Harmonic Discs of Solutions to the Complex Homogeneous Monge-Amp `ere\n Equation

2014/08/28 by Julius Ross, Ross, Julius, David Witt Nyström +1 · 1 citation
Mathematics · #32Q15 #32W20 #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1408.6663

openalex publication_date 2014/08/28 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We study regularity properties of solutions to the Dirichlet problem for the\ncomplex Homogeneous Monge-Amp `ere equation. We show that for certain boundary\ndata on mathbb P¹ the solution \Φ to this Dirichlet problem is\nconnected via a Legendre transform to an associated flow in the complex plane\ncalled the Hele-Shaw flow. Using this we determine precisely the harmonic discs\nassociated to \Φ. We then give examples for which these discs are not dense\nin the product, and also prove that this situation persists after small\nperturbations of the boundary data.\n

Citations

• Potential methods in variational inequalities

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