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Large genus limit of energy minimizing compact minimal surfaces in the 3-sphere

We consider conformal immersions $$f$$ from a 2-dimensional manifold $$M$$ into the 3-dimensional sphere which are minimal or have constant mean curvature $$H$$. These surfaces admit a generalized Weierstrass representation and their Willmore energies, defined as $$\mathcal{W}(f) = \int_{M}(H^2 + 1)dA$$, are given in terms of their Weierstrass data.
Following a generalized Whitham flow, this method allows for the construction of high genus examples of the Lawson surfaces $$\xi_{g,1}$$ and leads to the following estimate on their areas:
$$\mathcal{A}(\xi_{g,1}) = \mathcal{W}(\xi_{g,1}) = 8\pi\left( 1-\frac{\ln 2}{2(g+1)} + O\left(\frac{1}{(g+1)^3}\right) \right).$$

The project aims at finding a pathway towards an implicit function theorem argument in order to confirm the Kusner conjecture in the case of symmetric compact surfaces of fixed high genus, which states that the minimizer of the Willmore energy among them is the Lawson surface $$\xi_{g,1}$$. This will rely on the construction of high genus limits of various families of minimal and constant mean curvature surfaces, together with the computation of their Willmore energies.

## Publications

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 of the self-duality equation. In this paper we construct such solutions for strongly parabolic $\mathfrak{sl}(2,\mathbb C)$ Higgs fields on a $4$-punctured sphere with parabolic weights $t \sim 0$ using loop groups methods through an implicit function theorem argument. We identify the rescaled limit hyper-K\"ahler moduli space at $t=0$ to be (the completion of) the nilpotent orbit in $\mathfrak{sl}(2, \mathbb C)$ equipped 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. Due to the implicit function theorem, Taylor expansions of these quantities can be computed at $t=0$. By construction they have closed form expressions in terms of Multiple-Polylogarithms and their geometric properties lead to some identities of $\Omega$-values which we believe deserve further investigations.

 Link to preprint version

The Lawson surfaces $\xi_{1,g}$ of genus $g$ are constructed by rotating and reflecting the Plateau solution $f_t$ with respect to a particular geodesic $4$-gon $\Gamma_t$ along its boundary, where $t= \tfrac{1}{2g+2}$ is an angle of  $\Gamma_t$.  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 $\eta_t$ for every $f_t$ with $t\in(0, \tfrac{1}{4}]$. Moreover, the coefficients of $\eta_t$ are shown to depend real analytically on $t$. This implies that the Taylor approximation of the DPW potential $\eta_t$ and of the area obtained at $t=0$ found in \cite{HHT2} determines these quantities for  all $\xi_{1,g}$. In particular, this leads to an algorithm to conformally parametrize all Lawson surfaces $\xi_{1,g}$.

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

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For every $g \gg 1$, we show the existence of a complete and smooth family  of closed constant mean curvature surfaces $f_\varphi^g,$ $\varphi \in [0, \tfrac{\pi}{2}],$ in the round $3$-sphere deforming the Lawson surface $\xi_{1, g}$ to a doubly covered geodesic 2-sphere with monotonically increasing Willmore energy. To do so we use an implicit function theorem argument in the parameter $s= \tfrac{1}{2(g+1)}$. This allows us to give an iterative algorithm to compute the power series expansion of the DPW potential and area of $f_\varphi^g$  at $s= 0$ explicitly.  In particular,  we obtain for large genus Lawson surfaces $\xi_{1,g}$, due to the real analytic dependence of its area  and DPW potential on $s,$  a scheme to explicitly compute the coefficients of the power series in $s$ in terms of multilogarithms. Remarkably, the third order coefficient of the area expansion coincides numerically with $\tfrac{9}{4}\zeta(3),$ where $\zeta$ is the Riemann $\zeta$ function (while the first and second order term were shown to be $\log(2)$  and $0$ respectively in \cite{HHT}).

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For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann surface $\Sigma$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal O}^{\oplus 2}_{\Sigma}$ 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 $g$. This also answers a question of \cite{CDHL}. The construction carries over to all very stable and compatible real holomorphic structures over the topologically trivial rank two bundle on $\Sigma$, and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.

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## Team Members

M.Sc. Balázs Márk Békési
Doctoral student
Leibniz-Universität Hannover
balazs.bekesi(at)math.uni-hannover.de

Dr. Lynn Heller
Leibniz-Universität Hannover
lynn.heller(at)math.uni-hannover.de

Dr. Thomas Raujouan
Researcher
Leibniz-Universität Hannover
raujouan(at)math.uni-hannover.de

M.Sc. Max Schult
Doctoral student
Leibniz-Universität Hannover
max.schult(at)math.uni-hannover.de