Generic regularity of area minimising hypersurfaces and mean curvature flows

Descripción del proyecto

El avance de la geometría a través de la regularidad genérica

En matemáticas, una forma de entender la geometría de un espacio es estudiar las superficies «óptimas» que contiene. Sin embargo, resulta que en dimensiones superiores estas superficies pueden desarrollar singularidades en que se rompe la descripción matemática «habitual». Eso ha bloqueado durante mucho tiempo los avances en la resolución de algunos de los mayores problemas de este campo, desde la comprensión de la curvatura hasta la demostración de teoremas clave sobre el espacio y la masa. El proyecto GENREG, financiado por el Consejo Europeo de Investigación, aborda este reto a través del concepto de regularidad genérica: la idea de que, bajo pequeñas perturbaciones, las estructuras geométricas pueden suavizarse o incluso quedar totalmente libres de singularidades. El equipo de GENREG pretende extender este principio a múltiples áreas de la geometría y la topología (desde las hipersuperficies que minimizan el área hasta los flujos de curvatura media) allanando el camino para importantes avances en matemáticas y física teórica.

Objetivo

Major advances in geometry and topology have been achieved by studying critical points and gradient flows for natural energies, but these analytic methods are hindered when singularities occur. In fact, singularities are the main obstacle in the use of area minimisation in the proof of the positive mass theorem up to dimension 7 and the use of Ricci flow with surgery in the proof of the 3-dimensional Poincaré conjecture. A key observation in geometry and physics is that generic solutions, obtained by small perturbations, can exhibit simpler singularities or even none at all. This phenomenon, called generic regularity, can yield outstanding results. The recent generic regularity breakthroughs by the PI-led group will allow to address fundamental open problems in three areas:

For area-minimising hypersurfaces, we aim to extend generic regularity to all dimensions, building on the PI's work in up to 10 dimensions. This would establish the positive mass theorem in all dimensions, bypassing technical analysis of the singular set of minimisers. It would also allow the resolution of other well-known problems related to scalar curvature.

For mean curvature flow singularities, which are unavoidable, generic flows are expected to encounter only the simplest types. Work of the PI has proven this in 3 and 4 dimensions up to the problem of “multiplicity”. Bamler–Kleiner recently excluded multiplicity in 3 dimensions. Our goal is to prove that multiplicity generically cannot occur in higher dimensions. This would mark major progress towards the Schoenflies conjecture, a main open problem in 4-dimensional topology.

For special Lagrangian submanifolds, fundamental objects in symplectic geometry, we would geometrise Lagrangians in Calabi-Yau manifolds by establishing generic Lagrangian mean curvature flows through singularities. The anticipated contributions to mirror symmetry are expected to impact fields spanning algebra, geometry, topology, and theoretical physics.