Geodesic
Harmonic discs of solutions to the complex homogeneous Monge-Ampère equation
Ross, Julius1 ; Nyström, David Witt1
1 DPMMS, University of Cambridge Cambridge UK
Publications Mathématiques de l'IHÉS, Tome 122 (2015), pp. 315-335

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We study regularity properties of solutions to the Dirichlet problem for the complex Homogeneous Monge-Ampère equation. We show that for certain boundary data on P1 the solution Φ to this Dirichlet problem is connected via a Legendre transform to an associated flow in the complex plane called the Hele-Shaw flow. Using this we determine precisely the harmonic discs associated to Φ. We then give examples for which these discs are not dense in the product, and also prove that this situation persists after small perturbations of the boundary data.

DOI : 10.1007/s10240-015-0074-0
Keywords: Weak Solution, Dirichlet Problem, Regular Solution, Boundary Data, Boundary Component

Ross, Julius 1 ; Nyström, David Witt 1

1 DPMMS, University of Cambridge Cambridge UK 
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     title = {Harmonic discs of solutions to the complex homogeneous {Monge-Amp\`ere} equation},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {315--335},
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Ross, Julius; Nyström, David Witt. Harmonic discs of solutions to the complex homogeneous Monge-Ampère equation. Publications Mathématiques de l'IHÉS, Tome 122 (2015), pp. 315-335. doi: 10.1007/s10240-015-0074-0

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