Plurisuperharmonicity of reciprocal energy function on Teichmüller space and Weil-Petersson metric

@article{Kim2019PlurisuperharmonicityOR,
  title={Plurisuperharmonicity of reciprocal energy function on Teichm{\"u}ller space and Weil-Petersson metric},
  author={Inkang Kim and Xueyuan Wan and Genkai Zhang},
  journal={arXiv: Differential Geometry},
  year={2019}
}
We consider harmonic maps$u(z): \mathcal{X}_z\to N$ in a fixed homotopy class from Riemann surfaces $\mathcal{X}_z$ of genus $g\geq 2$ varying in the Teichmu{}ller space $\mathcal T$ to a Riemannian manifold $N$ with non-positive Hermitian sectional curvature. The energy function $E(z)=E(u(z))$ can be viewed as a function on $\mathcal T$ and we study its first and the second variations. We prove that the reciprocal energy function $E(z)^{-1}$ is plurisuperharmonic on Teichmuller space. We also… Expand
Second Variation of Energy Functions associated to Families of Canonically Polarized Manifolds
Let $\pi:\mathcal{X}\to S$ be a holomorphic family of canonically polarized manifolds over a complex manifold $S$, and $f:\mathcal{X}\to N$ a smooth map into a Riemannian manifold $N$. Consider theExpand
Convexity of energy function associated to the harmonic maps between surfaces
For a fixed smooth map $u_0$ between two Riemann surfaces $\Sigma$ and $S$ with non-zero degree, we consider the energy function on Teichmuller space $\mc{T}$ of $\Sigma$ that assigns to a complexExpand
Plurisubharmonicity of the Dirichlet energy and deformations of polarized manifolds
  • Che-Hung Huang
  • Mathematics
  • 2021
We show that if {Mt}t∈∆ is a polarized family of compact Kähler manifolds over the open unit disk ∆, if N is a Riemannian manifold of nonpositive complexified sectional curvature, and if {φt ∶ Mt →Expand

References

SHOWING 1-10 OF 39 REFERENCES
Plurisubharmonicity and geodesic convexity of energy function on Teichmüller space
Let $\pi:\mc{X}\to \mc{T}$ be Teichm\"uller curve over Teichm\"uller space $\mc{T}$, such that the fiber $\mc{X}_z=\pi^{-1}(z)$ is exactly the Riemann surface given by the complex structure $z\inExpand
Convexity of energy function associated to the harmonic maps between surfaces
For a fixed smooth map $u_0$ between two Riemann surfaces $\Sigma$ and $S$ with non-zero degree, we consider the energy function on Teichmuller space $\mc{T}$ of $\Sigma$ that assigns to a complexExpand
The moduli space of Riemann surfaces is Kähler hyperbolic
Let $\cM_{g,n}$ be the moduli space of Riemann surfaces of genus $g$ with $n$ punctures. From a complex perspective, moduli space is hyperbolic. For example, $\cM_{g,n}$ is abundantly populated byExpand
Positivity of relative canonical bundles and applications
Given a family $f:\mathcal{X} \to S$ of canonically polarized manifolds, the unique Kähler–Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundleExpand
A new proof that Teichmüller space is a cell
A new proof is given, using the energy of a harmonic map, that Teichmüller space is a cell. In [2] the authors developed a new approach to Teichmüller's famous theorem on the dimension of theExpand
Variation of geodesic length functions in families of Kähler-Einstein manifolds and applications to Teichmüller space
In the study of Teichmuller spaces the second variation of the logarithm of the geodesic length function plays a central role. So far, it was accessible only in a rather indirect way. We treat theExpand
Deformations of metrics and associated harmonic maps
In this paper we establish a theorem ((3.1) below) concerning the dependence of harmonic maps on the Riomatmian metrics used to define them. Our method is aa application of the implicit functionExpand
Hermitian Curvature and Plurisubharmonicity of Energy on Teichmüller Space
Let M be a closed Riemann surface, N a Riemannian manifold of Hermitian non-positive curvature, f : M → N a continuous map, and E the function on the Teichmüller space of M that assigns to a complexExpand
Canonical Metrics on the Moduli Space of Riemann Surfaces II
One of the main purpose of this paper is to compare those well-known canonical and complete metrics on the Teichmuller and the moduli spaces of Riemann surfaces. We use as bridge two new metrics, theExpand
Liouville Action and Weil-Petersson Metric on Deformation Spaces, Global Kleinian Reciprocity and Holography
AbstractWe rigorously define the Liouville action functional for the finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associatedExpand
...
1
2
3
4
...