Related papers: Recent progress in K\"ahler geometry
Let $(M,J,\Omega)$ be a closed polarized complex manifold of K\"ahler type. Let $G$ be the maximal compact subgroup of the automorphism group of $(M,J)$. On the space of K\"ahler metrics that are invariant under $G$ and represent the…
In this work, we first establish short time existence and Shi's type estimate of second Ricci flow on complete noncompact Hermitian manifolds. As an application, we use the second Ricci flow to discuss the existence of Kaehler-Einstein…
Over the past two decades the theory of the Weil-Petersson metric has been extended to general Teichm\"uller spaces of infinite type, including for example the universal Teichm\"uller space. In this paper we give a survey of the main…
In this survey article, we discuss some recent progress on geometric analysis on manifold with ends. In the final section, we construct manifolds with ends with oscillating volume functions which may turn out to have a different heat kernel…
I survey some of the developments in the theory of Ricci flow and its applications from the past decade. I focus mainly on the understanding of Ricci flows that are permitted to have unbounded curvature in the sense that the curvature can…
We improve the understanding of both finite time and infinite time singularities of the modified K\"ahler-Ricci flow as initiated by the second author of this paper in [26]. This is done by relating the modified K\"ahler-Ricci flow with the…
Given a semi-Riemannian $4$-manifold $(M,g)$ with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of K\"ahler metrics $g_K$ is constructed, defined on an…
We use the transverse K\"ahler-Ricci flow on the canonical foliation of a closed Vaisman manifold to deform the Vaisman metric into another Vaisman metric with a transverse K\"ahler-Einstein structure. We also study the main features of…
We prove a compactness theorem for K\"ahler metrics with various bounds on Ricci curvature and the $\mathcal I$ functional. We explore applications of our result to the continuity method and the Calabi flow.
This paper is a survey of some recent progress on the study of Calabi's extremal K\"ahler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we…
We consider the space of Kahler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are…
In this paper, we define a class of new geometric flows on a complete Riemannian manifold. The new flow is related to the generalized (third order) Landau-Lifishitz equation. On the other hand it could be thought of a special case of the…
The existence of K\"ahler-Einstein metrics on a compact K\"ahler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has…
In this paper, we construct a set of new functionals of Ricci curvature on any Kaehler manifolds which are invariant under holomorphic transfermations in Kaehler Einstein manifolds and essentially decreasing under the Kaehler Ricci flow.…
We give an up-to-date overview of geometric and topological properties of cosymplectic and coKaehler manifolds. We also mention some of their applications to time-dependent mechanics.
In this expository paper we review on the existence problem of Einstein-Maxwell K\"ahler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein-Maxwell K\"ahler metrics, and give extensions of…
We give an overview of some recent interactions between the geometry of K3 surfaces and their Ricci-flat Kahler metrics and the dynamical study of K3 automorphisms with positive entropy.
We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a…
Given a compact K\"ahler manifold (X,\omega_0), according to Mabuchi, the set of K\"ahler forms cohomologous to \omega_0 has the natural structure of an infinite dimensional Riemannian manifold. We address the question whether points in…
These lecture notes give an introduction to the Kahler-Ricci flow. They are based on lectures given by the authors at the conference "Analytic Aspects of Complex Algebraic Geometry", held at the Centre International de Rencontres…