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Positive Scalar Curvature and Lagrangian Mean Curvature Flow

Final Report Summary - PSC AND LMCF (Positive Scalar Curvature and Lagrangian Mean Curvature Flow)

The goal of the project was to study minimal surfaces and their interacations with the ambient geometry.

We pioneered the use of min-max methods to further explore this connection and in doing so not only we solved several open problems in Geometry but we also developed further a technique (that we named Almgren-Pitts Min-max Theory) that can now be used to solve several other problems.

The most famous problem that we solved was the Willmore Conjecture, proposed 60 years ago. The conjectured stipulated that the Clifford torus is the shape with least bending energy among all tori and, while proposed first in the context of pure mathematics, it has been also been proposed independently by Biophysicists due to their observations of lipid membranes.

We solved the conjecture using in a novel way the theory of minimal surfaces combined with min-max theory. That new point of view allowed us to solve several other open problems that I now explain.

The first was the Freedman-He-Wang conjecture that stipulated what should be the most energy efficient conguration of a link in space. This problem had been open since the 80's and until our work came, there was not even a viable approach to attack it.

The second was to show that a manifold that is positively curved admits an infinite number of surfaces that are in equilibrium position (minimal surfaces). This problem had been conjectured by Yau in the 80's and before our work the best known result was that there is at least one.

Assuming the existence of these surfaces, Gromov conjectured that their areas should satisfy a Weyl Law, i.e. their asymptotic limit should depend only on the volume of the ambient manifold. We also solved this conjecture combining ideas from min-max theory with some topological arguments.