Solutions to degenerate complex Hessian equations

Lu Hoang Chinh

Solutions to degenerate complex Hessian equations

Complex Variables 2012-10-23 v4 Analysis of PDEs Differential Geometry

Authors: Lu Hoang Chinh

Abstract

Let $(X,\omega)$ be an $n$ -dimensional compact K\"{a}hler manifold. We study degenerate complex Hessian equations of the form $(\omega+dd^c\varphi)^m\wedge \omega^{n-m}=F(x,\varphi)\omega^n.$ Under some natural conditions on $F$ , this equation has a unique continuous solution. When $(X,\omega)$ is rational homogeneous we further show that the solution is H\"{o}lder continuous.

Keywords

kähler manifolds painlevé equations partial differential equations

Cite

@article{arxiv.1202.2436,
  title  = {Solutions to degenerate complex Hessian equations},
  author = {Lu Hoang Chinh},
  journal= {arXiv preprint arXiv:1202.2436},
  year   = {2012}
}

Comments

We slightly modify the proof of Theorem 5.1 and we prove in Proposition 3.20 that the class $\mathcal{P}_m(X,\omega)$ is stable under decreasing sequences

Solutions to degenerate complex Hessian equations

Abstract

Keywords

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