arXiv subjects

Tamás Darvas

Publications and source records attributed to Tamás Darvas.

42 records · Page 3Linked to original sources

arXiv2015

The Mabuchi Completion of the Space of Kähler Potentials

Suppose $(X,ω)$ is a compact Kähler manifold. Following Mabuchi, the space of smooth Kähler potentials $\mathcal H$ can be endowed with a Riemannian structure, which induces an infinite dimensional path length metric space $(\mathcal H,d)$. We prove that the metric completion of $(\mathcal H,d)$ can be identified with $(\mathcal E^2(X,ω),\tilde d)$, and this latter space is a complete non-positively curved geodesic metric space. In obtaining this result, we will rely on envelope techniques which allow for a treatment in a very general context. Profiting from this, we will characterize the pairs of potentials in $\text{PSH}(X,ω)$ that can be connected by weak geodesics and we will also give a characterization of $\mathcal E(X,ω)$ in this context.

math.DG↗

arXiv2015

Geodesic Rays and Kähler-Ricci Trajectories on Fano Manifolds

Suppose $(X,J,ω)$ is a Fano manifold and $t \to r_t$ is a diverging Kähler-Ricci trajectory. We construct a bounded geodesic ray $t \to u_t$ weakly asymptotic to $t \to r_t$, along which Ding's $\mathcal F$-functional decreases, partially confirming a folklore conjecture. In absence of non-trivial holomorphic vector fields this proves the equivalence between geodesic stability of the $\mathcal F$-functional and existence of Kähler-Einstein metrics. We also explore applications of our construction to Tian's $α$-invariant.

math.DG↗

arXiv2014

The Mabuchi Geometry of Finite Energy Classes

We introduce different Finsler metrics on the space of smooth Kähler potentials that will induce a natural geometry on various finite energy classes $\mathcal E_{\tilde χ}(X,ω)$. Motivated by questions raised by R. Berman, V. Guedj and Y. Rubinstein, we characterize the underlying topology of these spaces in terms of convergence in energy and give applications of our results to existence of Kähler-Einstein metrics on Fano manifolds.

math.DG↗

arXiv2014

Kiselman's principle, the Dirichlet problem for the Monge-Ampere equation, and rooftop obstacle problems

First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data. This yields a new relation between solutions of the Dirichlet problem for the homogeneous real and complex Monge-Ampere equations and Kiselman's minimum principle. More generally, it establishes partial regularity for a Bremermann envelope whether or not it solves the Monge-Ampere equation. Second, we prove the second order regularity of the solution of the free-boundary problem for the Laplace equation with a rooftop obstacle, based on a new a priori estimate on the size of balls that lie above the non-contact set. As an application, we prove that convex- and plurisubharmonic-envelopes of rooftop obstacles have bounded second derivatives.

math.AP↗

arXiv2013

Morse theory and geodesics in the space of Kähler metrics

Given a compact Kähler manifold $(X,ω_0)$ let $\mathcal H_{0}$ be the set of Kähler forms cohomologous to $ω_0$. As observed by Mabuchi \cite{m}, this space has the structure of an infinite dimensional Riemannian manifold, if one identifies it with a totally geodesic subspace of $\mathcal H$, the set of Kähler potentials of $ω_0$. Following Donaldson's research program, existence and regularity of geodesics in this space is of fundamental interest. In this paper, supposing enough regularity of a geodesic $u:[0,1]\to \mathcal H$, connecting $u_0 \in \mathcal H$ with $u_1 \in \mathcal H$, we establish a Morse theoretic result relating the critical points of $u_1-u_0$ to the critical points of $\dot u_0 = du/dt|_{t=0}$. As an application of this result, we prove that on all Kähler manifolds, connecting Kähler potentials with smooth geodesics is not possible in general. In particular, in the case $X \neq \Bbb C P^1$, we will also prove that the set of pairs of potentials that can not be connected with smooth geodesics has nonempty interior. This is an improvement upon the findings of \cite{lv} and \cite{dl}.

math.CV↗

arXiv2012

Weak geodesics in the space of Kähler metrics

Given a compact Kähler manifold (X,ω_0), according to Mabuchi, the set of Kähler forms cohomologous to ω_0 has the natural structure of an infinite dimensional Riemannian manifold. We address the question whether points in this space can be joined by a geodesic, and strengthening previous findings of the second author with Vivas, we show that this cannot always be done even with a certain type of generalized geodesics.

math.CV↗