Weierstrass Relation for a Submanifold $S^k$ in $\EE^n$ Coming from Submanifold Dirac Operator

Generalized Weierstrass Relations and Frobenius reciprocity

This article investigates local properties of the further generalized Weierstrass relations for a spin manifold $S$ immersed in a higher dimensional spin manifold $M$ from viewpoint of study of submanifold quantum mechanics. We show that kernel of a certain Dirac operator defined over $S$, which we call submanifold Dirac operator, gives the data of the immersion. In the derivation, the simple Frobenius reciprocity of Clifford algebras $S$ and $M$ plays important roles.

The Weierstrass representation for pluriminimal submanifolds

In this note we prove a Weierstrass representation formula for pluriminimal submanifolds of euclidean spaces. We use this formula to produce new families of examples of pluriminimal submanifolds. We also prove that any affine algebraic manifold can be pluriminimally embedded into some euclidean space in a non holomorphic manner.

Submanifold Differential Operators in $\Cal D$-Module Theory II: Generalized Weierstrass and Frenet-Serret Relations as Dirac Equations

This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in $n$-dimensional euclidean space $\EE^n$ to a surface or a space curve as physical models. These Dirac operators are identified with operators of the Frenet-Serret relation for a space curve case and of the generalized Weierstrass relation for a conformal surface case and completely represent the submanifolds. For example, the analytic index of Dirac operator of a space curve is identified with its writhing number. As another example, the ope…

Submanifold Dirac Operator with Torsion

The submanifold Dirac operator has been studied for this decade, which is closely related to Frenet-Serret and generalized Weierstrass relations. In this article, we will give a submanifold Dirac operator defined over a surface immersed in $\EE^4$ with U(1)-gauge field as torsion in the sense of the Frenet-Serret relation, which also has data of immersion of the surface in $\EE^4$.s

Submanifold Differential Operators in $\Cal D$-Module Theory I : Schrödinger Operators

For this quarter of century, differential operators in a lower dimensional submanifold embedded or immersed in real $n$-dimensional euclidean space $\EE^n$ have been studied as quantum mechanical models, which are realized as restriction of the operators in $\EE^n$ to the submanifold. For this decade, the Dirac operators in the submanifold have been investigated in such a scheme , which are identified with operators of the Frenet-Serret relation for a space curve case and of the generalized Weierstrass relation for a conformal surface case. These Dirac operators are concerned well in the diffe…

Berezin-Toeplitz 算子的量子传播

Quantum propagation for Berezin-Toeplitz operators

本文研究实值主符号的 Berezin-Toeplitz 算子生成的量子传播子及自治情形下平滑谱投影的渐近行为，引入浸入拉格朗日子流形相关的量子态，主要贡献在于涉及预量子丛与典范丛提升的几何量，并与 Maslov 指标相关。