The Monge-Amp\`{e}re equation on almost complex manifolds
Szymon Plis
TL;DR
This paper investigates the Monge-Ampère equation on almost complex manifolds, establishing the existence and uniqueness of smooth solutions to the Dirichlet problem in strictly pseudoconvex domains.
Contribution
It provides the first existence and uniqueness results for smooth solutions of the Monge-Ampère equation in this geometric setting.
Findings
Existence of unique smooth solutions in strictly pseudoconvex domains
Extension of classical Monge-Ampère theory to almost complex manifolds
New techniques for solving nonlinear PDEs in complex geometry
Abstract
We study the Dirichlet problem for the Monge-Amp\`ere equation on almost complex manifolds. We obtain the existence of the unique smooth solution of this problem in strictly pseudoconvex domains.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Holomorphic and Operator Theory