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Search: MSC category 53C55 ( Hermitian and Kahlerian manifolds [See also 32Cxx] )

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1. CJM Online first

Yuan, Rirong
On a class of fully nonlinear elliptic equations containing gradient terms on compact Hermitian manifolds
In this paper we study a class of second order fully nonlinear elliptic equations containing gradient terms on compact Hermitian manifolds and obtain a priori estimates under proper assumptions close to optimal. The analysis developed here should be useful to deal with other Hessian equations containing gradient terms in other contexts.

Keywords:Sasakian manifold, Hermitian manifold, subsolution, extra concavity condition, fully nonlinear elliptic equation containing gradient term on complex manifold
Categories:35J15, 53C55, 53C25, 35J25

2. CJM 2016 (vol 69 pp. 220)

Zheng, Tao
The Chern-Ricci Flow on Oeljeklaus-Toma Manifolds
We study the Chern-Ricci flow, an evolution equation of Hermitian metrics, on a family of Oeljeklaus-Toma (OT-) manifolds which are non-Kähler compact complex manifolds with negative Kodaira dimension. We prove that, after an initial conformal change, the flow converges, in the Gromov-Hausdorff sense, to a torus with a flat Riemannian metric determined by the OT-manifolds themselves.

Keywords:Chern-Ricci flow, Oeljeklaus-Toma manifold, Calabi-type estimate, Gromov-Hausdorff convergence
Categories:53C44, 53C55, 32W20, 32J18, 32M17

3. CJM 2013 (vol 66 pp. 1413)

Zhang, Xi; Zhang, Xiangwen
Generalized Kähler--Einstein Metrics and Energy Functionals
In this paper, we consider a generalized Kähler-Einstein equation on Kähler manifold $M$. Using the twisted $\mathcal K$-energy introduced by Song and Tian, we show that the existence of generalized Kähler-Einstein metrics with semi-positive twisting $(1, 1)$-form $\theta $ is also closely related to the properness of the twisted $\mathcal K$-energy functional. Under the condition that the twisting form $\theta $ is strictly positive at a point or $M$ admits no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of generalized Kähler-Einstein metric implies a Moser-Trudinger type inequality.

Keywords:complex Monge--Ampère equation, energy functional, generalized Kähler--Einstein metric, Moser--Trudinger type inequality
Categories:53C55, 32W20

4. CJM 2009 (vol 62 pp. 3)

Anchouche, Boudjemâa
On the Asymptotic Behavior of Complete Kähler Metrics of Positive Ricci Curvature
Let $( X,g) $ be a complete noncompact Kähler manifold, of dimension $n\geq2,$ with positive Ricci curvature and of standard type (see the definition below). N. Mok proved that $X$ can be compactified, \emph{i.e.,} $X$ is biholomorphic to a quasi-projective variety$.$ The aim of this paper is to prove that the $L^{2}$ holomorphic sections of the line bundle $K_{X}^{-q}$ and the volume form of the metric $g$ have no essential singularities near the divisor at infinity. As a consequence we obtain a comparison between the volume forms of the Kähler metric $g$ and of the Fubini--Study metric induced on $X$. In the case of $\dim_{\mathbb{C} }X=2,$ we establish a relation between the number of components of the divisor $D$ and the dimension of the groups $H^{i}( \overline{X}, \Omega_{\overline{X}}^{1}( \log D) )$.

Categories:53C55, 32A10

5. CJM 2000 (vol 52 pp. 757)

Hanani, Abdellah
Le problème de Neumann pour certaines équations du type de Monge-Ampère sur une variété riemannienne
Let $(M_n,g)$ be a strictly convex riemannian manifold with $C^{\infty}$ boundary. We prove the existence\break of classical solution for the nonlinear elliptic partial differential equation of Monge-Amp\`ere:\break $\det (-u\delta^i_j + \nabla^i_ju) = F(x,\nabla u;u)$ in $M$ with a Neumann condition on the boundary of the form $\frac{\partial u}{\partial \nu} = \varphi (x,u)$, where $F \in C^{\infty} (TM \times \bbR)$ is an everywhere strictly positive function satisfying some assumptions, $\nu$ stands for the unit normal vector field and $\varphi \in C^{\infty} (\partial M \times \bbR)$ is a non-decreasing function in $u$.

Keywords:connexion de Levi-Civita, équations de Monge-Ampère, problème de Neumann, estimées a priori, méthode de continuité
Categories:35J60, 53C55, 58G30

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