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Critical-point-free energy for fractional-Toledo representations

This paper constructs irreducible reductive representations of surface groups into $\PU(2,1)$ with non-integral Toledo invariants for which the associated energy function on Teichmüller space possesses no critical points, thereby demonstrating that the branched-minimal-surface forgetful map is not surjective in these components.

Original authors: Rivu Bardhan, Anu Dhochak, Pradip Kumar

Published 2026-08-18

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CC BY 4.0

Original authors: Rivu Bardhan, Anu Dhochak, Pradip Kumar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the rules of geometry are bent, not by magic, but by the deep, hidden laws of mathematics. In this realm, mathematicians study shapes that curve inward like the inside of a saddle, stretching out forever without ever closing up. These are called hyperbolic spaces, and they are the natural home for understanding the complex structures of surfaces, like the skin of a donut with many holes. When mathematicians map these surfaces into such curved spaces, they look for the most efficient way to do it, a path that uses the least amount of energy possible. This concept of energy is not about electricity or fuel, but a mathematical measure of how much a map stretches or distorts the surface as it travels through the curved space. For decades, a fundamental question has lingered: does every such map always have a point of perfect balance, a spot where the energy is at its lowest, like a ball settling into the bottom of a bowl?

A team of researchers has now answered this question with a surprising twist. They have constructed a specific type of mathematical map where no such point of balance exists. In their work, they created a scenario where the energy function, which usually settles down to a minimum, simply keeps flowing without ever finding a resting place. This discovery is significant because it reveals a hidden limitation in how we understand the relationship between the shape of a surface and the space it inhabits. It shows that when the connection between the surface and the space is slightly "broken" or incomplete, the usual rules of stability can fail completely. The researchers proved that for certain complex surfaces and specific ways of mapping them, the energy never stops changing, meaning there is no single, perfect configuration to be found.

To understand how they reached this conclusion, one must first look at the tools they used. The researchers started with a known, stable map that worked perfectly on a smaller surface. This map was already famous for being efficient and well-behaved, like a tightrope walker who never loses their balance. They then took a larger surface, one with more holes than the original, and created a special bridge between the two. This bridge, which they call a pinch, effectively collapses the extra holes of the larger surface down to nothing, squashing them into the smaller one. By using this bridge to carry the stable map from the small surface to the large one, they created a new map for the larger surface.

At first glance, this new map looks just like the old one because it travels through the same path in the curved space. However, there is a crucial difference. Because the bridge squashed parts of the larger surface into nothing, the new map is no longer a faithful reflection of the entire larger surface; it has forgotten about the parts that were pinched away. This loss of information is the key. The researchers showed that this specific type of forgetting creates a geometric barrier. The map is forced to stretch in a way that prevents it from ever finding a stable, low-energy state. It is as if the map is constantly trying to settle, but the very act of pinching the surface creates a slope that keeps pushing it away from equilibrium.

The team demonstrated that this phenomenon happens under very specific conditions. They needed the surface to have a certain number of holes, and the way the map connected to the curved space had to involve a particular kind of fractional relationship, a number that cannot be written as a whole integer. When these conditions were met, and the surface was large enough, the energy function had no critical points at all. In mathematical terms, a critical point is where the slope of the energy landscape is flat, indicating a minimum or a maximum. The researchers proved that for their constructed maps, the landscape is never flat; it is always sloping, meaning the energy is always changing and never settles.

This finding challenges a previous belief that such maps would always find a minimum energy state, provided the surface and the space were connected in a robust way. The researchers showed that while the map still travels through the same curved space, the fact that it is not a perfect, one-to-one reflection of the original surface is enough to break the stability. They did not just suggest this might happen; they provided a rigorous proof that it does happen. Their work constructs a concrete example where the energy function is not well-behaved, proving that the existence of a minimum is not guaranteed in all cases.

