• Corpus ID: 204734616

High energy harmonic maps and degeneration of minimal surfaces

@article{Ouyang2019HighEH,
  title={High energy harmonic maps and degeneration of minimal surfaces},
  author={Charles Ouyang},
  journal={arXiv: Differential Geometry},
  year={2019}
}
  • Charles Ouyang
  • Published 15 October 2019
  • Mathematics
  • arXiv: Differential Geometry
Let $S$ be a closed surface of genus $g \geq 2$ and let $\rho$ be a maximal $\mathrm{PSL}(2, \mathbb{R}) \times \mathrm{PSL}(2, \mathbb{R})$ surface group representation. By a result of Schoen, there is a unique $\rho$-equivariant minimal surface $\widetilde{\Sigma}$ in $\mathbb{H}^{2} \times \mathbb{H}^{2}$. We study the induced metrics on these minimal surfaces and prove the limits are precisely mixed structures. In the second half of the paper, we provide a geometric interpretation: the… 

Length spectrum compactification of the $\mathrm{SO}_{0}(2,3)$-Hitchin component

We find a compactification of the $\mathrm{SO}_{0}(2,3)$-Hitchin component by studying the degeneration of the induced metric on the unique equivariant maximal surface in the 4-dimensional

Limits of Blaschke metrics

We find a compactification of the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component by studying the degeneration of the Blaschke metrics on the associated equivariant affine spheres. In the process, we

Riemannian metrics on the moduli space of GHMC anti-de Sitter structures

In this short note we explain how to adapt the construction of two Riemannian metrics on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component to the deformation space of globally hyperbolic anti-de

A closed ball compactification of a maximal component via cores of trees

We show that, in the character variety of surface group representations into the Lie group PSL(2,R) × PSL(2,R), the compactification of the maximal component introduced by the second author is a

Boundary of the Gothen components

In this short note we describe an interesting new phenomenon about the Sp(4,R)-character variety. Precisely, we show that the Hitchin component and all Gothen components share the same boundary in

References

SHOWING 1-10 OF 47 REFERENCES

On univalent harmonic maps between surfaces

Hence the energy defines a functional on the space of Lipshitz maps between M and M'. Critical points of this functional are called harmonic maps. These maps were studied by Bochner, Morrey, Rauch,

Cyclic surfaces and Hitchin components in rank 2

We prove that given a Hitchin representation in a real split rank 2 group $\mathsf G_0$, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we

Core and intersection number for group actions on trees

Elliptic Partial Differential Equations of Second Order

We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations

Character varieties and harmonic maps to R-trees

We show that the Korevaar-Schoen limit of the sequence of equivariant harmonic maps corresponding to a sequence of irreducible SL2(C) representations of the fundamental group of a compact Riemannian

Sobolev spaces and harmonic maps for metric space targets

When one studies variational problems for maps between Riemannian manifolds one must consider spaces which we denote Vr'(r2,X). Here ft is a compact domain in a Riemannian manifold, X is a second

Minimal surfaces and particles in 3-manifolds

We consider 3-dimensional anti-de Sitter manifolds with conical singularities along time-like lines, which is what in the physics literature is known as manifolds with particles. We show that the

On harmonic maps

TLDR
This work highlights the key questions of existence, uniqueness and regularity of harmonic maps between given manifolds, and surveys some of the main methods of global analysis for answering these questions.

Extremal length geometry of teichmüller space

Assume τ is a point in the Teichmuller space of a Riemann surface which is compact or obtainable from a compact surface by deleting a finite number of punctures. Let be extermal lengths of two