Dirichlet duality and the nonlinear dirichlet problem on riemannian manifolds

F. Reese Harvey Rice University H. Blaine Lawson, Jr. Stony Brook University

Differential Geometry mathscidoc:1609.10229

Journal of Differential Geometry, 88, (3), 395-482, 2011
In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in our previous paper [HL4], we define equations via closed subsets of the 2-jet bundle where each equation has a natural dual equation. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting with a constant coefficient equation as a model, which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model. For example, this covers all branches of the homogeneous complex Monge-Amp`ere equation on an almost complex hermitian manifold X. In general, for an equation F on a manifold X and a smooth domain  X, we give geometric conditions which imply that the Dirichlet problem on is uniquely solvable for all continuous boundary functions. We begin by introducing a weakened form of comparison which has the advantage that local implies global. We then introduce two fundamental concepts. The first is the notion of a monotonicity cone M for F. If X carries a global Msubharmonic function, then weak comparison implies full comparison. The second notion is that of boundary F-convexity, which is defined in terms of the asymptotics of F and is used to define barriers. In combining these notions the Dirichlet problem becomes uniquely solvable as claimed. This article also introduces the notion of local affine jet-equivalence for subequations. It is used in treating the cases above, but gives results for a much broader spectrum of equations on manifolds, including inhomogeneous equations and the Calabi-Yau equation on almost complex hermitian manifolds. A considerable portion of the paper is concerned with specific examples. They include a wide variety of equations which make sense on any riemannian manifold, and many which hold universally on almost complex or quaternionic hermitian manifolds, or topologically calibrated manifolds.
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  author={F. Reese Harvey, and H. Blaine Lawson, Jr.},
  booktitle={Journal of Differential Geometry},
F. Reese Harvey, and H. Blaine Lawson, Jr.. DIRICHLET DUALITY AND THE NONLINEAR DIRICHLET PROBLEM ON RIEMANNIAN MANIFOLDS. 2011. Vol. 88. In Journal of Differential Geometry. pp.395-482. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913204800524846890.
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