The implications of this result reach beyond just this single example. It helps mathematicians understand the boundaries of the "character variety," a vast space that catalogs all possible ways to map surfaces into these curved geometries. The researchers showed that there are points in this catalog that cannot be reached by looking at the most efficient, minimal maps. In other words, there are valid mathematical maps that simply do not correspond to any minimal surface. This fills a gap in our understanding of the landscape of these geometric possibilities, showing that the map from minimal surfaces to the space of all possible maps is not complete.

The construction relies on a delicate interplay between the geometry of the surface and the curvature of the space. The researchers used a known, stable map on a smaller surface and extended it to a larger one by collapsing the extra parts. This process, while simple to describe, creates a complex situation where the energy function behaves unexpectedly. They proved that the squared distance from the map to a specific stable surface within the curved space acts as a barrier. This barrier ensures that the energy cannot drop to a minimum, forcing the map to remain in a state of constant flux.

In the end, the work of Bardhan, Dochak, and Kumar provides a clear and definitive answer to a long-standing question. They have shown that for a specific class of representations, the energy function has no critical points. This means that the associated minimal surfaces, which are the geometric objects that usually represent these maps, do not exist for these cases. The result is a testament to the power of precise mathematical construction, revealing that even in the rigid world of geometry, there are scenarios where stability is impossible to achieve. The discovery does not break the laws of mathematics, but rather expands our knowledge of where those laws apply and where they allow for a different kind of behavior. It reminds us that in the deep study of shapes and spaces, the absence of a solution can be just as informative as the presence of one.

Technical Summary: Critical-Point-Free Energy for Fractional-Toledo Representations

Problem Statement
The paper investigates the energy function $E_\rho$ on the Teichmüller space $\mathcal{T}(S_g)$ associated with a reductive representation $\rho: \pi_1(S_g) \to PU(2, 1)$ , where $S_g$ is a closed oriented surface of genus $g \ge 2$ . The energy function is defined via equivariant harmonic maps $f_{\rho, J}: (\tilde{S}_g, J) \to \mathbb{CH}^2$ into complex hyperbolic space. A critical point of $E_\rho$ corresponds to a complex structure $J$ for which the Hopf differential vanishes, implying the harmonic map is a (possibly branched) minimal immersion.

The central question addressed is whether the energy function $E_\rho$ always possesses a critical point, particularly within the connected components of the $PU(2, 1)$ character variety indexed by non-integral Toledo invariants. While previous work (Goldman–Wentworth) established that $E_\sigma$ is a proper function (and thus attains a minimum) for convex-cocompact representations $\sigma$ , the behavior of $E_\rho$ for representations with non-trivial kernels or specific geometric constraints in non-integral Toledo components remained open. Specifically, the authors address whether the forgetful map from the moduli space of equivariant branched minimal surfaces to the character variety is surjective in these components.

Methodology
The authors construct a specific family of irreducible reductive representations $\rho_{g,h,d}$ and demonstrate that their associated energy functions have no critical points. The methodology proceeds in three main stages:

