Lutwak and Gardner regarding the intersection body operator I. Intersection bodies play a cardinal role in the dual Brunn-Minkowski theory, and so a natural question is to characterize the fixed and periodic points of I. We confirmed that IK = c K iff K is a centered Euclidean ball, and that I^2 K = c K iff K is a centered ellipsoid. To this end, we revisited Busemann's isoperimetric inequality for the volume of the intersection body and its monotonicity under Steiner symmetrization, characterizing the equality case.
In a solo work, we obtained a new proof of Royen's Gaussian Correlation Inequality using an inverse Brascamp-Lieb inequality for even functions which are more-log-concave than prescribed Gaussians, generalizing a recent inequality of Nakamura-Tsuji.
Our post-doc Bizeul obtained a new and simplified proof of the very recent groundbreaking confirmation of the Slicing Problem by Klartag-Lehec, which is intimately related to the KLS isoperimetric conjecture for convex bodies. Furthermore, Bizeul studied Gaussian-like isoperimetry in the form of the log-Sobolev inequality for general sub-Gaussian log-concave measures, which may be seen as a sub-Gaussian analogue of the KLS conjecture.
Our post-doc Xu and his collaborators established affine isoperimetric type inequalities for static convex domains in hyperbolic space, with equality being attained uniquely for hyperbolic ellipsoids. Furthermore, they established new Heintze-Karcher-type inequalities (a useful tool for establishing isoperimetric inequalities as shown by Gromov, Ros, and others) in sub-static warped product manifolds, and derived uniqueness results for a class of curvature equations.
In a joint work with our post-doc Xu, we established that all standard bubbles on R^n, S^n and H^n (hyperbolic space) are stable, an infinitesimal local form of minimizing perimeter. Moreover, our results apply to Mobius-flat non-standard bubbles and even more general partitions as well.
We also studied the isoperimetric problem on the n-dimensional cube. The isoperimetric conjecture on the 3D cube predicts that minimizers are enclosed by spheres about a corner, cylinders about an edge and coordinate planes. This has only been previously established for relative volumes close to 0, 1/2 and 1 by compactness arguments. After decades without progress, our analysis confirms the conjecture on a cube with side lengths (β,1,1) in a new range of relatives volumes v∈[0,1/2]. In particular, we confirm the conjecture for the standard cube (β=1) for all v≤0.120582 when β≤0.919431 for the entire range where spheres are conjectured to be minimizing, and also for all v∈[0,1/2]∖(1/π−β/4,1/π+β/4). As for higher dimensions, we showed that, surprisingly, the analogous conjecture is false when n≥10.
In two joint works with Ivaki, we investigated refinements of the Brunn-Minkowski inequality, a key tool for understanding isoperimetric properties of convex bodies K in R^n. The log-Brunn-Minkowski conjecture predicts that when K is origin-symmetric, it enjoys improved volumetric properties, as captured by the cone-volume measure V_K. It was shown by Brendle-Choi-Daskalopoulos that if V_K is a multiple of the Lebesgue measure, then K is necessarily a centered Euclidean ball, thereby resolving a longstanding conjecture of Firey. Using a local version of the Brunn–Minkowski inequality and centro-affine differential geometry, we obtained a new proof of Firey’s conjecture (which generalizes to a multitude of additional nonlinearities), and moreover, we confirmed the log-Minkowski conjecture whenever K enjoys very liberal curvature pinching estimates, improving upon all previously known results.
In a joint work with Shabelman and Yehudayoff, we resolved a longstanding conjecture from the 1990s of Lutwak and Gardner regarding the intersection body operator I. Intersection bodies play a cardinal role in the dual Brunn-Minkowski theory, and so a natural question is to characterize the fixed and periodic points of I. We confirmed that IK = c K iff K is a centered Euclidean ball, and that I^2 K = c K iff K is a centered ellipsoid. To this end, we revisited Busemann's isoperimetric inequality for the volume of the intersection body and its monotonicity under Steiner symmetrization, characterizing the equality case.
In a solo work, we obtained a new proof of Royen's Gaussian Correlation Inequality using an inverse Brascamp-Lieb inequality for even functions which are more-log-concave than prescribed Gaussians, generalizing a recent inequality of Nakamura-Tsuji.
Our post-doc Bizeul obtained a new and simplified proof of the very recent groundbreaking confirmation of the Slicing Problem by Klartag-Lehec, which is intimately related to the KLS isoperimetric conjecture for convex bodies. Furthermore, Bizeul studied Gaussian-like isoperimetry in the form of the log-Sobolev inequality for general sub-Gaussian log-concave measures, which may be seen as a sub-Gaussian analogue of the KLS conjecture.
