| 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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