Globality of the DPW construction for Smyth potentials in the case of SU1,1

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications Pub Date : 2024-11-13 DOI:10.1016/j.difgeo.2024.102211

Tadashi Udagawa

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引用次数: 0

Abstract

We construct harmonic maps into

{SU}_{1, 1} / U_{1}

starting from Smyth potentials ξ, by the DPW method. In this method, harmonic maps are obtained from the Iwasawa factorization of a solution L of

L^{- 1} d L = ξ

. However, the Iwasawa factorization in the case of a noncompact group is not always global. We show that L can be expressed in terms of Bessel functions and from the asymptotic expansion of Bessel functions we solve a Riemann-Hilbert problem to give a global Iwasawa factorization. In this way we give a more direct proof of the globality of our solution than in the work of Dorfmeister-Guest-Rossman [5], while avoiding the general isomonodromy theory used by Guest-Its-Lin [11], [12].

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SU1,1情况下斯密斯电势的DPW构造的全局性

我们通过 DPW 方法，从斯迈势 ξ 开始，构建进入 SU1,1/U1 的谐波映射。在这种方法中，谐波映射是从 L-1dL=ξ 的解 L 的岩泽因子化得到的。然而，在非紧密群的情况下，岩泽因式分解并不总是全局的。我们证明 L 可以用贝塞尔函数来表示，并通过贝塞尔函数的渐近展开求解黎曼-希尔伯特问题，从而给出全局岩泽因式分解。与 Dorfmeister-Guest-Rossman [5] 的研究相比，我们通过这种方法更直接地证明了我们的求解的全局性，同时避免了 Guest-Its-Lin [11], [12] 所使用的一般等单调性理论。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Differential Geometry and its Applications 数学-数学

CiteScore

1.00

自引率

20.00%

发文量

审稿时长

6-12 weeks

期刊介绍： Differential Geometry and its Applications publishes original research papers and survey papers in differential geometry and in all interdisciplinary areas in mathematics which use differential geometric methods and investigate geometrical structures. The following main areas are covered: differential equations on manifolds, global analysis, Lie groups, local and global differential geometry, the calculus of variations on manifolds, topology of manifolds, and mathematical physics.