Differential Geometry and Partial Differential Equations

Stanford University, Stanford CA

Investigators

Abstract

Abstract for DMS-0104163 PI: Richard M. Schoen Professor Schoen is proposing to study the existence of an extremal metric which defines the Bartnik quasilocal mass for a domain in a spacetime of general relativity. This would yield an existence theorem for asymptotically flat solutions of the static vacuum Einstein equations with suitable boundary conditions. It is part of a variational approach for constructing three dimensional geometries. Schoen's second project involves the construction of special lagrangian, and more generally minimal lagrangian, submanifolds in Calabi-Yau and Kaehler-Einstein manifolds. The approach is to construct hamiltonian stationary submanifolds by direct volume minimization among lagrangian submanifolds, and to obtain sufficient regularity to show that they are minimal lagrangian. Schoen's third project is to further develop the harmonic map approach to prove the rigidity of smooth actions of lattices in semisimple Lie groups on compact manifolds. Professor Mutao Wang proposes to study the mean curvature flow for special classes of submanifolds of codimension greater than one. The major thrust is to obtain stability and regularity properties of the flow. Dr. Baozhang Yang will study singular behavior of Yang-Mills connections in arbitrary dimension. This study will include an investigation into the structure of blow-up sets and asymptotic behavior near singularities. This research project concerns the study of geometric shapes which optimize certain physical and geometric energies. For curved spacetimes in general relativity, there is no natural mass-energy density which can be assigned to the gravitational field, so Bartnik proposed to measure the gravitational mass of a region in a spacetime by minimizing the total mass of all physical spacetimes which contain this region as a subset. This minimal mass spacetime, if it can be shown to exist, will be a static solution of Einstein's equations. One of the goals of this project is to find a way to construct such static solutions, and to use them to study three dimensional geometry. It is expected that three dimensional spaces have natural geometries on them which are uniquely characterized by their curvature properties. Another goal of this project is to construct certain special surfaces in Calabi-Yau manifolds, the spaces of string theory. These (three dimensional) surfaces, called special lagrangian submanifolds, are analogous to soap films in that they are surfaces of least possible area. Finally, it is proposed to study the evolution problem for surfaces which is called the mean curvature evolution. This is an evolution problem which moves a surface in space in such a way that its area is decreased most rapidly. Understanding the behavior of this evolution is important for simplifying and smoothing complicated surfaces in an optimal way. Mathematically this is a difficult problem because the surfaces may develop singularities such as cone points and tears which must be accounted for.