Vadim V. Varlamov
Representations of Dirac-Hestenes and Dirac spinor fields via coordinates of surfaces conformally immersed into 4-dimensional complex space are proposed. A relation between time evolution of spinor fields and integrable deformations of surfaces is discussed.
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A representation of generalized Weierstrass formulae for an immersion of generic surfaces into a 4-dimensional complex space in terms of spinors treated as minimal left ideals of Clifford algebras is proposed. The relation between integrable deformations of surfaces via mVN-hierarchy and integrable deformations of spinor fields on the surface is also discussed.
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 give a generalized Weierstrass formula for a Lorentz surface conformally immersed in the four-dimensional space $\mathbb{R}^{2,2}$ using spinors and Lorentz numbers. We also study the immersions of a Lorentzian surface in {\bf the} Anti-de Sitter space (a pseudo-sphere in $\mathbb{R}^{2,2}$): we give a new spinor representation formula and deduce the conformal description of a flat Lorentzian surface in that space.
Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces immersed into Lorentzian manifolds and on surfaces conformally immersed into Minkowski spacetime is defined.
This is a survey of results on surfaces in noncommutative three-dimensional Lie groups obtained by using the Weierstrass (spinor) representation of surfaces. It is based on the talk given at the conference "Geometry related to the theory of integrable systems" (RIMS, Kyoto, September 2007).
In this paper we give a geometrically invariant spinorial representation of surfaces in four-dimensional space forms. In the Euclidean space, we obtain a representation formula which generalizes the Weierstrass representation formula of minimal surfaces. We also obtain as particular cases the spinorial characterizations of surfaces in $\R^3$ and in $S^3$ given by T. Friedrich and by B. Morel
In the literature, two approaches to the Weierstrass representation formula using spinors are known, one explicit, going back to Kusner & Schmitt, and generalized by Konopelchenko and Taimanov, and one abstract due to Friedrich, Bayard, Lawn and Roth. In this article, we show that both points of view are indeed equivalent, for surfaces in R^3, Nil_3 and R^4. The correspondence between the equations of both approaches is explicitly given and as a consequence we derive alternate (and simpler) proofs of these previous theorems.
In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Lie groups equipped with left invariant metrics. As applications, we get a spinorial proof of the Fundamental Theorem for submanifolds into Lie groups, we recover previously known representations of submanifolds in $\mathbb{R}^n$ and in the 3-dimensional Lie groups $S^3$ and $E(κ,τ),$ and we get a new spinorial representation for surfaces in the 3-dimensional semi-direct products: this achieves the spinorial representations of surfaces in the 3-dimensional homogeneous spaces. We finally indic…
Spinor representation in isotropic 3-space via Laguerre geometry
该研究在拉盖尔几何框架下开展各向同性3维空间的几何研究，建立基础曲面理论并定义旋转变换以得到共形曲面的旋量表示，还推导出两类平均曲率曲面的表示式，可构造大量显式示例。
The Weierstrass representation for spheres in $\R^3$ and, in particular, effective construction of immersions from data of spectral theory origin is discussed. These data are related to Dirac operators on a plane and on an infinite cylinder and these operators are just representations of Dirac operators acting in spinor bundles over the two-sphere which is naturally obtained as a completion of a plane or of a cylinder. Spheres described in terms of Dirac operators with one-dimensional potentials on a cylinder are completely studied and, in particular, for them a lower estimate of the Willmore …
The first time that the connection between isometric immersion of surfaces and solutions of the Dirac equation appeared in the literature was in the seminal paper of Thomas Friedrich in 1998. In consequence of that, several authors contributed to this topic hereafter, by obtaining the spinorial representation of Spin manifolds with arbitrary dimension and also by presenting a generalization of the Weierstrass representation map, for example. All these results assume that the manifolds and bundles involved carry a Spin structure, however this hypothesis is somehow restrictive, as for instance, …
We derive the Weierstrass (or spinor) representation for surfaces in three-dimensional Lie groups Nil, \tilde{SL}_2, and Sol with Thurston's geometries and establish the generating equations for minimal surfaces in these groups. By using the spectral properties of the corresponding Dirac operators we find analogs of the Willmore functional for surfaces in these geometries.
Surfaces via spinors and soliton equations
本文综述了三维与四维空间中曲面的 Weierstrass 表示及其与 Willmore 泛函的关系，并应用该表示构造了 Davey-Stewartson II 方程的新型解。
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…
We study the action of conformal transformations of the ambient space on the Dirac operator coming into the Weierstrass (or spinor) representation of a torus in the Euclidean four-space. It is showed that such an action generates a flow acting on the potential of the operator, that this flow is described by a nonlinear system of the Melnikov type and that it preserves the Floquet multipliers of the Dirac operator with double-periodic potential. However this flow is only almost isospectral since it does not preserve the spectral curve in general and its action may result in adding or removing m…
A classical result of Gelfand shows that the topologized spectrum of characters on commutative Banach algebra is homeomorphic to the underlying space. This fact is used in solving the Calderón problem in dimension 2 via the boundary control (BC) method. To apply the BC method in dimension 3, the algebra of complex holomorphic functions can be replaced by the space of harmonic quaternion fields, but this space is no longer an algebra and is not commutative. Nonetheless, it has been shown that a suitable notion of a spectrum exists and in the case when the underlying space is convex, the spectru…
Complex Quaternions and Superminimal Surfaces in Four-Space
利用复四元数代数，将共形超极小曲面表示为全纯零曲线因子分解，并证明超极小性等价于广义高斯映射的一个分量恒定。
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