arXiv subjects

Sebastian Heller

Publications and source records attributed to Sebastian Heller.

At least 19 recordsLinked to original sources

arXiv2026

Embedded special Legendrian surfaces in $\mathbb S^5$

We construct the first smooth embedded compact special Legendrian surfaces in $\mathbb S^5$ of genus greater than one. More precisely, for every sufficiently large integer $k$, we construct an embedded special Legendrian surface whose conformal structure is the Fermat curve of degree $k$ and genus $\tfrac12(k-1)(k-2)$. Our approach combines an elementary implicit function theorem with the description of special Legendrian surfaces via loop algebra-valued meromorphic connections and a characterization of the unitarizability locus in the ${SL}_{3}(\mathbb C)$-character variety of the thrice-punctured sphere.

math.DG↗

arXiv2026

The Enclosed Volume for Periodic Constant Mean Curvature Surfaces

We establish a general formula for the enclosed volume of constant mean curvature (CMC) surfaces in Euclidean three space with translational periods forming a lattice. The formula relates the volume to the surface area, a Wess-Zumino-Witten-type term, and a newly defined curvature term of the associated family of flat connections, thereby extending the classical Minkowski formula for closed CMC surfaces. Interpreting the volume as a gauge-invariant quantity, we apply the result to a variety of examples and provide explicit computations. As an application, we construct a counterexample to the isoperimetric problem in $\mathbb{T}^2 \times \mathbb{R}$, disproving the conjecture that minimizers are restricted to spheres, cylinders, or pairs of planes.

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arXiv2025

Semiclassical Limits of Strongly Parabolic Higgs Bundles and Hyperpolygon Spaces

We investigate the Hitchin hyperk\"ahler metric on the moduli space of strongly parabolic $\mathfrak{sl}(2,\C)$-Higgs bundles on the $n$-punctured Riemann sphere and its degeneration obtained by scaling the parabolic weights $t\alpha$ as $t\to0$. Using the parabolic Deligne--Hitchin moduli space, we show that twistor lines of hyperpolygon spaces arise as limiting initial data for twistor lines at small weights, and we construct the corresponding real-analytic families of $\lambda$-connections. On suitably shrinking regions of the moduli space, the rescaled Hitchin metric converges, in the semiclassical limit, to the hyperk\"ahler metric on the hyperpolygon space $\mathcal X_\alpha$, which thus serves as the natural finite-dimensional model for the degeneration of the infinite-dimensional hyperk\"ahler reduction. Moreover, higher-order corrections of the Hitchin metric in this semiclassical regime can be expressed explicitly in terms of iterated integrals of logarithmic differentials on the punctured sphere.

math.DG↗

arXiv2025

Transgressive Harmonic Maps and SU(1,1) Self-Duality Solutions

We establish a duality between harmonic maps from Riemann surfaces to hyperbolic 3-space $\mathbb{H}^3$ and harmonic maps from Riemann surfaces to de Sitter three-space $\operatorname{dS}_3$, best viewed as a generalized Gauss map. On the gauge theoretic side, it matches SU(2) and SU(1,1) solutions of Hitchin's self-duality equations via a signature flip along an eigenline of the Higgs field. Reversing this operation typically produces singular solutions, occurring where the eigenline becomes lightlike. Motivated by explicit model examples and this singular behavior, we extend this duality to a class of transgressive harmonic maps $f:M\to \mathbb{S}^3$: these are harmonic on the hemispheres equipped with the hyperbolic metric, intersect the equator orthogonally, and have vanishing Hopf differential along the crossing set. We construct large families by gluing and analyze their regularity, and as an application obtain $\tau$-real negative sections of the Deligne--Hitchin moduli space of arbitrarily large energy that are not twistor lines.

math.DG↗

arXiv2025

Application of Chern-Simons gauge theory to the enclosed volume of constant mean curvature surfaces in the 3-sphere

