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We prove exponential estimates for plurisubharmonic functions with respect to Monge-Ampère measures with Hölder continuous potential. As an application, we obtain several stochastic properties for the equilibrium measures associated to holomorphic maps on projective spaces. More precisely, we prove the exponential decay of correlations, the central limit theorem for general d.s.h. observables, and the large deviations theorem for bounded d.s.h. observables and Hölder continuous observables.
In this paper we obtain a stability theorem of generalized Kähler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kähler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kähler manifolds which admits deformations to generalized complex manifolds and obtain non-trivial generalized Kähler structures on Fano surfaces and toric Kähler manifolds. In particular, we show that every nonzero holomorphic Poisson structure on a Kähler manifold induces deformations of non-trivial generalized Kähler structures.
In this paper we study the problem of recovering the reflecting surface in a reflector system which consists of a point light source, a reflecting surface, and an object to be illuminated. This problem involves a fully nonlinear partial differential equation of Monge- Ampère type, subject to a nonlinear second boundary condition. A weak solution can be obtained by approximation by piecewise ellipsoidal surfaces. The regularity is a very complicated issue but we found precise conditions for it.
We obtain an estimate for the norm of the second fundamental form of stable H-surfaces in Riemannian 3-manifolds with bounded sectional curvature. Our estimate depends on the distance to the boundary of the surface and on the bound on the sectional curvature but not on the manifold itself. We give some applications, in particular we obtain an interior gradient estimate for H-sections in Killing submersions.