Introduction
Questions of interest
Indeed, there are a number of conjectures about the geometry of harmonic maps for Hitchin representations [Li19, Part 3]. The Labourie conjecture is already intriguing from the perspective of harmonic maps and minimal surfaces.
Overview of thesis and main results
2(𝛾), and the analysis of [Sag19] shows that the total energy of the harmonic maps in the collar behaves as . As in the introductory chapter, 𝑇 𝑀C=𝑇 𝑀⊗C is the complexification of the tangent bundle of 𝑀 and E := 𝑓∗𝑇 𝑀C is the pullback bundle.
Preliminaries
Harmonic maps
If the target is flat, equation (2.4) is reduced to the ordinary Laplace equation. -Sampson developed the heat flux method to prove the existence of harmonic maps in non-positive curvature [ES64].
Harmonic maps of Riemann surfaces
For now, assume that (𝑀 , 𝜈) is also a hyperbolic Riemann surface and that 𝜈 is compatible in the sense that if 𝑤 is a holomorphic coordinate on 𝑀, then 𝜈 = 𝜈(𝑤) |𝑑 𝑤|2. We consider expressions. In the coordinates, 𝐻 and 𝐿 are equal to the functions described by (2.8) in these coordinates, if we choose the orientations correctly.
Teichmüller theory
Let𝜂be a harmonic Beltrami form on (Σ, 𝜇), and let𝜙be the Hopf differential of the harmonic map at𝜇. Assume that 𝑀 is negatively curved and let 𝑓 : (Σ, 𝜇) → (𝑀 , 𝜈) be the unique harmonic map in the homotopy class of the identity.
The non-abelian Hodge correspondence
By a small and well-understood modification of the proof of the Courant-Lebesque lemma [Jos84, Lemma 3.1], we obtainℓ𝑌′(ℎ′. It is clear that the map is a fiber in the sense of the statement of Theorem 4B.
Infinite energy harmonic maps and AdS 3-manifolds
Introduction
There exists a Fuchsian representation 𝑗Σ such that the Hausdorff dimension of the limit group of 𝜌(Γ) is bounded below by that of 𝑗Σ. Consequently, there is nothing really new in the proof of the general existence result - it is an amalgamation of familiar ideas.
Representations of discrete groups
Take any equivariant map 𝑓 : ˜Σ → 𝑋 and fix a boundary neighborhood connected to the periphery and isometric to𝑈(𝜏). We construct a finite energy map in a neighborhood of each corner, equivariant with respect to the subgroup generated by 𝜌(𝛾𝑗) and then smoothly extend to a 𝜌-equivariant map on the (compact) complement of cusps.
Energy domination
Thus there exists a sequence contained in 𝑈 tending to the Halong limit that 𝐻(𝑓)/𝐻(ℎ) >1 and increases to sup𝐻(𝑓)/𝐻(ℎ). By Bochner's formula above, these forces 𝜅(𝑓∗𝑔) = −1, and so by Lemma 3.3.5 𝑓 maps H diffeomorphically to a completely geodesic plane.
Quadratic differentials with poles of order 2
For a closed arc 𝑐 on a hyperbolic surface, let ℓ(𝑐) denote the hyperbolic length of the geodesic representative. Any simple closed geodesic line that intersects itself once must pass through the entire vertical length of the ring.
Existence and classification of tame harmonic maps
There is only one such arc in the quotient, so 𝑓∞ maps to the core geodesy. From Proposition 3.3.2 and [Wol91b, page 516], the energy density 𝑓𝑘 is uniformly bounded on Σ in the plane cylinder metric.
Domination and AdS 3-manifolds
Working with arbitrary𝑋, we know from the proof of Proposition 3.3.2𝜓either the Lipschitz constant has 1 everywhere, or the Lipschitz constant is strictly less than 1 on every compact set. We now begin the proof of Theorem 4B. By generalizing the map Ψ from [Tho17, subsection 2.3]), we define the map. Since the starting point lies in 𝐵𝜖(𝑝 .. 2), we see the image belowℎof any segment lies in𝐵𝜏(𝑝.
From Proposition 6.4.5 we also know that the set ΣSR(𝑓𝜇,𝜈) is dense in Σ, and hence nonempty.
