Variational Methods in Singular Geometry

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Physical phenomena tend to obey the least action principle, namely moving in trajectories that make the action locally stationary. If the action is given by Dirichlet energy, then the stationary points are in some cases geodesics (shortest paths), and in other cases harmonic functions and various generalizations of these notions familiar from elementary physics. In this project, the PI will study stationary solutions either related to harmonic maps obtained by minimizing Dirichlet energy or best Lipschitz maps (sometimes called infinity harmonic) obtained by minimizing Lipschitz constants. Both problems have interesting applications to geometric topology and group theory. The project also includes significant training and mentoring of junior mathematicians (students, post-doctoral researchers, junior faculty) The work of Eells-Sampson in the 60's launched an explosion of research for harmonic maps between Riemannian manifolds. Many important applications followed in minimal surface theory, Kaehler geometry and rigidity of group actions on manifolds among others. More recently, the seminal work of Gromov-Schoen and Korevaar-Schoen on harmonic maps to metric space targets initiated major progress in understanding phenomena associated with singular spaces, like rigidity of groups acting on buildings and the completion of Teichmueller space. In the first part of the project PI will study several problems in harmonic map theory for singular geometry with applications to Teichmueller theory. In the second part PI will study the calculus of variations of functionals associated with the sup-norm of the gradient of maps between Riemannian manifolds. Such functionals yield solutions of fully non-linear degenerate PDE's with very challenging regularity properties and whose singular set gives geometric realizations of topological objects like geodesic foliations and laminations related to Thurston theory. The PI will also train graduate students and maintains a robust mentoring program for all junior researchers in his department. The latter provides guidance and support to post-doctoral researchers and junior faculty on a range of issues related to professional development.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.