Symmetric Pseudospherical Surfaces I: General Theory

关于伪球面曲面描述方程的回顾

A look on equations describing pseudospherical surfaces

本文回顾了伪球面曲面描述方程的发展，从Sasaki到当前柯西问题及几何影响的研究。

A class of quasilinear second order partial differential equations which describe spherical or pseudospherical surfaces

Second order partial differential equations which describe spherical surfaces (ss) or pseudospherical surfaces (pss) are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature $K = 1$ or $K = -1$, respectively, and they can be seen as the compatibility condition of an associated su(2)-valued or sl(2, R)-valued linear problem, also referred to as a zero curvature representation. Under certain assumptions we give a complete and explicit classiffcation of equations of the form $z_{tt} = A(z, z_x , z_t) z_{xx} + B(z, z_x , z_t ) z_{xt} + C(z, z_x …

Third order differential equations and local isometric immersions of pseudospherical surfaces

The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and non-local) and the presence associated linear problems. It thus contains many important known examples of integrable equations, like the sine-Gordon, Liouville, KdV, mKdV, Camassa-Holm and Degasperis-Procesi equations, and is also home to many new families of integrable equations. Our paper is concerned with the question of the local isometric immersion in ${\bf E}^{3}$ of the ps…

A class of third order partial differential equations describing spherical or pseudospherical surfaces

Third order equations, which describe spherical surfaces (ss) or pseudospherical surfaces (pss), of the form \[ ν\,z_{t}-λ\,z_{xxt}=A(z,z_{x},z_{xx})\,z_{xxx}+B(z,z_{x},z_{xx}) \] with $ν$, $λ$ $\in$ $\mathbb{R}$, $ν^2+λ^2\neq 0$, are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature $K=1$ or $K=-1$, respectively. Alternatively they can be seen as the compatibility condition of an associated $\mathfrak{su}(2)$-valued or $\mathfrak{sl}(2,\mathbb{R})$-valued linear problem, also referred to as a zero curvature representation. Under certain …

源自柯西问题的周期伪球面的存在性与唯一性

Existence and uniqueness of periodic pseudospherical surfaces emanating from Cauchy problems

该研究利用Kato半群方法证明描述伪球面的Novikov方程周期初值柯西问题解的适定性，进而得到周期伪球面的存在唯一性及相关几何量的正则性结果。

由一类三阶微分方程描述的伪球面的局部等距浸入

Local isometric immersions of pseudospherical surfaces described by a class of third order differential equations

本文研究由一类三阶微分方程定义的伪球面，证明其在三维欧氏空间中局部等距浸入的第二基本形式三元组仅依赖于解的零阶射流，具有通用性。

常曲率旋转网与周期性Bäcklund变换

Constant curvature rotational nets and periodic Bäcklund transforms

本文研究了常负高斯曲率旋转网的Bäcklund变换，构造了具有显式参数化的非旋转离散伪球面新例子，并给出了保持周期性的条件。

Wave maps and constant curvature surfaces: singularities and bifurcations

Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructing germs of Lorentzian-harmonic maps from their $k$-jets and use this construction to study the singularities of such maps. We also show how to construct pseudospherical surfaces with prescribed singularities using loop groups. We study the singularities of pseudospherical surface…

Backlund transformations and knots of constant torsion

The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vortex filament flow (also known as the Localized Induction Equation). In doing so, we obtain analytic constant-torsion representatives for a large number of knot types.

Camassa-Holm方程解的内蕴几何与破波现象

Intrinsic geometry and wave-breaking phenomena in solutions of the Camassa-Holm equation

该研究聚焦Camassa-Holm方程柯西问题对应的伪球面，分析全局解对曲面的影响与两类度量奇点，证明度量爆破等价于解在有限时间内破缺，为理解该方程的破波现象提供几何视角。

Parallel Mean Curvature Surfaces in Symmetric Spaces

We present a reduction of codimension theorem for surfaces with parallel mean curvature in symmetric spaces.

Singular minimal translation surfaces in Euclidean spaces endowed with semi-symmetric connections

In this paper, we study and classify singular minimal translation surfaces in a Euclidean space of dimension 3 endowed with a certain semi-symmetric (non-)metric connection.

Minimal translation surfaces with respect to semi-symmetric connections in $\mathbb{R}^3$ and $\mathbb{R}^3_1$

In this paper, we define and classify minimal translation surfaces with respect to a kind of semi-symmetric metric connections and a kind of semi-symmetric non-metric connections in $\mathbb{R}^3$ and $\mathbb{R}^3_1$.

