In this paper, we derive a partial result related to a question of Professor Yau: “Does a simply-connected complete Kähler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?” Main Theorem. Let M be a simply-connected complete Kähler manifold M with negative sectional curvature ≤ −1 and S∞(M) be the sphere at infinity of M . Then there is an explicit ...

In this note, we will show that the fundamental group of any negatively δ-pinched (δ > 14) manifold can’t be the fundamental group of a quasi-compact Kähler manifold. As a consequence of our proof, we also show that any nonuniform lattice in F4(−20) cannot be the fundamental group of a quasi-compact Kähler manifold. The corresponding result for uniform lattices was proved by Carlson and Hernánd...

We prove that a complete noncompact Kähler manifold Mof positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of C and we show that the manifold is topologically R2n. In particular, when M is a Kähler surface of positive bisectional curvature satisfying certain natural geometric growth conditions, it is biholomorphic to C2.

In this short note, we give a new proof of a theorem of Arezzo-Tian on the existence of smooth geodesic rays tamed by a special degeneration. In [7], S. K. Donaldson proposed an ambitious program to tackle the problem of the existence and uniqueness of extremal metrics on a Kähler manifold from the perspective of the infinite dimensional space of Kähler potentials. He observed that the existenc...

We have developed N = 1 supersymmetric nonlinear realization methods, which realize global symmetry breaking in N = 1 supersymmetric theories. The target space of nonlinear sigma models with a linear model origin is a Gorbit, which is a non-compact non-homogeneous Kähler manifold. We show that, if and only if the orbit is open, it includes a compact homogeneous Kähler manifold as a submanifold,...

In this paper we begin the study of polysymplectic manifolds, and of their relationship with PDE’s. This notion provides a generalization of symplectic manifolds which is very well suited for the geometric study of PDE’s with values in a smooth manifold. Some of the standard tools of analytical mechanics, such as the Legendre transformation and Hamilton’s equations, are shown to generalize to t...

A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kähler covering M̃ , with the deck transform acting on M̃ by homotheties. If compact, it admits a canonical Vaisman metric, due to Gauduchon. We show that a Hermitian Einstein-Weyl structure on a compact complex manifold is determined by its volume form. This result is a conformal analogue of Calabi’s theorem stating ...

The results of this paper are related to a well-known problem, attributed to Hopf, to the effect that the Euler number χ(M2n) of a compact Riemannian manifold M2n of negative sectional curvature must satisfy the inequality (−1)nχ(M2n) > 0. This conjecture is true in dimensions 2 and 4 [Ch] and it has been verified in the Kähler case for all n by Gromov [G] and Stern [S] (the work in [S] also us...

We study when Calabi-Yau supermanifolds M with one complex bosonic coordinate and two complex fermionic coordinates are super Ricci-flat, and find that if the bosonic manifold is compact, it must have constant scalar curvature. In [1], we found that super Ricci-flat Kähler manifolds with one fermionic dimension and an arbitrary number of bosonic dimensions exist above a bosonic manifold with a ...

In this note we briefly summarize the necessary tools from complex manifold theory in order to give an introduction to the basic results about Kähler and locally conformal Kähler manifolds. We look at many equivalent definitions of all three types of manifolds and examine in detail the parallels which arise in the theories. These parallels include the compatibility of metrics and fundamental 2-...