• 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… 
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References

SHOWING 1-10 OF 43 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,
Thurston's Work on Surfaces
This book is an exposition of Thurston’s theory of surfaces: measured foliations, the compactification of Teichmuller space and the classification of diffeomorphisms. The mathematical content is
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
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
...
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