Semi-symmetric Kahler surfaces

The aim of this paper is to describe Kahler surfaces which admit an opposite almost Hermitian structure satisfying the first Gray condition

Exploring the space of compact symmetric CMC surfaces

We map out the moduli space of Lawson symmetric constant mean curvature surfaces in the 3-sphere of genus $g>1$ by flowing numerically from Delaunay tori with even lobe count via the generalized Whitham flow.

Connections with skew-symmetric Ricci tensor on surfaces

Some known results on torsionfree connections with skew-symmetric Ricci tensor on surfaces are extended to connections with torsion, and Wong's canonical coordinate form of such connections is simplified.

Incompressible surfaces in rank one locally symmetric spaces

We show that cocompact lattices in rank one simple Lie groups of non-compact type distinct from SO(2m,1) (m>0) contain surface subgroups.

Rotationally symmetric harmonic diffeomorphisms between surfaces

In this paper, we show that the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.

Deformations of symmetric CMC surfaces in the 3-sphere

In this paper we numerically construct CMC deformations of the Lawson minimal surfaces $ξ_{g,1}$ using a spectral curve and a DPW approach to CMC surfaces in spaceforms.

The moduli space of Type~A surfaces with torsion and non-singular symmetric Ricci tensor

We examine the moduli spaces of Type~A connections on oriented and unoriented surfaces both with and without torsion in relation to the signature of the associated symmetric Ricci tensor. If the signature of the symmetric Ricci tensor is (1,1) or (0,2), the moduli spaces are smooth. If the signature is (2,0), there is an orbifold singularity.

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

Inspired by the work of Heller [12], we show that there exists a DPW potential for the Lawson surface $ξ_{k-1, l-1}$ from which it is possible to reconstruct the minimal immersion $f: ξ_{k-1, l-1} \to \mathbb{S}^3$ via the DPW method. Moreover, we extend the result to surfaces immersed in the 3-sphere with constant mean curvature which satisfy a certain symmetric condition.

On the curvature of symmetric products of a compact Riemann surface

Let $X$ be a compact connected Riemann surface of genus at least two. The main theorem of arxiv:1010.1488 says that for any positive integer $n \leq 2({\rm genus}(X)-1)$, the symmetric product $S^n(X)$ does not admit any Kähler metric satisfying the condition that all the holomorphic bisectional curvatures are nonnegative. Our aim here is to give a very simple and direct proof of this result of Bökstedt and Romão.

Local obstructions to projective surfaces admitting skew-symmetric Ricci tensor

The equation determining whether a projective structure admits a connection in its given projective class that has skew-symmetric Ricci tensor is an overdetermined system of semi-linear partial differential equations which we call the projective Einstein-Weyl (pEW) equation. In 2-dimensions, we give local obstructions for projective surfaces to admit such a connection in its projective class. The obstructions are the resultants of polynomial equations that have to be satisfied for there to admit any pEW solution.

Hamiltonian stationary Lagrangian surfaces in Hermitian symmetric spaces

This paper is the third of a series on Hamiltonian stationary Lagrangian surfaces. We present here the most general theory, valid for any Hermitian symmetric target space. Using well-chosen moving frame formalism, we show that the equations are equivalent to an integrable system, generalizing the C^2 subcase analyzed in the first article (arXiv:math.DG/0009202). This system shares many features with the harmonic map equation of surfaces into symmetric spaces, allowing us to develop a theory close to Dorfmeister, Pedit and Wu's, including for instance a Weierstrass-type representation. Notice t…

On symmetric Willmore surfaces in spheres I: the orientation preserving case

In this paper we provide a systematic discussion of how to incorporate orientation preserving symmetries into the treatment of Willmore surfaces via the loop group method. In this context we first develop a general treatment of Willmore surfaces admitting orientation preserving symmetries, and then show how to induce finite order rotational symmetries. We also prove, for the symmetric space which is the target space of the conformal Gauss map of Willmore surfaces in spheres, the longstanding conjecture of the existence of meromorphic invariant potentials for the conformal Gauss maps of all com…

关于局部对称多项式度量：黎曼曲面与芬斯勒曲面

On locally symmetric polynomial metrics: Riemannian and Finslerian surfaces

本文研究黎曼与芬斯勒曲面上的局部对称多项式度量，给出二维四次根度量正定的充要条件及三维情形的必要条件与例子，并借助MAPLE验证。

Navigating the Space of Symmetric CMC Surfaces

In this paper we introduce a flow on the spectral data for symmetric CMC surfaces in the $3$-sphere. The flow is designed in such a way that it changes the topology but fixes the intrinsic (metric) and certain extrinsic (periods) closing conditions of the CMC surfaces. For rational times we obtain closed (possibly branched) connected CMC surfaces of higher genus. We prove the short time existence of this flow near the spectral data of (a class of) CMC tori. In particular we prove that flowing the spectral data for the Clifford torus is equivalent to the flow of Plateau solutions by varying the…