Maximal surfaces and AdS 3-manifolds
Introduction
Near the end of the original paper [Case19], we found something strange: pairs of representations𝜌. Below, let (𝑋 , 𝜈) be a Hadamard manifold with isometry group𝐺. 2)-equivariant 1-Lipschitz map𝑔defined on the convex hull of the limit set of𝜌. 1(𝑆𝑔,𝑛)) of the stretch locus of an optimal Lipschitz map is exactly the limit of the convex kernel.
A maximal spatial surface 𝐹 : (Σ˜,𝜇˜) → (H×, 𝜎 ⊕ (−𝜈)) is called smooth if the Hopf differentials of the harmonic maps have poles of order at most 2 at the edges.
The derivative of the energy functional
Throughout this chapter, by "hyperbolic metric on Σ" we mean a complete finite-volume hyperbolic metric, unless otherwise stated. It is a conformal cylinder, and when we say a "conformal cylinder for𝜇", we mean that the length is adjusted to be𝜏and the height is 1, so there are conformal mapsC →𝑈(𝜏). We then couple the uniform energy bounds on 𝜑𝜇 with Cheng's lemma to get uniform energy bounds, and then we appeal to elliptic theory (which . we've done many times at this point).
Therefore, for 𝑡 small enough, using the expression (4.8), we see that for small𝑡, the first term in (4.11) decays at most as .
Maximal surfaces: existence, uniqueness, deformations
Here, ℓ(𝐴𝑡 . 𝑥) is the length of the (possibly broken) segment 𝐴𝑥 ,𝑡, clearly bounded above by 1. We claim that there is a choice such that the homotopic class of the new curve is the same as the class ℎ𝑛 (𝛼). In the proof, we require control over the energy of harmonic mappings on top, as we change the source metric and representation.
The proof of Lemma 4.3.4 can then be made uniform: by examining the proof, the integral.
Anti-de Sitter 3-manifolds
With this in mind, given a locally strictly contracting mapping 𝑔 : 𝑉 → H×H with the properties above, time-like geodesics of the form𝐿𝑝,𝑔(𝑝) and 𝐿𝑞,𝑔(𝑞) never intersect. 2-equivariant domain that provides a fiberization on the union of the convex core with this funnel, contradicting our standing assumption. We can extend𝑔 outside the convex hull of the fixed limit by using 1-Lipschitz (𝜌.
With the most important theorems finished, we briefly move on to discuss the topology of the quotients.
Parabolic Higgs bundles
Given any two open sets Ω𝑖 containing 𝑝𝑖, we can radially reduce our Φ-disks to have . Let Δ denote the bond-induced Laplacian ∇F and 𝑅 = 𝑅𝑀 the curvature tensor of the Levi-Civita bond of 𝜈. The Jacobi operatorJ𝑓 =J:Γ(F) → Γ(F) is defined. 6.2) The Jacobi operator is a second-order strongly elliptic linear operator and is essentially self-adjoint in the sense that.
Arguing by contradiction, suppose that on an open subsetΩ ⊂ 𝐴 we have that for every 𝑝 ∈ Ω there exists 𝑞 ∈ 𝑓−1(𝑓(𝑝)) such that 𝜇(𝑝) and 𝑝(𝑝) ) 𝑓.
The factorization theorem
Introduction
Osserman ruled out true branching points and made progress towards the nonexistence of false branching points in [Oss70] and Gulliver showed that there are no false branching points in [Gul73]. In [GOR73], a branching dip is a map from a surface that is regular everywhere except for an isolated set of good branching points. Gulliver-Osserman-Royden uses the representation formula of Hartman and Wintner [HW53] to show that a minimal map is a well-branched submergence (see Propositions 2.2 and 2.4 in [GOR73]).
Let Σ be a 𝐶1 surface, 𝑀 a 𝐶1 manifold, and 𝑓 :Σ→ 𝑀a𝐶1well branched immersion with the unique continuation property and no true branch points.
Local properties of harmonic maps
Implicit in the proof of the unique continuation property for minimal maps is the following result (see [GOR73, Lemma 2.10]). If Ω1 is aΦ-disk then so is Ω2, and ℎ takes a natural coordinate𝑧onΩ1 to a natural coordinate𝑤onΩ2 in which 𝑤(ℎ(𝑧)) =𝑧. For any point 𝑝 ∉ Z, there exists a maximum radius𝑟𝑝 such that we can expand any natural coordinate centered at𝑝into the aΦ-disk of radius𝑟𝑝.
