Domination between non-Fuchsian representations and anti-de Sitter geometry

In this paper, the two main questions we address are the following.

1. (1)

   Given $\rho$, can we dominate it in the pullback or harmonic maps sense by a non-Fuchsian representation $j$?

2. (2)

   What geometry do dominating pairs encode?

Concerning (1), we provide a complete answer for certain classes of representations (Theorem C and Corollary C), and we also find that there is a menagerie of interesting examples that are worthy of further study (Theorem D). Our results toward (1), together with further context relating to harmonic maps, are contained in Section 1.2 below. At this stage, we can state one positive result, which extends [DT16, Theorem A]. Representations from a closed surface group to $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$ are organized into connected components according to an integer invariant called the Euler number (see Section 2.2). The set of possible Euler numbers is $[-2g+2,2g-2]\cap\mathbb{Z}$ and a representation $\rho$ is Fuchsian precisely when $\textrm{eu}(\rho)=\pm(2g-2)$, see [Gol80]. Note that when $\textrm{eu}(\rho)\neq 0$, the representation $\rho$ is reductive and the $\rho$-equivariant harmonic map is unique. Our first domination result concerns the case where this harmonic map is a branched immersion. Recall that a map $f$ between surfaces has a branch point at a point $p$ if one can find local coordinates $z$ and $w$ centered at $p$ and $f(p)$ respectively on which $f$ takes the form $z\mapsto w=z^{n}$. We say that $f$ is a branched immersion if it is an immersion apart from at a (discrete) set of branch points.

The assumption that the Euler class is positive is not restrictive, since its sign can be flipped by applying an outer automorphism of the fundamental group. Theorem A is essentially a consequence of Theorem C below. Pairs $(\rho,f)$ as above are constructed and parametrized using Propositions 1.4 and 1.6 below. Equivariant branched immersions are notable because they are equivalent to hyperbolic cone structures on $\Sigma$ (see [Tan94] for explanation). The equivariant harmonic map $h$ associated with $j$ is also a branched immersion, so Theorem A can be alternatively cast as a domination result between hyperbolic cone structures. The choices of $j$ of Euler number $\textrm{eu}(\rho)+k$ are parametrized by effective divisors of degree $k$ that dominate the branching divisor of $f$. For $k=2g-2-\textrm{eu}(\rho),$ there is just one choice, and this is the Fuchsian representation from [DT16, Theorem A].

Theorem C in fact produces many examples of domination data $(\rho,j,f,h)$ where $f$ is not a branched immersion, but there is no immediate geometric description for these maps, so we don’t include them in Theorem A. Nevertheless, these other examples are important for our applications to anti-de Sitter geometry.

For the second question (2), we state our main result here, but we first need to recall the basics on $\mathbb{A}\mathrm{d}\mathbb{S}^{3}$. The three-dimensional anti-de Sitter space $\mathbb{A}\mathrm{d}\mathbb{S}^{3}$ is a Lorentzian space form of constant negative sectional curvature, and it can be modeled on $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$ with the Killing metric. In this model, the group of space and time-orientation preserving isometries is $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})^{2}$, acting by right and left multiplication

$$(A,B)\cdot X=AXB^{-1}.$$

An anti-de Sitter (AdS) manifold is a Lorentzian manifold of constant negative sectional curvature; equivalently, up to scaling the metric, it is one that is locally isometric to $\mathbb{A}\mathrm{d}\mathbb{S}^{3}$.

For surface group representations $j,\rho:\Gamma\to\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$, we define $\Gamma_{j,\rho}<\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})^{2}$ by

$$\Gamma_{j,\rho}=\{(j(\gamma),\rho(\gamma)):\gamma\in\Gamma\}.$$

(1)

Kulkarni and Raymond proved that every group acting properly discontinuously on $\mathbb{A}\mathrm{d}\mathbb{S}^{3}$ identifies with some $\Gamma_{j,\rho}$, and that (up to switching factors) $j$ must be Fuchsian [KR85]. The quotient is a circle bundle over $\mathbb{H}^{2}/j(\Gamma)$, and the circle fibers are timelike geodesics. It was proved by Salein in [Sal00] (if direction) and Kassel in [Kas10] (only if direction) that, when $\Sigma$ is a closed surface of genus at least $2$, $\Gamma_{j,\rho}$ acts properly and discontinuously on $\mathbb{A}\mathrm{d}\mathbb{S}^{3}$ if and only if $j$ strictly dominates $\rho$ (see also [GK17, Theorem 1.8]). For surveys on these results, see [Tho17, Sections 0.1-0.2] and [Sag24, Sections 1.1-1.2].