具有半对称非度量联络的常截面曲率曲面

Constant sectional curvature surfaces with a semi-symmetric non-metric connection

本文研究赋予典范半对称非度量联络的三维欧氏空间中常截面曲率曲面，分类了母线与联络向量场正交/平行的柱面、轴平行于该场的锥形旋转曲面，并证明截面曲率为1/2时存在与轴正交相交的旋转曲面。

等变本原调和映入 $k-$对称空间及其在 Willmore 曲面中的应用

Equivariant primitive harmonic maps into $k-$symmetric spaces, with applications to Willmore surfaces

本文研究了等变本原调和映入 $k-$对称空间的构造，并应用于 Willmore 曲面，推广了 Lawson 极小 Klein 瓶得到了 $S^3$ 中的 Willmore Moebius 带。

若干极大对称$(2,3,5)$-分布的局部等价性 III：在平面上滚动的An-Nurowski曲面

Local equivalence of some maximally symmetric $(2,3,5)$-distributions III: An-Nurowski surface rolling on a plane

本研究分析An-Nurowski曲面与平面乘积上的圆扭量丛对应的极大对称$(2,3,5)$-分布，计算定义分裂$\frak{g}_2$李代数的相关向量场，推导得到$SL(3,\bR)$在子流形$\bR^2\times \bS^1$上的作用。

Y-奇异极小曲面的Morse指标

Morse index and nullity of rotationally symmetric $Y$-singular minimal surfaces

本文在Y-奇异性构成单个圆周的假设下，通过固定边界问题和Dirichlet-to-Neumann映射计算旋转对称极小Y-奇异曲面的Morse指标，并证明Y-悬链面的指标为1。

Symmetric affine surfaces with torsion

We study symmetric affine surfaces which have non-vanishing torsion tensor. We give a complete classification of the local geometries possible if the torsion is assumed parallel. This generalizes a previous result of Opozda in the torsion free setting; these geometries are all locally homogeneous. If the torsion is not parallel, we assume the underlying surface is locally homogeneous and provide a complete classification in this setting as well.

Affine metrics of locally strictly convex surfaces in affine 4-space

We introduce a new family of affine metrics on a locally strictly convex surface $M$ in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if $M$ is immersed in a locally strictly convex hyperquadric, then the symmetric and the antisymmetric planes coincide and contain the affine normal to the hyperquadric. In particular, any surface immersed in a locally strictly convex hyperquadric is affine semiumbilical with respect to the symmetric or antisymmetric equiaffine planes.

Closed symmetric 3-differentials on complex surfaces

We give a necessary and sufficient condition for a non-degenerate symmetric 3-differential with nonzero Blaschke curvature on a complex surface to be locally representable as a product of three closed holomorphic 1-forms. We give two versions of this condition corresponding to different choices of coordinates, one of which defines a coordinate-free differential operator, answering a question of Bogomolov and de Oliveira.

Twistors, 4-symmetric spaces and integrable systems

An order four automorphism of a Lie algebra gives rise to an integrable system discussed by Terng. We show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a Riemannian symmetric space. Specialising to 4-dimensional target, we find that surfaces with holomorphic mean curvature in 4-dimensional spaces with constant sectional or holomorphic sectional curvatures constitute an integrable system as do Hamiltonian stationary Lagrangian surfaces in a 4-dimensional Hermitian symmetric space (this last being a result of Hele…

On symmetric Willmore surfaces in spheres II: the orientation reversing case

In this paper we provide a systematic treatment of Willmore surfaces with orientation reversing symmetries and illustrate the theory by (old and new) examples. We apply our theory to isotropic Willmore two-spheres in $S^4$ and derive a necessary condition for such ( possibly branched) isotropic surfaces to descend to (possibly branched) maps from $\mathbb{R} P^2$ to $S^4$. The Veronese sphere and several other examples of non-branched Willmore immersions from $\mathbb{R} P^2$ to $S^4$ are derived as an illustration of the general theory. The Willmore immersions of $\mathbb{R} P^2$, just mentio…

Radially Symmetric Solutions To The Graphic Willmore Surface Equation

We show that a smooth radially symmetric solution $u$ to the graphic Willmore surface equation is either a constant or the defining function of a half sphere in ${\mathbb R}^3$. In particular, radially symmetric entire Willmore graphs in ${\mathbb R}^3$ must be flat. When $u$ is a smooth radial solution over a punctured disk $D(ρ)\backslash\{0\}$ and is in $C^1(D(ρ))$, we show that there exist a constant $λ$ and a function $β$ in $C^0(D(ρ))$ such that $u''(r) =\fracλ{2}\log r+β(r)$; moreover, the graph of $u$ is contained in a graphical region of an inverted catenoid which is uniquely determin…

The geometry of locally symmetric affine surfaces

We examine the local geometry of affine surfaces which are locally symmetric. There are 6 non-isomorphic local geometries. We realize these examples as Type A, Type B, and Type C geometries using a result of Opozda and classify the relevant geometries up to linear isomorphism. We examine the geodesic structures in this context. Particular attention is paid to the Lorentzian analogue of the hyperbolic plane and to the pseudosphere.

