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 Date:  Tuesday, December 08, 2009 Title:  THE DIRICHLET PROBLEM FOR FULLY NONLINEAR EQUATIONS ON RIEMANNIAN MANIFOLDS Abstract:  The aim of this lecture is to discuss a general approach to solving the Dirichlet problem for a variety of nonlinear partial differential equations which arise in geometry. Original motivation came from the fact that manifolds with geometric structure often carry large and useful families of non-standard subharmonic'' functions. For example, any almost complex manifold with hermitian metric carries plurisubharmonic functions. Moreover, it also carries Lagrangian subharmonic functions'' whose restrictions to Lagrangian submanifolds are subharmonic. Each Calabi-Yau manifold carries several interesting families. In fact every manifold with a calibration $\varphi$ carries $\varphi$-plurisubharmonic functions'' which are subharmonic on all calibrated submanifolds. In all cases the extremals in these families, the {\sl harmonic functions}, are interesting and often satisfy a basic non-linear second-order equation. I will discuss the Dirichlet Problem for such harmonic functions on bounded domains in a riemannian manifold. Existence and uniqueness will be established for these and other quite general second-order equations. The result holds for all continuous boundary data subject to a geometric {\sl F-convexity} of the boundary, defined entirely in terms of the equation {\sl F}. Examples include all branches of the homogeneous Monge-Amp\`ere equation over ${\bf R}$, ${\bf C}$ and ${\bf H}$, and all branches of the special lagrangian potential equation. Speaker:  Blaine Lawson Institution:  SUNY Stonybrook

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