Potential Theory on Almost Complex Manifolds

By F. R. Harvey, H. Lawson. 2011

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Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their "dual" objects, the plurisubharmonic functions. These functions are defined classically by requiring that the restriction to each pseudo-holomorphic curve is subharmonic. In this paper subharmonic functions are defined by applying the viscosity approach to a version of the complex hessian which exists intrinsically on any almost complex manifold. Three theorems are proved. The first is a restriction theorem which establishes the equivalence of our definition with the "classical" definition. In the second theorem, using our "viscosity" definitions, the Dirichlet problem is solved for the complex Monge-Amp\`ere equation in both the homogeneous and inhomogeneous forms. These two results are based on theorems found in two recent papers of the authors. Finally, it is shown that the plurisubharmonic functions considered here agree with the plurisubharmonic distributions. In particular, this proves a conjecture of Nefton Pali.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/024cd03f0be24305bb851f7ef3a66653de9be3b3

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Citations in the corpus

The Monge–Ampère equation for non-integrable almost complex structures (2016, result): “. Outline of the proof. We now discuss the proof of our main results, and relate this to the work of Pli´s [36] who, as noted above, investigated this problem on J-pseudoconvex domains (Harvey-Lawson [27] obtained weak solutions in the viscosity sense, and in a more general setting). We prove the a ”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, methodology): “s since they appear to be natural PDEs which easily ﬁt into our setting. The Dirichlet problem for the almost complex Monge-Amp`ere equation (1.8) has been solved in various settings by Harvey-Lawson [33] and Pli´s [47]. A related, but diﬀerent, equation in the almost complex case was investigated b”