László Lempert
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View article: Two variational problems in Kähler geometry
Two variational problems in Kähler geometry Open
On a Kähler manifold we consider the problems of maximizing/minimizing Monge--Ampère energy over certain subsets of the space of Kähler potentials. Under suitable assumptions we prove that solutions to these variational problems exist, are…
View article: Aron--Berner--type extension in complex Banach manifolds
Aron--Berner--type extension in complex Banach manifolds Open
Let $S$ be a compact Hausdorff space and $X$ a complex manifold. We consider the space $C(S,X)$ of continuous maps $S\to X$, and prove that any bounded holomorphic function on this space can be continued to a holomorphic function, possibly…
View article: To the geometry of spaces of plurisubharmonic functions on a Kähler manifold
To the geometry of spaces of plurisubharmonic functions on a Kähler manifold Open
Consider a compact Kähler manifold $(X,ω)$ and the space $\cal E(X,ω)=\cal E$ of $ω$--plurisubharmonic functions of full Monge--Ampère mass on it. We introduce a quantity $ρ[u,v]$ to measure the distance between $u, v\in\cal E$; $ρ[u,v]$ i…
View article: On the Bergman kernels of holomorphic vector bundles
On the Bergman kernels of holomorphic vector bundles Open
Consider a very ample line bundle $ E \to X$ over a compact complex manifold, endowed with a hermitian metric of curvature $-i ω$, and the space $\mathcal{O}(E)$ of its holomorphic sections. The Fubini--Study map associates with positive d…
View article: The principle of least action in the space of Kähler potentials
The principle of least action in the space of Kähler potentials Open
Given a compact Kähler manifold, the space $\mathcal H$ of its (relative) Kähler potentials is an infinite dimensional Fréchet manifold, on which Mabuchi and Semmes have introduced a natural connection $\nabla$. We study certain Lagrangian…
View article: On the adjoint action of the group of symplectic diffeomorphisms
On the adjoint action of the group of symplectic diffeomorphisms Open
We study the action of Hamiltonian diffeomorphisms of a compact symplectic manifold ($X,ω$) on $C^\infty(X)$ and on functions $C^\infty(X)\to \mathbb R$. We describe various properties of invariant convex functions on $C^\infty(X)$. Among …
View article: On Riemannian submersions
On Riemannian submersions Open
We prove that the image of a real analytic Riemannian manifold under a smooth Riemannian submersion is necessarily real analytic.
View article: Adapted complex and involutive structures
Adapted complex and involutive structures Open
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
View article: Spaces of Cauchy–Riemann Manifolds
Spaces of Cauchy–Riemann Manifolds Open
This work deals with the embeddability problem for three dimensional, compact, strongly pseudoconvex Cauchy-Riemann (CR) manifolds.Such a CR manifold is given by a compact manifold M without boundary, dim M = 3; a rank two subbundle H C TM…
View article: On complex Legendre duality
On complex Legendre duality Open
Complex Legendre duality is a generalization of Legendre transformation from Euclidean spaces to Kahler manifolds, that Berndtsson and collaborators have recently constructed. It is a local isometry of the space of Kahler potentials. We sh…
View article: Riemannian geometry in infinite dimensional spaces
Riemannian geometry in infinite dimensional spaces Open
We lay foundations of the subject in the title, on which we build in another paper devoted to isometries in spaces of Kähler metrics.
View article: Noncommutative potential theory
Noncommutative potential theory Open
We propose to view hermitian metrics on trivial holomorphic vector bundles $E\toΩ$ as noncommutative analogs of functions defined on the base $Ω$, and curvature as the notion corresponding to the Laplace operator or $\partial\overline\part…
View article: A proof of the Ohsawa–Takegoshi theorem with sharp estimates
A proof of the Ohsawa–Takegoshi theorem with sharp estimates Open
We give a proof of the Ohsawa-Takegoshi extension theorem with sharp estimates. The proof is based on ideas of Blocki to use variations of domains to simplify his proof of the Suita conjecture, and also uses positivity properties of direct…