Building on Hitchin's work of the Wess-Zumino-Witten term for harmonic maps into Lie groups, we derive a formula for the enclosed volume of a compact CMC surface $f$ in $\mathbb S^3$ in terms of a holonomy on the Chern-Simons bundle and the Willmore functional. By construction the enclosed volume only depends on the gauge classes of the associated family of flat connections of $f$. In this paper we show in various examples the effectiveness of this formula, in particular for surfaces of genus $g\geq2.$

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arXiv2024

Minimal surfaces and alternating multiple zetas

In this paper we show for every sufficiently large integer $g$ the existence of a complete family of closed and embedded constant mean curvature (CMC) surfaces deforming the Lawson surfaces $\xi_{1,g}$ parametrized by their conformal type. When specializing to the minimal case, we discover a pattern resulting in the coefficients of the involved expansions being alternating multiple zeta values (MZVs), which generalizes the notion of Riemann's zeta values to multiple integer variables. This allows us to extend a new existence proof of the Lawson surfaces $\xi_{1,g}$ to all $g\geq 3$ using complex analytic methods and to give closed form expressions of their area expansion up to order $7$. For example, the third order coefficient is $\tfrac{9}{4}\zeta(3)$ (the first and second order term were shown to be $\log(2)$ and $0$ respectively in \cite{HHT}). As a corollary, we obtain that the area of $\xi_{1,g}$ is monotonically increasing in their genus $g$ for all $g\geq 0.$

math.DG↗

arXiv2024

Minimal reflection surfaces in $\mathbb S^3.$ Combinatorics of curvature lines and minimal surfaces based on fundamental pentagons

We study compact minimal surfaces in the 3-sphere which are constructed by successive reflections from a minimal $n$-gon -- so-called minimal reflection surfaces. The minimal $n$-gon solves a free boundary problem in a fundamental piece of the respective reflection group. We investigate the combinatorics of the curvature lines of reflection surfaces, and construct new examples of minimal reflection surfaces based on pentagons. We end the paper by discussing the area of these minimal surfaces.

math.DG↗

arXiv2023

On the monodromy of holomorphic differential systems

First we survey and explain the strategy of some recent results that construct holomorphic $\text{sl}(2, \mathbb C)$-differential systems over some Riemann surfaces $Σ_g$ of genus $g\geq 2$, satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian \cite{BDHH} or some cocompact Kleinian subgroup $$Γ\subset \text{SL}(2, \mathbb C)$$ as in \cite{BDHH2}. As a consequence, there exist holomorphic maps from $Σ_g$ to the quotient space $\text{SL}(2, \mathbb C)/ Γ$, where $Γ\subset \text{SL}(2, \mathbb C)$ is a cocompact lattice, that do not factor through any elliptic curve \cite{BDHH2}. This answers positively a question of Ghys in \cite{Gh}; the question was also raised by Huckleberry and Winkelmann in \cite{HW}. Then we prove that when $M$ is a Riemann surface, a Torelli type theorem holds for the affine group scheme over $\mathbb C$ obtained from the category of holomorphic connections on {\it étale trivial} holomorphic bundles. After that, we explain how to compute in a simple way the holonomy of a holomorphic connection on a free vector bundle. Finally, for a compact Kähler manifold $M$, we investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over $M$. If $\varpi$ (respectively, $Θ$) stands for the affine group scheme over $\mathbb C$ obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism $v:{\mathcal O}(Θ)\longrightarrow {\mathcal O}(\varpi)$ of Hopf algebras. We present a description of the transpose of $v$ in terms of the iterated integrals.

math.DG↗

arXiv2023

Higgs bundles and SYZ geometry

Using non-Abelian Hodge theory for parabolic Higgs bundles, we construct infinitely many non-congruent hyperbolic affine spheres modeled on a thrice-punctured sphere with monodromy in $\mathrm{SL}_3(\mathbb{Z})$. These give rise to non-isometric semi-flat Calabi-Yau metrics on special Lagrangian torus bundles over an open ball in $\mathbb{R}^{3}$ minus a Y-vertex, thereby answering a question raised by Loftin, Yau, and Zaslow in [LYZ], [LYZerr].

