DPW Potentials for Compact Symmetric CMC Surfaces in $\mathbb{S}^3$

The spectral curve theory for $(k,l)-$symmetric CMC surfaces

Constant mean curvature surfaces in $S^3$ can be studied via their associated family of flat connections. In the case of tori this approach has led to a deep understanding of the moduli space of all CMC tori. For compact CMC surfaces of higher genus the theory is far more involved due to the non abelian nature of their fundamental group. In this paper we extend the spectral curve theory for tori developed in \cite{Hi, PiSt} and for genus $2$ surfaces \cite{He3} to CMC surfaces in $S^3$ of genus $g=k\cdot l$ with commuting $\mathbb Z_{k+1}$ and $\mathbb Z_{l+1}$ symmetries. We determine their a…

On the asymptotic Plateau's problem for CMC hypersurfaces on rank 1 symmetric spaces of noncompact type

Let $M$ be a Hadamard manifold with curvature bounded above by a negative constant $-α$, satisfying the "strict convexity condition", and assume that $M$ admits a "helicoidal" one-parameter subgroup $G$ of isometries of $M$. Then, given a compact topological $G-$shaped hypersurface $Γ$ in the asymptotic boundary of $M,$ and $|H|<\sqrtα$, we prove the existence of a complete properly embedded hypersurface whose mean curvature is equal to $H$ and whose asymptotic boundary is $Γ$. We are able, this way, to extend a previous theorem of B.Guan and J.Spruck on the hyperbolic space to any rank 1 symm…

A spectral curve approach to Lawson symmetric CMC surfaces of genus 2

Minimal and CMC surfaces in $S^3$ can be treated via their associated family of flat $\SL(2,\C)$-connections. In this the paper we parametrize the moduli space of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of Lawson's geometric construction. The parametrization uses Hitchin's abelianization procedure to write such connections explicitly in terms of flat line bundles on a complex 1-dimensional torus. This description is used to develop a spectral curve theory for the Lawson surface. This theory applies as well t…

Towards a classification of CMC-1 Trinoids in hyperbolic space via conjugate surfaces

We derive necessary conditions on the parameters of the ends of a CMC-1 trinoid in hyperbolic 3-space $H^{3}$ with symmetry plane by passing to its conjugate minimal surface. Together with Daniel's results, this yields a classification of generic symmetric trinoids. We also discuss the relation to other classification results of trinoids by Bobenko et al. and Umehara-Yamada. To obtain the result above, we show that the conjugate minimal surface of a catenoidal CMC-1 end in $H^{3}$ with symmetry plane is asymptotic to a suitable helicoid.

Spacelike spherically symmetric CMC hypersurfaces in Schwarzschild spacetimes (I): Construction

We solve spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in Schwarzschild spacetimes and analyze their asymptotic behavior near the coordinate singularity r = 2M. Furthermore, we join SS-CMC hypersurfaces in the Kruskal extension to obtain complete ones and discuss the smooth properties.

Spacelike spherically symmetric CMC foliation in the extended Schwarzschild spacetime

We first summarize the characterization of smooth spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in the Schwarzschild spacetime and Kruskal extension. Then use the characterization to prove special SS-CMC foliation property, and verify part of the conjecture by Malec and Ó Murchadha in their 2003 paper.

Gap results for free boundary CMC surfaces in conformally Euclidean three-balls

In this work, we consider $M=(\mathbb{B}^3_r,\bar{g})$ as the Euclidean three-ball with radius $r$ equipped with the metric $\bar{g}=e^{2h}\left\langle , \right\rangle$ conformal to the Euclidean metric. We show that if a free boundary CMC surface $Σ$ in $M$ satisfies a pinching condition on the length of the traceless second fundamental tensor which involves the support function of $Σ$, the positional conformal vector field $\vec{x}$ and its potential function $σ,$ then either $Σ$ is a disk or $Σ$ is an annulus rotationally symmetric. In a particular case, we construct an example of minimal s…

Triviality of the dressing isotropy for a Smyth-type potential and nonclosing of the resulting CMC surfaces

The quantum cohomology of CP^1 is generated by some potential (Frobenius manifold) that also has an interpretation as a potential of some harmonic map. Actually, the potential induces harmonic maps into three different symmetric spaces and each of these harmonic maps induces an immersion of an integrable surface. The final goal of this paper is to investigate if any of the associated immersions closes around the singularity at z=0 (none does). The crucial tool for deciding this question is the main technical result of the paper, which states that the dressing isotropy of the CP^1 potential is …