In her thesis [Jan22], Janigro studied singular anti-de Sitter $3$-manifolds that fiber via timelike geodesics over hyperbolic cone surfaces. Janigro defined the notion of a spin-cone AdS $3$-manifold, in which the singularities occur along timelike geodesics, generalizing a construction due to Barbot and Meusburger in the context of flat Lorentzian spacetimes, where they modeled massive particles with spin [BM12]. Janigro introduced a weak notion of domination for hyperbolic cone surfaces, which she related to spin-cone AdS $3$-manifolds. We extend Janigro’s work by showing that domination in the pullback sense gives rise to a singular AdS $3$-manifold (see Theorem 5.3). This extension is obtained by adding a singular set to the AdS $3$-manifold constructed by Janigro, thus making the geometric picture more precise. We will see in Section 5 that this singular set is closely related to the singular set of $h$, and we note that it also allows for more types of singularities than just spin-cone singularities. We then specialize to the case where the map $h$ is a branched immersion and analyze the resulting singularities in detail. We arrive at the following theorem.

Branched AdS manifolds are a special case of spin-cone AdS manifolds (see Definition 4.5). For clarity, in Sections 5 and 6, we provide further exposition on Janigro’s work in [Jan22] and explain how our work builds on it. It is worth noting that, in the case where $h$ is a global diffeomorphism, both $\mathbb{A}\mathrm{d}\mathbb{S}^{f,h}$ and $\mathbb{A}\mathrm{d}\mathbb{S}^{f,h}_{*}$ coincide with the anti–de Sitter space $\mathbb{A}\mathrm{d}\mathbb{S}^{3}$, and we thus recover previously known results about closed AdS $3$-manifolds.

Combining Theorems A and B, we obtain the following result, which provides a large class of examples of branched anti-de Sitter $3$-manifolds.

A current conjecture, attributed to Goldman, is that every faithful representation of non-zero and non-extremal Euler class uniformizes a branched hyperbolic structure, or, equivalently, satisfies the hypothesis of the corollary. This conjecture is linked in a circle of ideas around a question of Bowditch as well as Goldman’s conjecture about mapping class group dynamics on the $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$-character variety (see, for instance, [Far21]).

As indicated above, we will see in Theorem C that more examples of domination arise than in Theorem A (in particular, where $f$ is not a branched immersion but $h$ is), so Corollary B can also be expanded.

Here we give a detailed overview of our main results on harmonic maps and domination. See Sections 2.1-2.2 for preliminaries on harmonic maps.

For motivation, we recall the work of Deroin-Tholozan in [DT16]. Let $X$ be a closed Riemann surface of genus at least $2$ with fundamental group $\Gamma$ and let $\rho:\Gamma\to\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$ be a non-Fuchsian reductive representation with $\rho$-equivariant harmonic map $f:\widetilde{X}\to\mathbb{H}^{2}$. Every equivariant harmonic map from $\widetilde{X}$ determines a holomorphic quadratic differential on $X$ called the Hopf differential. By [Wol89], there exists a unique Fuchsian representation $j$ together with an equivariant harmonic diffeomorphism $h:\widetilde{X}\to\mathbb{H}^{2}$ with the same Hopf differential as $f$. In [DT16], it is shown that

$$f^{*}\sigma<h^{*}\sigma.$$

The map $F=f\circ h^{-1}:\mathbb{H}^{2}\to\mathbb{H}^{2}$ is $(j,\rho)$-equivariant and strictly contracting, and hence shows that $j$ strictly dominates $\rho$. Tholozan went on to prove that every strictly dominating pair arises in this fashion [Tho17]. Moreover, the combined works [DT16] and [Tho17] show the following.

At this point, Definition 1.1 is well motivated, and we can refine the problem (1): we look for pairs of equivariant harmonic maps with the same Hopf differential such that the domination inequality holds. In order to do so, we need to understand how harmonic maps with the same Hopf differential can differ. Via the non-abelian Hodge correspondence, we establish the following.

See Proposition 2.7 for the precise statement (which includes the case $\phi=0$). When $D=0$, one recovers Wolf’s parametrization of Teichmüller space [Wol89], and when $\phi=0$, one is led to Troyanov’s uniformization for hyperbolic branched metrics [Tro91]. Proposition 1.4 is probably known to some experts (compare with [Hit87, Theorem 10.8])–see Section 2.3 for explanation.

