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Mathematics Colloquium

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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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