Our post-doc Xu and his collaborators established affine isoperimetric type inequalities for static convex domains in hyperbolic space, with equality being attained uniquely for hyperbolic ellipsoids. Furthermore, they established new Heintze-Karcher-type inequalities (a useful tool for establishing isoperimetric inequalities as shown by Gromov, Ros, and others) in sub-static warped product manifolds, and derived uniqueness results for a class of curvature equations.
All of the above mentioned works constitute original research which go well beyond the previously known state-of-the-art. Below are a few examples:
- Our resolution of the triple- and quadruple-bubble isoperimetric conjectures on R^n and S^n, previously thought to be completely out-of-reach with existing methods, has been widely recognized by the Mathematical community as a truly groundbreaking result. This is witnessed by the media attention it received when it was first announced (including Quanta Magazine and Quanta's "2022's biggest breakthroughs in Math" youtube video), and the eventual publication in the prestigious journal Acta Mathematica.
- In our proof of the quintuple-bubble conjecture, we developed a novel spectral theory for the Jacobi operator on the interfaces of a bubble configuration. This theory was further enriched in our work on the stability of bubble configurations, in which we obtained new conjugated Brascamp-Lieb inequalities on partitions having conformally flat umbilical boundary, in analogy with the single-bubble setting. Using this, we obtained positive evidence, for the very first time, that the multi-bubble conjecture may be true in the hyperbolic setting.
- In our work on the isoperimetric inequality on the 3D cube, we confirmed the natural corresponding conjecture in a new explicit range of volume parameters, after decades without any progress on this problem beyond using compactness arguments.
- Our novel simple proof of Firey's conjecture extends to numerous non-linear Minkowski-type problems, and has already been used by numerous authors for resolving more general non-linear geometric PDEs.
- We obtained uniqueness in the Lp-Minkowski problem under the presently best-known curvature pinching estimates. To this end, we recast the problem in the language of centro-affine differential geometry, a novel approach in this context which was introduced in our previous work.
- We resolved a longstanding problem from the 1990's of characterizing the fixed and periodic points for the intersection-body operator. To this end, we developed a well-defined notion of continuous Steiner symmetrization for Lipschitz star-bodies.
- We obtained a novel simple proof of the Gaussian Correlation Inequality by applying an extension of the Nakamura-Tsuji inverse Brascamp-Lieb inequality. Our method was already extended by Nakamura-Tsuji to yield further captivating generalizations of the Gaussian Correlation Inequality.
We expect to continue uncovering further insights in our investigation of isoperimetric inequalities in various contexts, by using an assortment of tools including symmetrization, optimal-transport, differential geometry and variational analysis.
- Our resolution of the triple- and quadruple-bubble isoperimetric conjectures on R^n and S^n, previously thought to be completely out-of-reach with existing methods, has been widely recognized by the Mathematical community as a truly groundbreaking result. This is witnessed by the media attention it received when it was first announced (including Quanta Magazine and Quanta's "2022's biggest breakthroughs in Math" youtube video), and the eventual publication in the prestigious journal Acta Mathematica.
- In our proof of the quintuple-bubble conjecture, we developed a novel spectral theory for the Jacobi operator on the interfaces of a bubble configuration. This theory was further enriched in our work on the stability of bubble configurations, in which we obtained new conjugated Brascamp-Lieb inequalities on partitions having conformally flat umbilical boundary, in analogy with the single-bubble setting. Using this, we obtained positive evidence, for the very first time, that the multi-bubble conjecture may be true in the hyperbolic setting.
- In our work on the isoperimetric inequality on the 3D cube, we confirmed the natural corresponding conjecture in a new explicit range of volume parameters, after decades without any progress on this problem beyond using compactness arguments.
- Our novel simple proof of Firey's conjecture extends to numerous non-linear Minkowski-type problems, and has already been used by numerous authors for resolving more general non-linear geometric PDEs.
- We obtained uniqueness in the Lp-Minkowski problem under the presently best-known curvature pinching estimates. To this end, we recast the problem in the language of centro-affine differential geometry, a novel approach in this context which was introduced in our previous work.
- We resolved a longstanding problem from the 1990's of characterizing the fixed and periodic points for the intersection-body operator. To this end, we developed a well-defined notion of continuous Steiner symmetrization for Lipschitz star-bodies.
- We obtained a novel simple proof of the Gaussian Correlation Inequality by applying an extension of the Nakamura-Tsuji inverse Brascamp-Lieb inequality. Our method was already extended by Nakamura-Tsuji to yield further captivating generalizations of the Gaussian Correlation Inequality.
We expect to continue uncovering further insights in our investigation of isoperimetric inequalities in various contexts, by using an assortment of tools including symmetrization, optimal-transport, differential geometry and variational analysis.