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Sebastian Heller

Publications and source records attributed to Sebastian Heller.

At least 37 records · Page 2Linked to original sources
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.

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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 $(X, D,Φ)$, where $(X, D) \in {\mathcal T}_{g,d}$ is an element of the Teichmüller space of complex structures on $S_0$ with $d \geq 1$ ordered marked points $D\subset S_0= X$ and $Φ$ 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↗
arXiv2021

On the existence of holomorphic curves in compact quotients of $\mathrm{SL}(2,\mathbb C)$

We prove the existence of a pair $(Σ,\, Γ)$, where $Σ$ is a compact Riemann surface with $\text{genus}(Σ)\, \geq\, 2$, and $Γ\, \subset\, {\mathrm SL}(2, \mathbb C)$ is a cocompact lattice, such that there is a generically injective holomorphic map $Σ\, \longrightarrow\, {\mathrm SL}(2, \mathbb C)/Γ$. This gives an affirmative answer to a question raised by Huckleberry and Winkelmann \cite{HW} and by Ghys \cite{Gh}.

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arXiv2021

Area Estimates for High genus Lawson surfaces via DPW

Starting at a saddle tower surface, we give a new existence proof of the Lawson surfaces $ξ_{m,k}$ of high genus by deforming the corresponding DPW potential. As a byproduct, we obtain for fixed $m$ estimates on the area of $ ξ_{m,k}$ in terms of their genus $g=m k \gg1$.

math.DG↗
arXiv2021

Uniformization of branched surfaces and Higgs bundles

Given a compact Riemann surface $Σ$ of genus $g_Σ\, \geq\, 2$, and an effective divisor $D\, =\, \sum_i n_i x_i$ on $Σ$ with $\text{degree}(D)\, <\, 2(g_Σ-1)$, there is a unique cone metric on $Σ$ of constant negative curvature $-4$ such that the cone angle at each $x_i$ is $2πn_i$ (see McOwen and Troyanov [McO,Tr]). We describe the Higgs bundle corresponding to this uniformization associated to the above conical metric. We also give a family of Higgs bundles on $Σ$ parametrized by a nonempty open subset of $H^0(Σ,\,K_Σ^{\otimes 2}\otimes{\mathcal O}_Σ(-2D))$ that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin's results in [Hi1], for the case $D\,=\, 0$.

math.DG↗
arXiv2021

Branes and moduli spaces of Higgs bundles on smooth projective varieties

Given a smooth complex projective variety $M$ and a smooth closed curve $X \subset M$ such that the homomorphism of fundamental groups $π_1(X) \rightarrow π_1(M)$ is surjective, we study the restriction map of Higgs bundles, namely from Higgs bundles on $M$ to those on $X$. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on $M$ and $X$. We also consider the set-up where a finite group is acting on $M$ via holomorphic automorphisms or anti-holomorphic involutions, and the curve $X$ is preserved by the action. Branes are studied in this context.

math.AG↗
arXiv2021

On certain Tannakian categories of integrable connections over Kaehler manifolds

Given a compact Kaehler manifold X, it is shown that pairs of the form (E, D), where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on $E$, produce a neutral Tannakian category. The corresponding pro-algebraic affine group scheme is studied. In particular, it is shown that this pro-algebraic affine group scheme for a compact Riemann surface determines uniquely the isomorphism class of the Riemann surface.

math.AG↗
arXiv2021

Holomorphic $\mathfrak{sl}(2,\mathbb C)$-systems with Fuchsian monodromy (with an appendix by Takuro Mochizuki)

For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann surface $Σ$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal O}^{\oplus 2}_Σ$ admits holomorphic connections with $\text{SL}(2,{\mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. The construction carries over to all very stable and compatible real holomorphic structures for the topologically trivial rank two bundle over $Σ$ and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.

math.AG↗
arXiv2021

Constant mean curvature surfaces based on fundamental quadrilaterals

We describe the construction of CMC surfaces with symmetries in $\mathbb S^3$ and $\mathbb R^3$ using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The fundamental piece is constructed by the generalized Weierstrass representation using a geometric flow on the space of potentials.

math.DG↗
arXiv2020

Willmore spheres in the 3-sphere revisited

Bryant \cite{Bryant84} classified all Willmore spheres in $3$-space to be given by minimal surfaces in $\mathbb R^3$ with embedded planar ends. This note provides new explicit formulas for genus 0 minimal surfaces in $\mathbb R^3$ with $2k+1$ embedded planar ends for all $k\geq4.$ Peng and Xiao claimed these examples to exist in \cite{PengXiao2000}, but in the same paper they also claimed the existence of a minimal surface with 7 embedded planar ends, which was falsified by Bryant \cite{Bryant88}.

math.DG↗
arXiv2020

Higher solutions of Hitchin's self-duality equations

Solutions of Hitchin's self-duality equations corresponds to special real sections in the Deligne-Hitchin moduli space -- twistor lines. A question posed by Simpson in 1997 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would allow for complex analytic procedure to obtain all solutions of the self-duality equations. The purpose of this paper is to construct counter examples given by certain (branched) Willmore surfaces in $3$-space (with monodromy) via the generalized Whitham flow. Though these higher solutions do not give rise to global solutions of the self-duality equations on the whole Riemann surface $M$, they are solutions on an open dense subset of it. This suggest a deeper connection between Willmore surfaces, i.e., rank $4$ harmonic maps theory, with the rank $2$ self-duality theory.

math.DG↗
arXiv2020

Energy of sections of the Deligne-Hitchin twistor space

We study a natural functional on the space of holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We give a link to a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. Moreover, we prove that for a certain class of real holomorphic sections of the Deligne-Hitchin moduli space, the functional is basically given by the Willmore energy of corresponding (equivariant) conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne-Hitchin moduli space from the space of twistor lines.

math.DG↗
arXiv2019

Isothermic constrained Willmore tori in 3-space

We show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore energy below $8π$. In particular, every constrained Willmore torus with Willmore energy below $8π$ and non-rectangular conformal class is non-degenerated.

math.DG↗
arXiv2019

Minimal n-Noids in hyperbolic and anti-de Sitter 3-space

We construct minimal surfaces in hyperbolic and anti-de Sitter 3-space with the topology of a $n$-punctured sphere by loop group factorization methods. The end behavior of the surfaces is based on the asymptotics of Delaunay-type surfaces, i.e., rotational symmetric minimal cylinders. The minimal surfaces in $\mathrm{H}^3$ extend to Willmore surfaces in the conformal 3-sphere $\mathrm{S}^3=\mathrm{H}^3\cup\mathrm{S}^2\cup\mathrm{H}^3$.

math.DG↗
arXiv2019

Rectangular constrained Willmore minimizers and the Willmore conjecture

We show that the well-known family of $2$-lobed Delaunay tori $\;f^b\;$ in $\;S^3,\;$ parametrized by $\;b \in \mathbb R_{\geq1},\;$ uniquely minimizes the Willmore energy among all immersions from tori into $3$-space of conformal class $\;(a, b)\;$. As a corollary we obtain an alternate proof of the Willmore conjecture in $3$-space. This new strategy can be generalized to arbitrary codimensions provided a classification of isothermic constrained Willmore tori is possible and all $\;f^b\;$ remain stable in all codimensions.

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