We restrict ℎ to this Φ-disk, and as above we use ℎ to construct a natural coordinate 𝑤 on 𝐵𝛿/.
Holomorphic factorization
So ℎ(𝛾(𝑡)) can never be a zero 𝑡, and we can continue to the end point. So we can continue ℎ along 𝛾 as much as we want, and we extend to the border point 𝑝. In the above setting, we can choose to have our Φ disks small enough that no point𝑞 ∈𝐵𝛿(𝑝 . 1)\{𝑝.
We will first show that we can choose Φ-disks 𝑈𝑖 that satisfy condition (2) in the definition of.
Klein surfaces
In local holomorphic coordinates, the map𝜋 is of the form 𝑧 ↦→ 𝑧 or𝑧 ↦→ 𝑧𝑛, so it is certainly holomorphic. By the bicontinuity of the identity map between these Banach spaces, there exists𝐶𝑗 ≥ 1 such that for all𝑉 ∈Γ(F),. In the proof of the main theorems, we need clear expressions for the properties of the reproductive nuclei.
Here 𝑅 is the complex curvature tensor of 𝑀 and 𝜎2 is the density of the conformal metric 𝜇 on Σ𝜇.
Moduli spaces of harmonic surfaces
Introduction
The theory of harmonic maps of surfaces is well developed and has proven to be a useful tool in geometry and topology. In this chapter we consider moduli spaces of harmonic surfaces and study their generic qualitative behavior through transversality theory. Then there exists a neighborhood 𝑈 ⊂ 𝔐 containing (𝜇, 𝜈), so that the space of harmonic immersions in 𝑈 is open and closed.
Then there exists a neighborhood𝑈 ⊂ 𝔐 containing(𝜇, 𝜈) such that the space of harmonic embeddings in𝑈 is open and dense.
Moduli spaces
Using somewhere injectivity and a lemma from Moore [Moo06], we find that there is an open setΩon which𝑋is the real part of a holomorphic section of a special holomorphic line bundle L ⊂ E. By making use of the isolated singularity condition, we analytically continue the "imaginary part," so that 𝑋 is the real part of a globally defined meromorphic section 𝑍 of L. Let 𝜎𝑗 denote the maximum 0 and the largest cross-sectional curvature of 𝑀 in the image of 𝑓𝑗. For the analogue of the Eells-Lemaire- the result, applied to a suitable class of equivariant harmonic maps.
A version of the theorem should be true in these contexts, but it would take us too far here.
Reproducing kernels
The convergence result now follows from the Rellich-Kondrachov theorem, which gives a compact inclusion from𝑊2, 𝑝 →𝐶0, 𝛼 when 2−2/𝑝 > 𝛼. Meanwhile, continuing from the previous lemma, the left side converges to . Therefore, 𝑋 =𝑆−Ψ, and the expression for 𝑋 follows from the local expression for𝑆 stated above and from the fact thatΨ ∈𝐶0,𝛼.
Off the diagonal, the regularity in 𝑧 is the maximum regularity of 𝑈 and the vector bundle: of.
Somewhere injective harmonic maps
It is due to Sampson [Sam78, Theorem 3] that the set of regular points of an admissible harmonic map is open and dense. We then find a small tubular neighborhood𝑁 ⊂ 𝑀 of the submanifold 𝑓(Ω) in which the closest point projection 𝜋 : 𝑁 → 𝑓(Ω) is well defined. For the remainder of Sections 6.4 and 6.5, let us replace 𝔐 by the complement of the set of pairs of exceptional metrics onΣ.
From the proof of Theorem 6A, it is enough to show that it is close and connected.
Proof of the transversality lemma
To return to the path𝛾, we do the above procedure over all the coordinate maps, perturbing 𝛾 to a new path contained entirely in D\J and connecting the endpoints. If we choose a variation 𝜈¤ with support in 𝐷𝜖, then the induced variation of the pullback metric 𝑓∗𝜈 is supported in 𝑓−1(𝐷𝜖). In each case, we choose a different variation of the target metric to find our discrepancy.
We have essentially proved transversalityΘ, if we assume that the images of the harmonic map are tangential at 𝑧.
Immersions and embeddings
Unstable minimal surfaces in products
Introduction
Preliminaries
Minimal surfaces in products of R-trees
Unstable equivariant minimal surfaces in R 𝑛
The general case
