Two-forms on four-manifolds and elliptic equations

By Simon Donaldson. 2006

154 citations recorded by the source corpus.

We define a general class of elliptic equations for 2-forms on 4-manifolds, of which the complex Monge-Ampere equation is a prototype. We obtain some regularity results and discuss various connections (some speculative) with modern symplectic 4-manifold theory.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a3f81e2233843b03bb26c36490ccf59e22969997

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Uniqueness of symplectic structures (2012, background): “They are the subject of as yet unpublished work by Robin Krom [22], which is based on ideas of Donaldson in [13].”
Complex Monge-Ampere equations and totally real submanifolds (2009, background): “In [24], Donaldson proposed to generalize Yau’s theorems in a different direction; see [69,68, 65,66] for some recent developments.”
Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds (2007, background): “Part of our motivation for studying these cones is due to a question raised by Donaldson in [7].”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, background): “Donaldson’s Calabi– Yau equation for almost Kähler forms [21], which is not ostensibly of the form (1.”
The Monge–Ampère equation for non-integrable almost complex structures (2016, unknown): “[15] Donaldson, S.K. Two-forms on four-manifolds and elliptic equations, in Inspired by S.S. Chern, World Scientific, 2006.”
Estimates for the Complex Monge-Ampère Equation on Hermitian and Balanced Manifolds (2009, background): “For some related results, we refer the reader to: Calabi’s paper [Ca2] which inspired Yau’s computation; [PSS] for a concise proof of the parabolic version of this estimate; [To2] for a precise version of this estimate without assuming a C2 estimate; and [TWY] for a generalization of the third order”
Taming symplectic forms and the Calabi–Yau equation (2007, background): “The second author thanks Simon Donaldson for some useful and encouraging conversations, and his former advisor, D. H. Phong for his support and advice.”
On the J-anti-invariant cohomology of almost complex 4-manifolds (2011, methodology): “In [10], Donaldson introduced the symplectic version of the Calabi-Yau equation.”