Stability of variational problems in differential geometry

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The principle of least action says that the outcome of a physical model should always minimize a well-determined physical quantity, the action functional. This observation goes back to Fermat and Euler, and broadly speaking it says that, by the laws of nature, things are carried out in the most economical way. The phenomenon applies to many facets of physics, including Newtonian, Lagrangian and Hamiltonian mechanics, even general relativity. The abstract mathematical framework that studies such phenomena is the calculus of variations, and the principal investigator will apply this in the context of differential geometry. Namely, in differential geometry one often has a large collection of geometric objects (in this case Kahler metrics or Lagrangians) and is searching for special elements in this collection that have the nicest properties. These special elements often minimize a certain energy functional, and this is the starting point of the project. A thorough understanding of the problems at hand can lead to new insight into the shape of the universe, and it would help make exciting predictions in string theory and, more broadly, in theoretical physics.This project can be split into three main subjects: characterizing existence of constant scalar curvature metrics on Kahler manifolds; convexity and curvature properties of the L^p-Finsler geometry of the space of Kahler metrics, their finite dimensional approximations, and the structure of the associated space of geodesic rays; the metric structure of the space of positive Lagrangians. As a novelty, in the proposed study we will either specifically develop or use an adequate metric geometry, in hopes of understanding the underlying variational problems better. The metric spaces that the PI plans to use arise from the path length structure of infinite dimensional Finsler manifolds, and as such have a very rich geometry themselves. In the Kahler case it is hopeful that this will allow one to connect many notions of stability, including K-stability from the Yau-Tian-Donaldson conjecture, the energy properness of Tian, and geodesic stability, all conjectured to characterize existence of constant scalar curvature metrics. In the case of Lagrangian geometry much less is known. Following a recent program proposed by Solomon, the principal investigator intends to develop the underlying metric geometry further in order to formulate and prove stability conditions characterizing existence of special Lagrangians.