B. G. Konopelchenko, G. Landolfi
Generalizations of the Weierstrass formulae to generic surface immersed into $R^4$, $S^4$ and into multidimensional Riemann spaces are proposed. Integrable deformations of surfaces in these spaces via the modified Veselov-Novikov equation are discussed.
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The normal Gauss map of a minimal surface in the model space Sol of solvegeometry is a harmonic map with respect to a certain singular Riemannian metric on the extended complex plane.
Let C be a hyperelliptic Riemann surface. We show that the hyperelliptic Weierstrass points of C are non-degenerated critical points of Morse index +2 of the curvature function K of the Theta metric on C (called also Bergman metric). When the genus of C is two, we compute all critical points of K and we show in this case that K is a Morse function.
In this paper, following Sullivan, Kusner, and Schmitt, we study conformal immersions of Riemann surfaces into the three-dimensional Euclidean space. Regarding such immersions as special bundle maps from the tangent bundle of the surface to the cotangent bundle of the 2-dimensional sphere, we generalize the classical Weierstrass representation of minimal surfaces to the case of arbitrary conformal immersions. We study how such an immersion gives rise to a spin structure on the surface together with a pair of spinors and how the immersion itself can be studied by means of these spinors.
We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In particular, we discuss singularities of (1) semi-discrete surfaces with non-zero constant Gaussian curvature, (2) parallel surfaces of semi-discrete minimal and maximal surfaces, and (3) semi-discrete constant mean curvature $1$ surfaces in de Sitter $3$-space. We include comparisons w…
The Cantor Riemannium
本文提出坎托黎曼体，一种由波雷尔单生成延拓生成的度量空间，用于研究解析germ的几何性质。
A generalization of the classical one-dimensional Darboux transformation to arbitrary n-dimensional oriented Riemannian manifolds is constructed using an intrinsic formulation based on the properties of twisted Hodge Laplacians. The classical two-dimensional Moutard transformation is also generalized to non-compact oriented Riemannian manifolds of dimension n greater than one. New examples of quasi-exactly solvable multidimensional matrix Schrödinger operators on curved manifolds are obtained by applying the above results.
In this paper, we will construct an example of a closed Riemann surface $X$ that can be realized as a quotient of a triply periodic polyhedral surface $Π\subset \mathbb{R}^3$ where the Weierstrass points of $X$ coincide with the vertices of $Π.$ First we construct $Π$ by attaching Platonic solids in a periodic manner and consider the surface of this solid. Due to periodicity we can find a compact quotient of this surface, which has genus $g = 3.$ We claim that the resulting surface is not hyperelliptic and also that it is regular. By regular, we mean that the automorphism group of $X$ is trans…
We develop analytical methods for nonlinear Dirac equations. Examples of such equations include Dirac-harmonic maps with curvature term and the equations describing the generalized Weierstrass representation of surfaces in three-manifolds. We provide the key analytical steps, i.e., small energy regularity and removable singularity theorems and energy identities for solutions.
A Willmore surface $y:M\rightarrow S^{n+2}$ has a natural harmonic oriented conformal Gauss map $Gr_y:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$, which maps each point $p\in M$ to its oriented mean curvature 2-sphere at $p$. An easy observation shows that all conformal Gauss maps of Willmore surfaces satisfy a restricted nilpotency condition which will be called "strongly conformally harmonic." The goal of this paper is to characterize those strongly conformally harmonic maps from a Riemann surface $M$ to $SO^+ (1, n + 3)/{SO^+(1, 3) \times SO(n) }$ which are the conformal Gauss maps of s…
Demoulin surfaces in real projective $3$-space are investigated. Our result enable us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian $6$-symmetric space.
In this paper, we show that a complete embedded minimal surface in $\Real^3$ with finite topology and one end is conformal to a once-punctured compact Riemann surface. Moreover, using the conformality and embeddedness, we examine the Weierstrass data and conclude that every such surface has Weierstrass data asymptotic to that of the helicoid. More precisely, if $g$ is the stereographic projection of the Gauss map, then in a neighborhood of the puncture, $g(p) = \exp(iαz(p) + F(p))$, where $α\in \Real$, $z=x_3+ix_3^*$ is a holomorphic coordinate defined in this neighborhood and $F(p)$ is holomo…
In this paper, the Weierstrass technique for harmonic maps S^2 -> CP^(N-1) is employed in order to obtain surfaces immersed in multidimensional Euclidean spaces. It is shown that if the CP^(N-1) model equations are defined on the sphere S^2 and the associated action functional of this model is finite, then the generalized Weierstrass formula for immersion describes conformally parametrized surfaces in the su(N) algebra. In particular, for any holomorphic or antiholomorphic solution of this model the associated surface can be expressed in terms of an orthogonal projector of rank (N-1). The impl…
A conformal map from a Riemann surface to a Euclidean space of dimension greater than or equal to three is explained by using the Clifford algebra, in a similar fashion to quaternionic holomorphic geometry of surfaces in the Euclidean three- or four-space. The Weierstrass representation, the spin transform, the Darboux transforms, surfaces of parallel mean curvature vector, families of flat connections associated with a harmonic map from a Riemann surface to a sphere are explained. The degree of the spinor bundle associated with a conformal immersion is calculated. Analogues of a polar surface…
Complete minimal submanifolds of the non-compact Riemannian symmetric spaces SL_n(R)/SO(n), Sp(n,R)/U(n), SO*(2n)/U(n), SU*(2n)/Sp(n) via complex-valued eigenfunctions
本文通过复值特征函数构造了非紧黎曼对称空间的二维完整极小子流形的新多维家族，展示了其核心方法与主要贡献。
Runge and Mergelyan theorems on families of open Riemann surfaces
本文针对带有连续或光滑依赖于参数的Hölder类复结构族的开定向曲面，构造了到任意Oka流形的全纯映射族，并证明了Runge逼近、Mergelyan逼近和Weierstrass插值定理，应用于构造定向全纯浸入和共形极小浸入族。
These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains M in R^3 with infinite topology (equivalently, with an infinite number of ends). This final classification result by Meeks, Perez, and Ros states that such an M must be congruent to a homothetic scaling of one of the classical examples found by Riemann in 1860. These examples {\cal R}_s, 0<s<\infty, are defined in terms of the Weierstrass {\cal P}-functions {\cal P}_t on the rectangular elliptic curve {\C} / {< 1, t\sqrt{-1}>}, are singly-pe…
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