Now we can make our problem (1) more precise: for which pairs $(\phi,D_{1})$ and $(\phi,D_{2})$ do we have domination? The problems turns out to be quite delicate, but we find a complete solution when $(\phi,D_{2})$ gives rise to a branched harmonic immersion. Proposition 1.6, although not necessary for the proof, tells us when we have a branched immersion.

When $\phi=0$, the harmonic map is weakly conformal and automatically a branched immersion. The conditions $2D<(\phi)$ and $(\phi)<2D$ correspond to positive and negative Euler numbers respectfully. The key elements of Proposition 1.6 are contained in the literature, but the result has never been stated as above. The “only if” direction is observed in [BBDH21, Lemma 3.2], and the “if” direction follows from [SG24, Theorem 4.1, Proposition 6.5] (which improves [BBDH21, Theorems 3.4 and 4.1]). On route to Theorem C, we establish all the tools necessary to prove Proposition 1.6, so for the convenience of the reader we write out a proof. We will also prove a version of Proposition 1.6 for surfaces with boundary (see Lemma 3.8).

Finally, we state our solution to the domination problem for branched harmonic immersions. The theorem concerns dominating arbitrary maps by branched immersions, and the corollary is about dominating branched immersions themselves.

The assumption on the degrees is not necessary, but just keeps the statement cleaner. If $\deg D_{2}\geq 2g-2$, then we can flip back to the case $\deg D_{2}\leq 2g-2$ by using an outer automorphism of the fundamental group. A harmonic map with data $(\phi,D)$ becomes a map with data $(\phi,(\phi)-D)$, and the pullback metric is unaffected, and thus the condition $D_{2}<D_{1}$ becomes $(\phi)-D_{2}<D_{1}$. Similar if $\deg D_{1}\geq 2g-2.$ It is worth noting that if $h$ is not a branched immersion, the condition $D_{2}<D_{1}$ does not guarantee domination; see Proposition 3.6.

The main result toward the proof of Theorem C is Proposition 3.2, which is of independent interest. It provides an inequality between functions satisfying a Bochner-type equation, generalizing the key inequality from [SY97, Section 1.8] and [DT16, Lemma 2.6]. This proposition is also a key step in establishing Proposition 1.6.

With our application to anti-de Sitter geometry in mind, we are mainly concerned with dominating by branched immersions, which puts us in the case $|\textrm{eu}(\rho)|<|\textrm{eu}(j)|$. That being said, after Theorem C, it is natural to inquire about the equality case $|\textrm{eu}(\rho)|=|\textrm{eu}(j)|.$ Since it’s not focused toward an application in this paper, we don’t carry out a general treatment, but we just give examples that point to future directions.

In (2), the underlying representation has Euler number zero.

We list a few research directions that follow this work.

In Sections 2 and 3, we study the domination problem for harmonic maps, while in Sections 4, 5, and 6, we study anti-de Sitter geometry and construct singular anti-de Sitter $3$-manifolds from dominating pairs. Sections 2-3 and 4-6 can essentially be read independently.

In more detail, Section 2 provides preliminaries on harmonic maps and carefully states and proves Proposition 2.7, which characterizes harmonic maps via holomorphic data $(\phi,D)$. In Section 3, we address the domination problem and prove Proposition 1.6 as well as Theorems A, C, and D. In Section 4, we introduce spin-cone AdS $3$-manifolds, and in Section 5, we construct singular AdS $3$-manifolds from general dominating pairs (Theorem 5.3). Finally, in Section 6, we specialize to branched immersions and show that, in this case, the examples from Section 5 are branched AdS $3$-manifolds, thereby proving Theorem B.

This project stems from discussions with Andrea Seppi, to whom we are very grateful, and to whom we thank further for helpful comments on the first draft. We also thank Francesco Bonsante for sharing the thesis [Jan22].

Let $\Sigma_{g}$ be a closed oriented surface of genus at least $2$ and let $X$ be a Riemann surface structure on $\Sigma_{g}$ with universal cover $\tilde{X}$. Let $(M,\sigma)$ be a Riemannian manifold and let $f:\widetilde{X}\to M$ be a $C^{2}$ map. The derivative $df$ defines a section of the endomorphism bundle $T^{*}\widetilde{X}\otimes f^{*}TM$. Complexifying $T^{*}\widetilde{X}\otimes f^{*}TM$, let $df=\partial f+\overline{\partial}f$ be the decomposition into $(1,0)$ and $(0,1)$ parts. We denote by $\nabla$ the connection on $f^{*}TM\otimes\mathbb{C}$ induced by the Levi-Civita connection of $\sigma$, and we also use $\nabla$ for its extension to $f^{*}TM\otimes\mathbb{C}$-valued forms. The definition below depends on the Riemann surface structure $X$ and the metric $\sigma.$

