Complex Hessian Equation on K\"ahler Manifold

By Z. Hou. 2008

60 citations recorded by the source corpus.

In this paper, complex Hessian equation over K\"ahler manifold was studied. Under the condition that the underline K\"ahler manifold has non-negative holomorphic bisectional curvature, the existence and regularity of the solution was proved.

Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b65d9a74af0ac2d50b4bd946602ed759a98be3c7

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In Applied Mathematics

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

A second order estimate for complex Hessian equations on a compact Kähler manifold (2010, background): “2); he [8] also proved the existence of a smooth admissible solution of (1.”
Degenerate complex Hessian equations on compact K\"ahler manifolds (2014, methodology): “In [25] and [23] the authors independently proved that equation (1.”
A second order estimate for general complex Hessian equations (2015, background): “The corresponding problem on compact Kähler or Hermitian manifolds has also been studied extensively; see, for example, [Dinew and Kołodziej 2014; Hou 2009; Kołodziej and Nguyen 2016; Lu and Nguyen 2015; Zhang 2010].”
A priori estimates for complex Hessian equations (2011, methodology): “and consider the Hessian equation on a compact Kähler manifold X , as in [Hou 2009; Hou et al. 2010; Dinew and Kołodziej 2012]:”
The Monge-Ampère equation for (n − 1)-plurisubharmonic functions on a compact Kähler manifold (2013, methodology): “11) (ω + √ −1∂∂u) ∧ ω = eω, for fixed 2 6 k 6 n− 1, was proved by Dinew-Ko lodziej [11], who used the second order estimate of Hou-Ma-Wu [31] (see also [3, 10, 30, 32, 33, 34, 36, 39, 59] for earlier and related work on these equations).”
Liouville and Calabi-Yau type theorems for complex Hessian equations (2012, methodology): “.1) with some strictly positive functions fj satisfying ||f 1/m j ||C2(X) ≤ C for some uniform constant C (in particular the uniform norm of fj is also under control and so is the uniform norm of uj ([Hou], [DK])), yet Cj := supX|∇uj| → ∞. After passing to a subsequence one may assume that the point”