Construction of the Representation:
The construction begins with Bronstein's almost-Fuchsian representations $\rho_0: \pi_1(S_h) \to PU(2, 1)$ for a surface of genus $h$ . These representations admit a $\rho_0$ -equivariant holomorphic embedding $F: \tilde{S}_h \to \mathbb{CH}^2$ whose image $Y$ is a complete, minimal, almost-Fuchsian surface with a small second fundamental form ( $\|II_F\|_{op} \le \eta < 1/2$ ).
The target representation $\rho$ is defined by precomposing $\rho_0$ with an epimorphism $P_*: \pi_1(S_g) \to \pi_1(S_h)$ induced by an orientation-preserving degree-one pinch map $P: S_g \to S_h$ (which collapses $g-h$ handles). Thus, $\rho = \rho_0 \circ P_*$ . This representation is irreducible and reductive (sharing the same image as $\rho_0$ ) but is not faithful, possessing a non-trivial kernel.
Quantitative Geometric Trapping:
The authors establish a "branched trapping" lemma. They utilize Bronstein's normal-exponential theorem to define a squared-distance function $\Psi(x) = d(x, Y)^2$ on $\mathbb{CH}^2$ . They prove a uniform pointwise inequality for the trace of the Hessian of $\Psi$ on any two-plane in the tangent space of $\mathbb{CH}^2$ .
Specifically, for a weakly conformal harmonic map $f$ from a closed Riemann surface $M$ to $\mathbb{CH}^2$ , if the image of the fundamental group preserves $Y$ , the composition $u = \Psi \circ f$ satisfies a differential inequality $\Delta_M u \ge c u$ (where $c > 0$ depends on the distance to $Y$ ). By the maximum principle, if $u$ is non-negative and defined on a compact surface, $u$ must be constant. If the map is non-constant, this forces $u \equiv 0$ , implying the image of $f$ lies entirely within the minimal surface $Y$ .
Topological Contradiction:
Assuming a critical point $J$ exists for $E_\rho$ , the associated harmonic map $f$ must be weakly conformal. By the trapping lemma, $f$ maps into $Y$ . Since $F: \tilde{S}_h \to Y$ is an embedding, $f$ lifts to a map $H: \tilde{S}_g \to \tilde{S}_h$ . The equivariance of $f$ and $F$ implies $H$ is equivariant with respect to $P_*$ .
The map $H$ is shown to be a non-constant, weakly conformal harmonic map between Riemann surfaces, which implies it is holomorphic or antiholomorphic. Since the degree of the pinch map $P$ is 1, the degree of $H$ is also 1. A non-constant holomorphic map of degree 1 between compact Riemann surfaces is a biholomorphism, which implies the genus of the domain ( $g$ ) must equal the genus of the target ( $h$ ). This contradicts the construction assumption that $g > h$ .

Key Contributions and Results
The primary result is Theorem 1.1, which states:
For every positive integer $d$ such that $3 \nmid d$ , there exists a threshold $h_0(d)$ such that for all $h \ge h_0(d)$ and all $g > h$ , there exists an irreducible reductive representation $\rho_{g,h,d}: \pi_1(S_g) \to PU(2, 1)$ with Toledo invariant:
$\tau(\rho_{g,h,d}) = \frac{2h - 2 - 2d}{3} \notin \mathbb{Z}$
such that the set of critical points of the energy function is empty:
$\text{Crit}(E_{\rho_{g,h,d}}) = \emptyset$

Significance and Claims
The paper claims several specific theoretical implications:

Failure of Properness: The result demonstrates that the energy function $E_\rho$ is not proper for representations that are not convex-cocompact as representations of the domain group, even if their image subgroup is convex-cocompact. The loss of faithfulness (non-trivial kernel) leads to a drastic failure of the energy function to attain an infimum.
Non-Surjectivity of the Forgetful Map: The construction provides explicit points in the $PU(2, 1)$ character variety (specifically in non-integral Toledo components) that lie outside the image of the forgetful map from the moduli space of equivariant branched minimal surfaces. This resolves an image problem raised in prior literature [LM19].
Role of Non-Integrality: The authors clarify that non-integrality of the Toledo invariant alone does not preclude the existence of minimal surfaces (as such surfaces are known to exist in other fractional-Toledo components). The obstruction here arises specifically from the combination of the geometric barrier provided by the almost-Fuchsian surface and the degree-one pinch map.
Technical Innovation: The paper formulates a "uniform two-plane trace inequality" for the squared-distance barrier that applies to weakly conformal harmonic maps even at branch points, avoiding the need for limiting arguments at singularities.

In summary, the paper constructs a class of representations where the geometric constraints imposed by the image subgroup and the topological constraints of the domain group are incompatible with the existence of a critical point for the energy function, thereby proving the non-surjectivity of the forgetful map in specific non-integral components of the character variety.

Technical Summary: Critical-Point-Free Energy for Fractional-Toledo Representations

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