In this paper, we are concerned with equivariant harmonic maps from $\widetilde{X}$ to the space $(\mathbb{H}^{2},\sigma)$, where $\mathbb{H}^{2}$ is the $2$-dimensional upper half-plane and $\sigma$ is a hyperbolic metric of constant curvature $-1$. The starting point is the existence theorem of Donaldson [Don87] and Corlette [Cor88]. A representation $\rho:\Gamma\to\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$ is irreducible if it is not contained in a parabolic subgroup, and $\rho$ is reductive if the Zariski closure of $\rho(\Gamma)$ is a reductive group. Geometrically, and perhaps more intuitively, $\rho$ is irreducible if the induced action of $\Gamma$ on the Gromov boundary $\partial_{\infty}\mathbb{H}$ has no global fixed point, and $\rho$ is reductive if $\rho(\Gamma)$ preserves a geodesic in $\mathbb{H}^{2}.$

By Sampson’s argument in [Sam78, Theorem 3], if $\rho$ is irreducible, then $f$ is unique. When $\rho$ is just reductive and not irreducible, again by [Sam78, Theorem 3], there is a $1$-parameter family of harmonic maps, each of which is a parametrization onto the invariant geodesic, and all of the harmonic maps are related by an isometric translation along the geodesic. Since the harmonic maps are related by isometries, the pullback metric never depends on the choice of harmonic map.

Given a harmonic map from a Riemann surface, one can associate a number of analytic objects. For a detailed reference, see [SY97, Section 1-2]. To begin, let $X$ be a Riemann surface as above and fix a metric $\nu$ on $X$ that’s compatible with the Riemann surface structure. We use $\nu$ as well to denote the lift to a metric on the universal cover $\widetilde{X}$.

Keeping things general for now, let $\rho:\Gamma\to\textrm{Isom}(M,\sigma)$ be an action by isometries and let $f:\widetilde{X}\to(M,\sigma)$ be a $\rho$-equivariant $C^{2}$ map. The (possibly degenerate) pullback metric $f^{*}\sigma$ extends bilinearly to the complexified tangent bundle of $\widetilde{X}$ and then decomposes into types as

$$f^{*}\sigma=2\sigma(\partial f,\overline{\partial}f)+\sigma(\partial f,\partial f)+\sigma(\overline{\partial}f,\overline{\partial}f).$$

(2)

We write $2\sigma(\partial f,\overline{\partial}f)=e(f)\nu,$ where $e(f)$ is called the energy density function. As well, the quadratic differential $\phi(f):=\sigma(\partial f,\partial f)$ is called the Hopf differential of $f$ (note also that $\overline{\phi}(f)=\sigma(\overline{\partial}f,\overline{\partial}f)$). Since $f$ is $\rho$-equivariant, $e(f),\phi(f),$ $\overline{\phi}(f)$, and $f^{*}\sigma$ are invariant under the action of $\Gamma$ and hence descend to $X$. The formula (2) is rewritten as

$$f^{*}\sigma=e(f)\nu+\phi(f)+\overline{\phi}(f).$$

(3)

The significance of $e(f)$ stems from the fact that the equation $\nabla^{0,1}\partial f=0$ can be seen as the Euler-Lagrange equation for the Dirichlet energy

$$\mathcal{E}(X,f)=\int_{\Sigma_{g}}e(f)dA_{\nu},$$

where $dA_{\nu}$ is the area form of $\nu$ (note $\mathcal{E}(X,f)$ does not depend on the choice of $\nu$). That is, harmonic maps are equivalently critical points of $\mathcal{E}(X,\cdot)$. As for the Hopf differential $\phi(f)$, it is holomorphic when $f$ is harmonic, and when the target has dimension at most $2,$ so for example when $M=\mathbb{H}^{2},$ the converse holds as well.

