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1. CJM 2013 (vol 66 pp. 1413)
Generalized KÃ¤hler--Einstein Metrics and Energy Functionals In this paper, we consider a generalized
KÃ¤hler-Einstein equation on KÃ¤hler manifold $M$. Using the
twisted $\mathcal K$-energy introduced by Song and Tian, we show
that the existence of generalized KÃ¤hler-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
KÃ¤hler-Einstein metric implies a Moser-Trudinger type inequality.
Keywords:complex Monge--AmpÃ¨re equation, energy functional, generalized KÃ¤hler--Einstein metric, Moser--Trudinger type inequality Categories:53C55, 32W20 |
2. CJM 2009 (vol 62 pp. 3)
On the Asymptotic Behavior of Complete KÃ¤hler Metrics of Positive Ricci Curvature Let $( X,g) $ be a complete noncompact KÃ¤hler 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 KÃ¤hler 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 |
3. CJM 2000 (vol 52 pp. 757)
Le problÃ¨me de Neumann pour certaines Ã©quations du type de Monge-AmpÃ¨re sur une variÃ©tÃ© riemannienne |
Le problÃ¨me de Neumann pour certaines Ã©quations du type de Monge-AmpÃ¨re sur une variÃ©tÃ© 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, Ã©quations de Monge-AmpÃ¨re, problÃ¨me de Neumann, estimÃ©es a priori, mÃ©thode de continuitÃ© Categories:35J60, 53C55, 58G30 |