math.DG↗

arXiv2023

Real Slices of ${\rm SL}(r,\mathbb{C})$-Opers

Through the action of an anti-holomorphic involution $σ$ (a real structure) on a Riemann surface $X$, we consider the induced actions on ${\rm SL}(r,\mathbb{C})$-opers and study the real slices fixed by such actions. By constructing this involution for different descriptions of the space of ${\rm SL}(r,\mathbb{C})$-opers, we are able to give a natural parametrization of the fixed point locus via differentials on the Riemann surface, which in turn allows us to study their geometric properties.

math.DG↗

arXiv2023

Holomorphic Higgs bundles over the Teichm\"uller space

We study which representations $\rho$ of the fundamental group of a compact oriented surface $X$ admit Higgs data that depend holomorphically on the Riemann surface $\Sigma\,=\, (X,\, J)$ via non-abelian Hodge correspondence. For representations $\rho$ into $\mathrm{SL}(2,\mathbb C)$ we show that holomorphic dependency is equivalent to $\rho$ being unitary. For higher ranks this equivalence fails -- we show the existence of non-unitary and irreducible representations of the fundamental group into $\mathrm{SL}(n,\mathbb C)$ admitting Higgs data that are holomorphic in $\Sigma$, for $n$ large enough.

math.DG↗

arXiv2023

Parabolic opers and differential operators

Parabolic SL(r,C)-opers were defined and investigated in [BDP] in the set-up of vector bundles on curves with a parabolic structure over a divisor. Here we introduce and study holomorphic differential operators between parabolic vector bundles over curves. We consider the parabolic SL(r,C)-opers on a Riemann surface X with given singular divisor S and with fixed parabolic weights satisfying the condition that all parabolic weights at any point $x_i$ in S are integral multiples of $\frac{1}{2N_i+1}$, where $N_i > 1$ are fixed integers. We prove that this space of opers is canonically identified with the affine space of holomorphic differential operators of order r between two natural parabolic line bundles on X (depending only on the divisor S and the weights $N_i$) satisfying the conditions that the principal symbol of the differential operators is the constant function 1 and the sub-principal symbol vanishes identically. The vanishing of the sub-principal symbol ensures that the logarithmic connection on the rank r bundle is actually a logarithmic SL(r, C)-connection.

math.AG↗

arXiv2022

Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere

The non-abelian Hodge correspondence is a real analytic map between the moduli space of stable Higgs bundles and the deRham moduli space of irreducible flat connections mediated by solutions to the self-duality equations. In this paper we construct self-duality solutions for strongly parabolic $\mathfrak{sl}(2,\mathbb C)$ Higgs fields on a $4$-punctured sphere with parabolic weights $t \sim 0$ using complex analytic methods. We identify the rescaled limit hyper-K\"ahler moduli space $\mathcal M_t$ at $t=0$ to be the completion of the nilpotent orbit in $\mathfrak{sl}(2, \mathbb C)$ modulo a $\mathbb Z_2\times\mathbb Z_2$ action, equipped with the Eguchi-Hanson metric. Our methods and computations are based on the twistor approach to the self-duality equations using Deligne and Simpson's $\lambda$-connections interpretation. By construction we can compute the Taylor expansions of the holomorphic symplectic form $\varpi_t$ on $\mathcal M_t$ at $t=0$ which turn out to have closed form expressions in terms of multiple polylogarithms (MPLs). The geometric properties of $\mathcal M_t$ lead to some identities of certain MPLs which we believe deserve further investigations.

math.DG↗

arXiv2022

Branched SL(r,C)-opers

We define the branched analog of SL(r,C)-opers and investigate their properties. For the usual SL(r,C)-opers, the underlying holomorphic vector bundle is independent of the opers. For the branched SL(r,C)-opers, the underlying holomorphic vector bundle depends on the oper. Given a branched SL(r,C)-oper, we associate to it another holomorphic vector bundle equipped with a logarithmic connection. This holomorphic vector bundle does not depend on the branched oper. We characterize the branched SL(r,C)-opers in terms of the logarithmic connections on this fixed holomorphic vector bundle.