Real forms of complex surfaces of constant mean curvature

It is known that complex constant mean curvature ({\sc CMC} for short) immersions in $\mathbb C^3$ are natural complexifications of {\sc CMC}-immersions in $\mathbb R^3$. In this paper, conversely we consider {\it real form surfaces} of a complex {\sc CMC}-immersion, which are defined from real forms of the twisted $\mathfrak{sl}(2, \mathbb C)$ loop algebra $Λ\mathfrak{sl}(2, \mathbb C)_σ$, and classify all such surfaces according to the classification of real forms of $Λ\mathfrak{sl}(2, \mathbb C)_σ$. There are seven classes of surfaces, which are called {\it integrable surfaces}, and all int…

CMC tori of revolution in $\mathbb{S}^3$: additional data on the spectra of their Jacobi operators

We prove a theorem about elliptic operators with symmetric potential functions, defined on a function space over a closed loop. The result is similar to a known result for a function space on an interval with Dirichlet boundary conditions. These theorems provide accurate numerical methods for finding the spectra of those operators over either type of function space. As an application, we numerically compute the Morse index of constant mean curvature tori of revolution in the unit 3-sphere $\mathbb{S}^3$, confirming that every such torus has Morse index at least five, and showing that other kno…

An extension of Ruh-Vilms Theorem for hypersurfaces in symmetric spaces and some applications

The main purpose of the paper is twofold: First, to extend a well known theorem of Ruh-Vilms in the Euclidean space to symmetric spaces and, secondly, to apply this result to extend Hoffman-Osserman-Schoen Theorem (HOS Theorem) to 3-dimensional symmetric spaces. Precisely, it is defined a Gauss map of a hypersurface M^{n-1} immersed in a symmetric space N^n taking values in the unit pseudo sphere S^m of the Lie algebra g of the isometry group of N, dim(g)=m+1, and it is proved that M has CMC if and only if its Gauss map is harmonic. As an application, it is proved that if dim(N)=3 and the imag…

Dirichlet problem for the constant mean curvature equation and CMC foliation in the extended Schwarzschild spacetime

We prove the existence and uniqueness of the Dirichlet problem for spacelike, spherically symmetric, constant mean curvature equation with symmetric boundary data in the extended Schwarzschild spacetime. As an application, we completely solve the CMC foliation conjecture which is posted by Malec and O Murchadha in 2003.

Boundary behaviors of spacelike constant mean curvature surfaces in Schwarzschild spacetime

We prove that a spacelike spherical symmetric constant mean curvature (SSCMC) surface and a general spacelike constant mean curvature (CMC) surface with certain boundary condition at the future null-infinity in Schwarzschild spacetime are asymptotically hyperbolic in the sense of Wang \cite{Wang2001} and Chruściel-Herzlich \cite{ChruscielHerzlich} respectively. Near the future null-infinity ($s=0$), we derive that the boundary data of spacelike CMC surfaces can be expressed as those on $\mathbb{S}^{2}$ up to three order and obtain a compatibility condition for fourth order derivatives near $s=…

通过基于基点的规范化扩展标架研究具有有限 uniton 数的内对称空间调和映射

Harmonic maps of finite uniton number into inner symmetric spaces via based normalized extended frames

本文发展了从黎曼曲面到内对称空间的有限 uniton 数调和映射的环路群描述，将基于 Bruhat 胞的理论转化为基于 Birkhoff 胞的 DPW 理论，并证明了此类调和映射与特定规范化势之间的一一对应关系，为构造所有此类调和映射提供了直接方法。

Constant mean curvature $1/2$ surfaces in $\mathbb{H}^2\times\mathbb{R}$ asymptotic to the ends of horizontal catenoids

This thesis lies in the field of constant mean curvature (cmc) hypersurfaces and specifically cmc $1/2$ surfaces in the three-manifold $\mathbb{H}^2\times\mathbb{R}$. The value $1/2$ is the critical mean curvature for $\mathbb{H}^2\times\mathbb{R}$, in that there do no exist closed cmc surfaces with mean curvature $1/2$ or less. Daniel and Hauswirth have constructed a one-parameter family of complete, cmc $1/2$ annuli that are symmetric about a reflection in the horizontal plane $\mathbb{H}^2 \times \{0\}$, the horizontal catenoids. In this thesis we prove that these catenoids converge to a si…

通过环群方法研究二维复双曲二次曲面的极小拉格朗日面

Minimal Lagrangian surfaces in the two-dimensional complex hyperbolic quadric via the loop group method

本文研究复双曲二次曲面中的极小拉格朗日面，通过环群方法证明极小性由平坦连接的环表示特征化，并建立与反德西特3空间中类空最大面的对应关系，进而发展出DPW型表示并构造出具体的例子，包括R-等价和径向对称面。