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…

On Kaehler structures over symmetric products of a Riemann surface

Given a positive integer $n$ and a compact connected Riemann surface $X$, we prove that the symmetric product $S^n(X)$ admits a Kaehler form of nonnegative holomorphic bisectional curvature if and only if $\text{genus}(X) \leq 1$. If $n$ is greater than or equal to the gonality of $X$, we prove that $S^n(X)$ does not admit any Kaehler form of nonpositive holomorphic sectional curvature.

Chern scalar curvature and symmetric products of compact Riemann surfaces

Let $X$ be a compact connected Riemann surface of genus $g\geq 0$, and let ${\rm Sym}^d(X)$, $d \ge 1$, denote the $d$-fold symmetric product of $X$. We show that ${\rm Sym}^d(X)$ admits a Hermitian metric with negative Chern scalar curvature if and only if $g \geq 2$, and positive Chern scalar curvature if and only if $d > g$.

Bergman kernel on Riemann surfaces and Kaehler metric on symmetric products

Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer $k\geq 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle $Ω^{\otimes k}_X$, where $Ø$ is the holomorphic cotangent bundle of $X$. Our first main result estimates the corresponding Bergman metric on $X$ in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of $X$ into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of $Ω^{\otimes k}_X$. The Fubini-Study…

Generating Axially Symmetric Minimal Hyper-surfaces in R^{1,3}

It is shown that, somewhat similar to the case of classical Baecklund transformations for surfaces of constant negative curvature, infinitely many axially symmetric minimal hypersurfaces in 4-dimensional Minkowski-space can be obtained, in a non-trivial way, from any given one by combining the scaling symmetries of the equations in light cone coordinates with a non-obvious symmetry (the analogue of Bianchis original transformation) - which can be shown to be involutive/correspond to a space-reflection.

对称极大曲面方程

The symmetric maximal surface equation

本文研究对称极大曲面方程，建立其带退化边界条件的光滑解存在性，证明解最大化相关面积泛函，为双曲空间极小图面积极小化问题的洛伦兹类比结果。

非紧型对称空间中的不稳定极小曲面

Unstable minimal surfaces in symmetric spaces of non-compact type

本文证明了亏格至少为3的闭曲面在分裂实半单李群作用下存在不稳定极小映射，并利用高能极小映射指数的新下界，在特定参数下推翻了Labourie猜想。

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant.

$\mathrm{Sp}(4, \mathbb{R})$对称空间中具有多项式增长的平面极小曲面

Planar minimal surfaces with polynomial growth in the $\mathrm{Sp}(4, \mathbb{R})$-symmetric space

研究$\mathrm{Sp}(4,\mathbb{R})$对称空间中多项式增长的平面极小曲面的渐近几何，建立wild Higgs丛Hitchin分量与类光多边形边界极大曲面的同胚，并证明其为等变极大曲面的局部极限。

曲率、宏观维度与曲面的对称积

Curvature, macroscopic dimensions, and symmetric products of surfaces

研究曲面对称积的曲率和辛非球面性，揭示其在正标量曲率闭合黎曼流形研究中的作用，区分Gromov提出的两种宏观维度概念，并探讨Gromov-Lawson和Gromov猜想在Kähler投影设定下的联系。

Trapped surface formation for spherically symmetric Einstein-Maxwell-charged scalar field system with double null foliation

In this paper, under spherical symmetry we prove a trapped surface formation criterion for the Einstein-Maxwell-charged scalar field system. We generalize an approach introduced by Christodoulou for studying the Einstein-scalar field. In appendix, with double null foliation we also reprove Christodoulou's result and for Minkowskian incoming characteristic initial data, we improve Christodoulou's bound.

Surfaces with radially symmetric prescribed Gauss curvature

We study conformally flat surfaces with prescribed Gaussian curvature, described by solutions $u$ of the PDE: $Δu(x)+K(x)\exp(2u(x))=0$, with $K(x)$ the Gauss curvature function at $x\in\RR^2$. We assume that the integral curvature is finite. For radially symmetric $K$ we introduce the notion of a least integrally curved surface, and also the notion of when such a surface is critical. With respect to these notions we analyze the radial symmetry of $u$ for the whole spectrum of possible integral curvature values. Under a mild integrability condition which rules out harmonic non-radial behavior …