Problems in general relativity and geometric flows

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Professor Mu-Tao Wang proposes to apply the method of geometric analysis to study several problems in general relativity (GR) and geometric flows. He will further investigate properties of quasilocal mass and energy-momentum which he recently discovered with Shing-Tung Yau. Immediate goals include solving the optimal isometric embedding equation, understanding monotone and sub-additive properties of the quasilocal mass, and anchoring the definition of quasilocal angular momentum. Professor Wang plans to continue his research on the mean curvature flow of Lagrangian submanifolds. Immediate goals include proving the global existence and convergence of the flow in two cases (1) graphs of symplectomorphisms of Kaehler-Einstein manifolds (2) exact Lagrangian submanifolds of cotangent bundles, and applying these results to Lagrangian isotopy problems.Due to the non-local feature of gravitational energy, the ADM or Bondi mass only finds its expression at infinity where gravitation is weak. However, most observable physical models are finitely extended regions and measurement of mass/energy on such a region is essential in many fundamental issues in GR. Professor Wang's newly discovered quasi-local energy evaluates the total energy contained in any bounded region of the universe even if the effect of gravitation is very strong. The proposed research will further our understanding of gravitational energy in large scale and of problems related to black holes. Professor Wang purposes to study the ``mean curvature flow" of surfaces. This flow is like the heat equation which dissipates curvature distributions and deforms a surface into an equilibrium state. He plans to apply this flow to connect a general surface to ones with optimal shape. This procedure will help us understand the geometry and topology of such surfaces. It is also important in applied areas such as computer graphics and image processing.