We now set $M=\mathbb{H}^{2}$. In this special case, there are more analytic quantities to probe the harmonic map. Working in local coordinates, where $\sigma=\sigma(w)|dw|^{2}$, $\nu=\nu(z)|dz|^{2}$, we set

$$H(f)=\frac{\sigma(f(z))}{\nu(z)}\Big|\frac{\partial f}{\partial z}\Big|^{2},\hskip 2.84526ptL(f)=\frac{\sigma(f(z))}{\nu(z)}\Big|\frac{\partial f}{\partial\overline{z}}\Big|^{2}.$$

The function $H(f)$ is called the holomorphic energy and $L(f)$ is called the anti-holomorphic energy. They satisfy

$$e(f)=H(f)+L(f),\hskip 2.84526pt|\phi(f)|^{2}\nu^{-2}=H(f)L(f),$$

and the vanishing of the Jacobian of $f$ is equivalent to the vanishing of the function

$$J(f):=H(f)-L(f).$$

When $f$ is harmonic, the functions $H(f)$ and $L(f)$ are either identically zero or have isolated zeros and satisfy the Bochner formulae: away from the zeros of $H(f),$

$$\frac{1}{2}\Delta_{\nu}\log H(f)=-\kappa_{\sigma}(H(f)-L(f))+\kappa_{\nu},$$

(4)

where $\kappa_{\sigma}$ and $\kappa_{\nu}$ are the sectional curvatures of $\sigma$ and $\nu$, respectively (so $\kappa_{\sigma}=-1$), and away from the zeros of $L(f),$

$$\frac{1}{2}\Delta_{\nu}\log L(f)=-\kappa_{\sigma}(L(f)-H(f))+\kappa_{\nu}$$

(see [SY97, Section 1.7]). Note that the Bochner formula for $L(f)$ can be seen as a consequence of the Bochner formula for $H(f)$ and the equation $|\phi(f)|^{2}\nu^{-2}=H(f)L(f)$. Note that for a non-irreducible representation, again using that harmonic maps are related by isometries, $H(f)$ is independent of the choice of harmonic map.

We end this subsection with a discussion on Euler numbers. The Euler number of a representation $\rho,$ denoted $\textrm{eu}(\rho),$ can be defined in many ways; for instance, using characteristic classes, or as an obstruction to lifting $\rho$ to $\widetilde{\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})}$. Here, we give the most naive definition:

$$\textrm{eu}(\rho):=\frac{1}{2\pi}\int_{\Sigma_{g}}dA_{f^{*}\sigma},$$

(5)

where $f:\widetilde{\Sigma}_{g}\to\mathbb{H}^{2}$ is any $\rho$-equivariant $C^{2}$ map, and $dA_{f^{*}\sigma}$ is the area form of $f^{*}\sigma$ (which might be degenerate). By an application of Stoke’s theorem, the integral above indeed depends only on $\rho$ and not on $f$. It is also useful to recall the equality of $2$-forms

$$dA_{f^{*}\sigma}=J(f)dA_{\nu}.$$

(6)

The following lemma allows us to make Definition 2.4 below.

Starting from the characterization (5), a classical argument, which involves the Bochner formula, (6), and the Gauss-Bonnet theorem, can be applied nearly verbatim to prove the following. The argument can be found in [SY97, pp. 11-12].

If we reverse the orientation of $\Sigma_{g}$ or precompose $\rho$ with an outer automorphism of $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$, then the Euler numbers flip sign. For this reason, we often restrict to the case $\textrm{eu}(\rho)\geq 0$. For $\textrm{eu}(\rho)\leq 0$, one has results analogous to above, with $H(f)$ replaced with $L(f).$

The purpose of this section is to prove Proposition 2.7 below (or, Proposition 1.4). We use Higgs bundles and the non-abelian Hodge correspondence; Higgs bundles will not come up again, so the unfamiliar reader might benefit from skipping the proofs on a first reading.

For a non-zero holomorphic section $\alpha$ of a holomorphic line bundle on $X$, we denote the divisor by $(\alpha).$ Let $\mathcal{K}$ be the canonical bundle of $X,$ so that $H^{0}(X,\mathcal{K}^{2})$ is the space of holomorphic quadratic differentials on $X$.

To parametrize representations with non-positive Euler numbers, one takes pairs $(\phi,D)$ such that $\deg D\geq 2g-2$ and $D\geq(\phi)$; simply use an outer automorphism. About the case $\deg D=2g-2$, the equation (7) shows that if $\deg D=2g-2$, i.e, $\textrm{eu}(\rho)=0$, then no reductive representation can carry a harmonic map $f$ with $\phi(f)=0$, for then we would have $H(f)=L(f)=0$.

Proposition 2.7 should be known to experts. The parametrization of the character variety by pairs $(\phi,D)$ is proved in Hitchin’s original paper [Hit87, Theorem 10.8] (outside of Euler number $0$, although comments are made on that case), but the characterization in terms of zeros of the holomorphic energy does not appear to be recorded. We essentially redo Hitchin’s proof in a different language, and explain, from our point of view, how $D$ comes from the holomorphic energy.