Joint with Analysis Seminar.

Abstract: Fully nonlinear elliptic and parabolic equations on manifolds play central roles in some important problems in real and complex geometry. A key ingredient in solving such equations is to establish apriori  estimates up to second order. For general Riemannian manifolds, or Kaehler/Hermitian manifolds in the  complex case, one encounters difficulties caused by the curvature (as well as torsion in the Hermian case) of the manifolds. 

In this talk we report some results in our effort to overcome these obstacles over the past  few years. We shall emphasize on understanding the roles of subsolutions and concavity of the equation based on which our techniques were developed. We are interested both in equations on closed manifolds, and the Dirichlet problem for equations on manifolds with boundary of arbitrary geometry. 

For the Dirichlet problem on manifolds with boundary, we prove that under some fundamental structure conditions which were first proposed by Caffarelli-Nirenberg-Spruck and are now standard in the literature, there exist a smooth solution provided that there is a C2 subsolution. 

For equations on closed manifolds, there have appeared two different notations of weak subsolutions, the C-subsolution introduced by Gabor Szekelyshidi (JDG, 2018) and "tangent cone at infinity" condition by myself (Duke J Math, 2014). We show for type I cones the two notations coincide. We also construct examples showing for the Dirichlet problem that the subsolution condition can not be replaced by the weaker versions.