arXiv research

We prove that every projective special Kähler manifold with \emph{regular boundary behaviour} is complete and defines a family of complete quaternionic Kähler manifolds depending on a parameter $c\ge 0$. We also show that, irrespective of its boundary behaviour, every complete projective special Kähler manifold with \e…

We show the existence of complete negative Kähler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler-Einstein metric using the normalized Kähler-Ricci flow.

Kähler manifold loop space inherits Kähler structure and is complete.

The paper studies rigidity results for harmonic forms on Kähler manifolds.

In this paper, we prove a Liouville theorem for holomorphic functions on a class of complete Gauduchon manifolds. This generalizes a result of Yau for complete Kähler manifolds to the complete non-Kähler case.

New vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.

We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component H of a hypersurface {h=1} defined by a homogeneous cubic polynomial such that -d^2 h is a complete Riemannian metric on H defines a complete projective special Kahler manifold and any complete projective special Kahler m…

This paper extends Nevanlinna's unicity theorems to complete Kahler manifolds.

We extend Calabi ansatz over Kähler-Einstein manifolds to Sasaki-Einstein manifolds. As an application we prove the existence of a complete scalar-flat Kähler metric on Kähler cone manifolds over Sasaki-Einstein manifolds. In particular there exists a complete scalar-flat Kähler metric on the toric Kähler cone manifold…

The $L^2$-$\partial\overline\partial$-Lemma is extended to complete Kähler manifolds with a gap in the spectrum.

New theorems compare Laplacian on Kähler manifolds.

Survey on bottom of spectrum of Hodge Laplacian on complete noncompact Kähler manifolds

In this article we study the limiting behavior of the Kähler Ricci flow on complete non-compact Kähler manifolds. We provide sufficient conditions under which a complete non-compact gradient Kähler-Ricci soliton is biholomorphic to $\ce^n$. We also discuss the uniformization conjecture by Yau \cite{Y} for complete non-…

We prove that a complete Kähler manifold with holomorphic curvature bounded between two negative constants admits a unique complete Kähler-Einstein metric. We also show this metric and the Kobayashi-Royden metric are both uniformly equivalent to the background Kähler metric. Furthermore, all three metrics are shown to …

We show that for any complete connected Kähler manifold the index of the group of complex affine transformations in the group of c-projective transformations is at most two unless the Kähler manifold is isometric to complex projective space equipped with a positive constant multiple of the Fubini-Study metric. This est…

Study on scalar-flat Kahler 4-manifolds with a continuous symmetry.

Study extends holomorphic forms on noncompact Kahler manifolds.

Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.

The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.

The paper constructs Einstein metrics on holomorphic bundles.

In this paper we obtain three results concerning the geometry of complete noncompact positively curved Kähler manifolds at infinity. The first one states that the order of volume growth of a complete noncompact Kähler manifold with positive bisectional curvature is at least half of the real dimension (i.e., the complex…

The paper describes geodesics on a Kähler cone of the Heisenberg group.

In this paper we survey the recent developments of the Ricci flows on complete noncompact Kähler manifolds and their applications in geometry.

Study equivalence of metrics on noncompact Kähler manifolds with Bergman kernel properties.

The paper proves conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.

Study proves existence of Kähler-Einstein metrics and Ricci flat Kähler metrics in 4-manifolds.

We determine all complete projective special real surfaces. By the supergravity r-map, they give rise to complete projective special Kähler manifolds of dimension 6, which are distinguished by the image of their scalar curvature function. By the supergravity c-map, the latter manifolds define in turn complete quaternio…

In this paper, we construct local and global solutions to the Kähler-Ricci flow from a non-collapsed Kähler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the Gromov-Hausdorff limit of sequence of complete noncompact non-collapsed Kähler manif…

Paper solves complex equations on noncompact manifolds.

Extends construction of Kähler-Einstein metrics to noncompact manifolds.

Let $M$ be a compact Kähler manifold and $N$ be a subvariety with codimension greater than or equal to 2. We show that there are no complete Kähler--Einstein metrics on $M-N$. As an application, let $E$ be an exceptional divisor of $M$. Then $M-E$ cannot admit any complete Kähler--Einstein metric if blow-down of $E$ is…

This paper classifies Kähler manifolds with specific Einstein-type properties.

Stochastic Schwarz lemma on Kähler manifolds via couplings.

We define a class of geometric flows on a complete Kähler manifold to unify some physical and mechanical models such as the motion equations of vortex filament, complex-valued mKdV equations, derivative nonlinear Schrödinger equations etc. Furthermore, we consider the existence for these flows from $S^1$ into a complet…

We prove compactification theorems for some complete Kähler manifolds with nonnegative Ricci curvature. Among other things, we prove that a complete noncompact Kähler Ricci flat manifold with maximal volume growth and quadratic curvature decay is a crepant resolution of a normal affine algebraic variety. Furthermore, s…

In this article, we prove a Liouville property of holomorphic maps from a complete Kahler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected Kahler manifold with a certain assumption on the sectional curvature.

Finite volume ends found in quaternionic Kähler manifolds.

We classify all complete projective special real manifolds with reducible cubic potential, obtaining four series. For two of the series the manifolds are homogeneous, for the two others the respective automorphism group acts with co-homogeneity one. Complete projective special real manifolds give rise to complete quate…

The paper explores Ricci forms on noncompact complex manifolds, including Kähler-Einstein and canonical metrics.

Researchers create non-homogeneous finite-volume ends on quaternionic Kähler manifolds.

Flow analysis leads to metric completion in Kähler geometry.

The study examines complete Kähler manifolds with nonnegative Ricci curvature and discovers rigidity properties.

We prove that a complete noncompact Kähler manifold $M^{n}$of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {\bf C}$^{n}$ and we show that the manifold is topologically {\bf R}$^{2n}$. In particular, when $M^{n}$ is a Kähler surface of positive bisecti…

Study estimates the first eigenvalue on Kähler manifolds with specific curvature conditions.

In this paper, we study the global Kähler-Ricci flow on a complete non-compact Kähler manifold. We prove the following result. Assume that $(M,g_0)$ is a complete non-compact Kähler manifold such that there is a potential function $f$ of the Ricci tensor, i.e., $R_{i\bar{j}}(g_0)=f_{i\bar{j}}.$ Assume that the quan…

Let $X$ be a compact Kähler manifold and $S$ a subvariety of $X$ with higher co-dimension. The aim is to study complete constant scalar curvature Kähler metrics on non-compact Kähler manifold $X-S$ with Poincaré--Mok--Yau asymptotic property (see Definition \ref{def}). In this paper, the methods of Calabi's ansatz and …

In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds c…

New criterion found for Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.