Bounded properties of curvatures of perturbed Ricci metric on curve moduli

2014 ◽  
Vol 25 (12) ◽  
pp. 1450113
Author(s):  
Xiaorui Zhu
Keyword(s):  
Finite Volume ◽  
Ricci Curvature ◽  
Moduli Space ◽  
Point Of View ◽  
Bisectional Curvature ◽  
Bounded Geometry ◽  
Holomorphic Bisectional Curvature ◽  
Chern Form ◽  
Thick Part ◽  
Kahler Metric

As is well-known, the Weil–Petersson metric ωWP on the moduli space ℳg has negative Ricci curvature. Hence, its negative first Chern form defines the so-called Ricci metric ωτ. Their combination [Formula: see text], C > 0, introduced by Liu–Sun–Yau, is called the perturbed Ricci metric. It is a complete Kähler metric with finite volume. Furthermore, it has bounded geometry. In this paper, we investigate the finiteness of this new metric from another point of view. More precisely, we will prove in the thick part of ℳg, the holomorphic bisectional curvature of [Formula: see text] is bounded by a constant depending only on the thick constant and C0 when C ≥ (3g - 3)C0, but not on the genus g.

scholarly journals THE DEGREE OF HOLOMORPHIC APPROXIMATION ON A TOTALLY REAL SET

2009 ◽  
Vol 79 (1) ◽  
pp. 171-176
Author(s):  
SAID ASSERDA
Keyword(s):  
Compact Support ◽  
Infinite Order ◽  
Holomorphic Functions ◽  
Degree Of Approximation ◽  
Gevrey Class ◽  
Bisectional Curvature ◽  
Bounded Geometry ◽  
Holomorphic Bisectional Curvature ◽  
Open Set ◽  
Totally Real

AbstractLet E be a totally real set on a Stein open set Ω on a complete noncompact Kähler manifold (M,g) with nonnegative holomorphic bisectional curvature such that (Ω,g) has bounded geometry at E. Then every function f in a Cp class with compact support on Ω and $\overline \partial $-flat on E up to order p−1,p≥2 (respectively, in a Gevrey class of order s>1, with compact support on Ω and $\overline \partial $-flat on E up to infinite order) can be approximated on compacts subsets of E by holomorphic functions fk on Ω with degree of approximation equal k−p/2 (respectively, exp (−c(s)k1/2(s−1)) ).

scholarly journals Normal functions and the height of Gross-Schoen cycles

2014 ◽  
Vol 214 ◽  
pp. 53-77 ◽  
Author(s):  
Robin De Jong
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
The Self ◽  
Compact Type ◽  
Stable Curves ◽  
Normal Functions ◽  
Chern Form ◽  
Bloch Line ◽  
Pointed Curve ◽  
Dualizing Sheaf

AbstractWe prove a variant of a formula due to Zhang relating the Beilinson– Bloch height of the Gross–Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach, the height of the Gross–Schoen cycle occurs as the degree of a suitable Bloch line bundle. We show that the Chern form of this line bundle is nonnegative, and we calculate its class in the Picard group of the moduli space of pointed stable curves of compact type. The basic tools are normal functions and biextensions associated to the cohomology of the universal Jacobian.

scholarly journals Hyperbolic Unfoldings of Minimal Hypersurfaces

2018 ◽  
Vol 6 (1) ◽  
pp. 96-128 ◽  
Author(s):  
Joachim Lohkamp
Keyword(s):  
Point Of View ◽  
Hyperbolic Spaces ◽  
Intrinsic Geometry ◽  
Minimal Hypersurfaces ◽  
Bounded Geometry ◽  
New Concepts ◽  
Gromov Hyperbolic ◽  
Area Minimizing ◽  
Conformal Deformations ◽  
Gromov Boundary

Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\Ʃ. These are canonical conformal deformations of H\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.

scholarly journals Normal functions and the height of Gross-Schoen cycles

2014 ◽  
Vol 214 ◽  
pp. 53-77
Author(s):  
Robin De Jong
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
The Self ◽  
Compact Type ◽  
Stable Curves ◽  
Normal Functions ◽  
Chern Form ◽  
Bloch Line ◽  
Pointed Curve ◽  
Dualizing Sheaf

AbstractWe prove a variant of a formula due to Zhang relating the Beilinson– Bloch height of the Gross–Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach, the height of the Gross–Schoen cycle occurs as the degree of a suitable Bloch line bundle. We show that the Chern form of this line bundle is nonnegative, and we calculate its class in the Picard group of the moduli space of pointed stable curves of compact type. The basic tools are normal functions and biextensions associated to the cohomology of the universal Jacobian.

Sign in / Sign up

Export Citation Format

Share Document