A Bernstein result and counterexample for entire solutions to Donaldson's equation
Micah Warren
TL;DR
This paper proves convex entire solutions to Donaldson's equation are quadratic, introduces counterexamples that are not quadratic, and uncovers new solutions to complex Monge-Ampère equations, advancing understanding of these complex geometric PDEs.
Contribution
It establishes a Bernstein-type result for Donaldson's equation and provides explicit counterexamples, expanding the class of known solutions.
Findings
Convex entire solutions to Donaldson's equation are quadratic.
Existence of non-quadratic entire solutions to Donaldson's equation.
Discovery of new entire solutions to complex Monge-Ampère equations.
Abstract
We show that convex entire solutions to Donaldson's equation are quadratic, using a result of Weiyong He. We also exhibit entire solutions to the Donaldson equation that are not of the form discussed by He. In the process we discover some non-trivial entire solutions to complex Monge-Amp\`{e}re equations.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Algebraic Geometry and Number Theory