Let Sg be a closed surface of genus g and Mg be the moduli space of Sg endowed with the Weil-Petersson metric. In this paper we investigate the Weil-Petersson curvatures of Mg for large genus g. First, we study the asymptotic behavior of the extremal Weil-Petersson holomorphic sectional curvatures at certain thick surfaces in Mg as g → ∞. Then we prove two curvature properties on the whole spac...

In this paper, we derive a partial result related to a question of 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 bounded con...

Grauert showed that it is possible to construct complete Kähler metrics on the complement of complex analytic sets in a domain holomorphy. In this note, we study holomorphic sectional curvatures such principal divisor $$\mathbb {C}^n$$ , $$n \ge 1$$ . addition, also how metric and its curvature behave when corresponding divisors vary continuously.

In this paper, we extend the general maximum principle in [NT3] to the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we exhibit complete Riemannian manifolds with bounded nonnegative sectional curvature of dimension greater than three such that the Ricci flow does not preserve the nonnegativity of t...

One of the main purpose of this paper is to compare those well-known canonical and complete metrics on the Teichmüller and the moduli spaces of Riemann surfaces. We use as bridge two new metrics, the Ricci metric and the perturbed Ricci metric. We will prove that these metrics are equivalent to those classical complete metrics. For this purpose we study in detail the asymptotic behaviors and th...

Let p : X → S be a smooth Kähler fibration and E → X a Hermitian holomorphic vector bundle. As motivated by the work of Berdtsson([Bern09]), by using basic Hodge theory, we derive several general curvature formulas for the direct image p∗(KX/S ⊗E) for general Hermitian holomorphic vector bundle E in a very simple way. A straightforward application is that, if the Hermitian vector bundle E is Na...