math.AG↗

arXiv2022

Infinitesimal deformations of parabolic connections and parabolic opers

We compute the infinitesimal deformations of quadruples $(X, S, E_*, D)$, where $(X, S)$ is a compact Riemann surface with $n$ marked points, $E_*$ is a parabolic vector bundle on $X$ with parabolic structure over $S$, and $D$ is a parabolic connection on $E_*$. Using it we compute the infinitesimal deformations of $(X, S, D)$, where $D$ is a parabolic SL(r, C)-oper on $(X, S)$. It is shown that the monodromy map, from the moduli space of triples $(X, S, D)$, where $D$ is a parabolic SL(r, C)-oper on $(X, S)$, to the SL(r, C)-character variety of X - S, is an immersion.

math.AG↗

arXiv2022

Fuchsian DPW potentials for Lawson surfaces

The Lawson surfaces $ξ_{1,g}$ of genus $g$ are constructed by rotating and reflecting the Plateau solution $f_t$ with respect to a particular geodesic $4$-gon $Γ_t$ along its boundary, where $t= \tfrac{1}{2g+2}$ is an angle of $Γ_t$. In this paper we combine the existence and regularity of the Plateau solution $f_t$ in $t \in (0, \tfrac{1}{4})$ with topological information about the moduli space of Fuchsian systems on the 4-puncture sphere to obtain existence of a Fuchsian DPW potential $η_t$ for every $f_t$ with $t\in(0, \tfrac{1}{4}]$. Moreover, the coefficients of $η_t$ are shown to depend real analytically on $t$. This implies that the Taylor approximation of the DPW potential $η_t$ and of the area obtained at $t=0$ found in \cite{HHT2} determines these quantities for all $ξ_{1,g}$. In particular, this leads to an algorithm to conformally parametrize all Lawson surfaces $ξ_{1,g}$.

math.DG↗

arXiv2022

Real projective structures on Riemann surfaces and new hyper-Kähler manifolds

The twistor space of the moduli space of solutions of Hitchin's self-duality equations can be identified with the Deligne-Hitchin moduli space of $λ$-connections. We use real projective structures on Riemann surfaces to prove the existence of new components of real holomorphic sections of the Deligne-Hitchin moduli space. Applying the twistorial construction we show the existence of new hyper-Kähler manifolds associated to any compact Riemann surface of genus $g\geq2$. These hyper-Kähler manifolds can be considered as moduli spaces of (certain) singular solutions of the self-duality equations.

math.DG↗

arXiv2022

On the monodromy map for the logarithmic differential systems

We study the monodromy map for logarithmic $\mathfrak g$-differential systems over an oriented surface $S_0$ of genus $g$, with $\mathfrak g$ being the Lie algebra of a complex reductive affine algebraic group $G$. These logarithmic $\mathfrak g$-differential systems are triples of the form $(X, D,\Phi)$, where $(X, D) \in {\mathcal T}_{g,d}$ is an element of the Teichm\"uller space of complex structures on $S_0$ with $d \geq 1$ ordered marked points $D\subset S_0= X$ and $\Phi$ is a logarithmic connection on the trivial holomorphic principal $G$-bundle $X \times G$ over $X$ whose polar part is contained in the divisor $D$. We prove that the monodromy map from the space of logarithmic $\mathfrak g$-differential systems to the character variety of $G$-representations of the fundamental group of $S_0\setminus D$ is an immersion at the generic point, in the following two cases: A) $g \geq 2$, $d \geq 1$, and $\dim_{\mathbb C}G \geq d+2$; B) $g=1$ and $\dim_{\mathbb C}G \geq d$. The above monodromy map is nowhere an immersion in the following two cases: 1) $g=0$ and $d \geq 4$; 2) $g\geq 1$ and $\dim_{\mathbb C}G < \frac{d+3g-3}{g}$. This extends to the logarithmic case the main results in \cite{CDHL}, \cite{BD} dealing with nonsingular holomorphic $\mathfrak g$-differential systems (which corresponds to the case of $d\,=\,0$).

math.AG↗