Related papers: On holomorphic functions on negatively curved mani…

In this paper we extend Yau's celebrated Liouville theorem to the biharmonic case. Namely, we show that in a complete Riemannian manifold with a pole and nonnegative Ricci curvature, any biharmonic function of subquadratic growth must be…

We prove the following results: An almost Hermitian manifold of indefinite metric is of pointwise constant holomorphic sectional curvature if the holomorphic sectional curvature is bounded from above and from below. If the antiholomorphic…

In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding's method in [13]. With Cheeger-Colding theory, we obtain the Laplacian comparison…

On a compact K\"ahler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a K\"ahler metric with semi-negative holomorphic sectional curvature.…

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity result under a given inequality involving the Weyl curvature and the traceless Ricci curvature. Moveover, under an inequality involving…

Let Y be an infinite covering space of a projective manifold M in P^N of dimension n geq 2. Let C be the intersection with M of at most n-1 generic hypersurfaces of degree d in P^N. The preimage X of C in Y is a connected submanifold. Let…

In this paper we show that, under some curvature assumptions the integral of distance function on a compact Riemannian manifold is bounded below by the product of diameter, volume and a constant only depending on the dimension.

For a complete noncompact Riemannian manifold with nonnegative Ricci curvature, we show that bounded biharmonic functions are constant and the space consists of biharmonic functions with polynomial growth of a fixed rate is finite…

In the present paper, we show that given a compact K\"ahler manifold $(X,\omega)$ with a K\"ahler metric $\omega$, and a complex submanifold $V\subset X$ of positive dimension, if $V$ has a holomorphic retraction structure in $X$, then any…

The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(\lambda) = \exp(i(\lambda_1\psi_1 + ... + \lambda_n\psi_n)) for \lambda in R^n, gives rise to a manifold M of orthogonal projections…

A recent theorem of Diverio--Trapani and Wu--Yau asserts that a compact K\"ahler manifold with a K\"ahler metric of quasi-negative holomorphic sectional curvature is projective and canonically polarized. This confirms a long-standing…

In his recent work \cite{Y1}, X. Yang proved a conjecture raised by Yau in 1982 (\cite{Yau82}), which states that any compact K\"{a}hler manifold with positive holomorphic sectional curvature must be projective. In this note, we prove that…

This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions.…

Consider the moduli functor of canonically polarized complex manifolds with Hilbert polynomial h, and let M_h be the corresponding coarse quasi-projective moduli scheme. We show that M_h is Brody hyperbolic in the following sense: Assume…

This paper proves that the universal covering of a compact K\"{a}hler manifold with small positive sectional curvature in a certain sense is contractible.

We prove that any complete non-compact K\"ahler surface with positive sectional curvature is biholomorphic to $\mathbb{C}^2$, establishing the two dimensional case of the weaker form of Yau's uniformisation conjecture. In contrast to all…

It is shown that a contraction on a Hilbert space is complex symmetric if and only if the values of its characteristic function are all symmetric with respect to a fixed conjugation. Applications are given to the description of complex…

We show that non-collapsed Gromov-Hausdorff limits of polarized Kahler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union…

A known Hardy-Littlewood theorem asserts that if both the function and its conjugate are of bounded variation, then their Fourier series are absolutely convergent. It is proved in the paper that the same result holds true for functions on…

The solution of the Calabi Conjecture by Yau implies that every K\"ahler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $\textrm{SU}(n)$, and that these metrics are parametrized by the positive cone in…