This thesis deals with the optimal transport problem, in particular with regularity properties shared by optimal transport maps. The first part of this manuscript provides a new proof of the Caffarelli contraction theorem, stating that the optimal transport map from the gaussian measure to a measure with a uniformly log-concave density with respect...
Let (X, ω) be a compact Hermitian manifold of complex dimension n. I shall discuss some recent results concerning weak solutions to the complex Monge-Ampère equation and, the more general, complex Hessian equation (ω+ddc φ)k∧ωn−k =cfωn,(where 0≤f belongs to some Lp space) including existence, stability and H ̈older continuity. They were obtained in...
Given a Kahler manifold (X, ω), finding smooth solutions to the equation (ø +i∂ ̄∂u)n=føn goes back to Yau’s solution of the Calabi conjecture in the seventies. In joint work with E. Di Nezza and C.H. Lu, we proposed to solve this same equation with the added constraint that u∈PSH(X, ω) has prescribed singularity type. As it turns out, this problem...
Published in Geometric and Functional Analysis
We obtain a necessary and sufficient condition of existence of a Kähler–Einstein metric on a G × G-equivariant Fano compactification of a complex connected reductive group G in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the continuity method and its translation in...
The main result of this work is a necessary and sufficient condition for the existence of a Kähler-Einstein metric on a smooth and Fano bi-equivariant compactification of a complex connected reductive group. Examples of such varieties include wonderful compactifications of adjoint semisimple groups.The tools needed to study the existence of Kähler-...
Let us consider a projective manifold and Ω a volume form. We define the gradient flow associated to the problem of Ω-balanced metrics in the quantum formalism, the \Omega−balacing flow.At the limit of the quantization,we prove that the \Omega$-balancing flow converges towards a natural flow in K\"ahler geometry, the$\Omega$-K\"ahler flow. We study...
Published in Analysis
In this paper we consider fully nonlinear elliptic equations of the form including the Monge–Ampère, the Hessian and the Weingarten equations and give conditions which ensure that a singular set E of a solution u is removable. This is shown by proving suitable maximum principles along the lines of the Aleksandrov–Bakel´man maximum principle.
We study the Navier-Stokes and Euler equations of incompressible hydrodynamics in two and three spatial dimensions and show how the constraint of incompressiblility leads to equations of Monge--Amp\ère type for the stream function, when the Laplacian of the pressure is known. In two dimensions a K\\ähler geometry is described, which is associated w...
Published in Advances in Mathematics
In dimension n ⩾ 3 , we define a generalization of the classical two-dimensional partial Legendre transform, that reduces interior regularity of the generalized Monge–Ampère equation det D 2 u = k ( x , u , Du ) to regularity of a divergence form quasilinear system of special form. This is then used to obtain smoothness of C 2 , 1 solutions, having...