S^n×R和H^n×R中CMC超曲面的Abresch-Rosenberg型张量与间隙定理

An Abresch-Rosenberg type tensor and a gap theorem for CMC hypersurfaces in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$

该研究构造了S^n×R和H^n×R中CMC超曲面的Abresch-Rosenberg型对称无迹张量，结合积分Simons恒等式得到显式间隙定理，还确定了阈值的二阶大h渐近行为。

On symmetries of constant mean curvature surfaces

We start the investigation of immersions $Ψ$ of a simply connected domain $D$ into three dimensional Euclidean space $R^3$, which have constant mean curvature (CMC-immersions), and allow for a group of automorphisms of $D$ which leave the image $Ψ(D)$ invariant. On one hand, this leads to a detailed description of symmetric CMC-surfaces and the associated symmetry groups. On the other hand, it allows us to start the classification of CMC-immersions of an arbitrary, compact or noncompact Riemann surface $M$ into $R^3$ in terms of Weierstrass-type data, as introduced by Pedit, Wu, and one of the…

Rotational Surfaces in S^3 with constant mean curvature

Very recently Ben Andrews and Haizhong Li showed that every embedded cmc torus in the three dimensional sphere is axially symmetric. There is a two-parametric family of axially symmetric cmc surfaces; more precisely, for every real number H and every C > 2 (H+\sqrt{1+H^2}) there is an axially symmetry surface Σ_{H,C} with mean curvature H. In 2010, Perdomo showed that for every H between cot(π/m) and (m^2-2)/(2(m^2-1)^1/2), there exists an embedded axially symmetric example with non constant principal curvatures that is invariant under the ciclic group Z_m. Andrews and Li, showed that these ex…

Willmore surfaces in spheres via loop groups $I$: generic cases and some examples

In this paper we deal with the global properties of Willmore surfaces in spheres via the harmonic conformal Gauss map using loop groups. We first derive a global description of those harmonic maps which can be realized as conformal Gauss maps of some Willmore surfaces (Theorem 3.4, Theorem 3.11 and Theorem 3.18). Then we introduce the DPW procedure for these harmonic maps, and state appropriate versions of the Iwasawa decomposition and the Birkhoff decomposition Theorems. In particular, we show how the harmonic maps associated with Willmore surfaces can be constructed in terms of loop groups. …

具有对称边界的超曲面的对称性

Symmetry of hypersurfaces with symmetric boundary

本文证明具有对称边界和接触角的最小及常平均曲率超曲面内部具有对称性，并推广至高阶平均曲率和Helfrich型超曲面。

Generalized Doubling Constructions for Constant Mean Curvature Hypersurfaces in the (n+1)-Sphere

The (n+1)-sphere contains a simple family of constant mean curvature (CMC) hypersurfaces which are products of lower-dimensional spheres called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically non-trivial CMC hypersurfaces resembling a pair of neighbouring generalized Clifford tori connected to each other by small catenoidal bridges at a sufficiently symmetric configuration of points can be constructed by perturbative PDE methods. That is, one can create an approximate solution by gluing a rescaled catenoid into the neighbourhood of each point; and then o…

通过环群方法研究二维复二次曲面的极小拉格朗日面

Minimal Lagrangian surfaces in the two dimensional complex quadric via the loop group method

本文通过环群方法研究复二次曲面中的极小拉格朗日面，建立了与$\mathbb S^{3}$中极小面的对应关系，并给出了多种显式例子。

关于Bryant对应关系的规范理论推广

A gauge theoretical generalization of Bryant's correspondence

本文提出了一种规范理论下的Bryant对应关系推广，将空间状全纯浸入与共形浸入通过特定的双线性形式联系起来。

The nonlinear stability of rotationally symmetric spaces with low regularity

We consider rotationally symmetric spaces with low regularity, which we regard as integral currents spaces or manifolds with Sobolev regularity and are assumed to have nonnegative scalar curvature. Relying on the flat distance and on Sobolev norms, we establish several nonlinear stability estimates about the ``distance'' between a rotationally symmetric manifold and the Euclidian space, which are stated in terms of the ADM mass of the manifold. Importantly, we make explicit the dependencies and scales involved in this problem, particularly the ADM mass, the depth, and the CMC reference hypersu…

$\mathbb{S}^2$上的极值领域：几何与分析方法

Extremal domains in $\mathbb{S}^2$: Geometric and Analytic methods

本文研究$\mathbb{S}^2$上支持过定问题的极值领域，通过几何与分析方法证明其对称性及特定非线性情况下的旋转对称性。