12
12

Jun 28, 2018
06/18

by
Slawomir Kolodziej; Ngoc Cuong Nguyen

texts

#
eye 12

#
favorite 0

#
comment 0

We prove the existence of weak solutions of complex $m-$Hessian equations on compact Hermitian manifolds for the nonnegative right hand side belonging to $L^p, p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has been recently solved by Szekelyhidi and Zhang. We also give a stability result for such solutions.

Topics: Differential Geometry, Mathematics, Complex Variables

Source: http://arxiv.org/abs/1507.06755

60
60

Sep 20, 2013
09/13

by
Vincent Guedj; Slawomir Kolodziej; Ahmed Zeriahi

texts

#
eye 60

#
favorite 0

#
comment 0

We study the regularity of solutions to complex Monge-Amp\`ere equations $(dd^c u)^n=f dV$, on bounded strongly pseudoconvex domains $ \Omega \subset \C^n$. We show, under a mild technical assumption, that the unique solution $u$ to such an equation is H\"older continuous if the boundary values $\phi$ are H\"older continuous and the density $f$ belongs to $L^p(\Omega)$ for some $p>1$. This improves previous results by Bedford-Taylor and Kolodziej.

Source: http://arxiv.org/abs/math/0607314v1

57
57

Sep 19, 2013
09/13

by
Slawomir Dinew; Slawomir Kolodziej

texts

#
eye 57

#
favorite 0

#
comment 0

We discuss pluripotential aspects of the Monge-Amp\`ere equations on compact Hermitian manifolds and prove $L^{\infty}$ estimates for any metric, as well as the existence of weak solutions under an extra assumption.

Source: http://arxiv.org/abs/0910.3937v1

61
61

Sep 23, 2013
09/13

by
Slawomir Kolodziej; Gang Tian

texts

#
eye 61

#
favorite 0

#
comment 0

We prove uniform sup-norm estimates for the Monge-Ampere equation with respect to a family of Kahler metrics which degenerate towards a pull-back of a metric from a lower dimensional manifold. This is then used to show the existence of generalized Kahler-Einstein metrics as the limits of the Kahler-Ricci flow for some holomorphic fibrations (in the spirit of Song and Tian "The Kahler-Ricci flow on surfaces of positive Kodaira dimension", arXiv:math/0602150).

Source: http://arxiv.org/abs/0710.1144v1

253
253

Jul 20, 2013
07/13

by
Slawomir Dinew; Slawomir Kolodziej

texts

#
eye 253

#
favorite 0

#
comment 0

We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex Hessian equation on a compact K\"ahler manifold. This terminates the program, initiated by Hou, Ma and Wu, of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak...

Source: http://arxiv.org/abs/1203.3995v1

70
70

Sep 23, 2013
09/13

by
Jean-Pierre Demailly; Slawomir Dinew; Vincent Guedj; Hoang Hiep Pham; Slawomir Kolodziej; Ahmed Zeriahi

texts

#
eye 70

#
favorite 0

#
comment 0

Let $(X,\omega)$ be a compact K\"ahler manifold. We obtain uniform H\"older regularity for solutions to the complex Monge-Amp\`ere equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $\MAH(X,\omega)$ of the complex Monge-Amp\`ere operator acting on $\omega$-plurisubharmonic H\"older continuous functions. We show that this set is convex, by sharpening Ko{\l}odziej's...

Source: http://arxiv.org/abs/1112.1388v1

57
57

Sep 24, 2013
09/13

by
Slawomir Dinew; Slawomir Kolodziej

texts

#
eye 57

#
favorite 0

#
comment 0

We prove some $L^{\infty}$ a priori estimates as well as existence and stability theorems for the weak solutions of the complex Hessian equations in domains of $C^n$ and on compact K\"ahler manifolds. We also show optimal $L^p$ integrability for m-subharmonic functions with compact singularities, thus partially confirming a conjecture of Blocki. Finally we obtain a local regularity result for $W^{2,p}$ solutions of the real and complex Hessian equations under suitable regularity...

Source: http://arxiv.org/abs/1112.3063v1

70
70

Sep 18, 2013
09/13

by
Sławomir Kołodziej

texts

#
eye 70

#
favorite 0

#
comment 0

We prove that on compact K\"ahler manifolds solutions to the complex Monge-Amp\`ere equation, with the the right hand side in $L^p, p>1,$ are H\"older continuous.

Source: http://arxiv.org/abs/math/0611051v1