阪大-阪市大‐神戸大-九大合同幾何学セミナー (第9回) 


Katrin Leschke 教授 連続講義


開催日 平成26年9月11日(木)-12日(金)
場所 大阪市立大学杉本キャンバス 共通研究棟3階 401室(第3セミナー室)

Wayne Rossman(神戸大学)


援助 平成25年度科学研究費補助金 基盤研究(C)  
研究課題名 「正則写像から受け継がれる超共形写像の性質とその応用」
(研究代表者:守屋 克洋,研究分担者:大仁田 義裕)
後援 大阪市立大学数学研究所(OCAMI)

Professor Katrin Leschke (Department of Mathematics, University of Leicester, UK)
Mr. Yuta Ogata (D1, Department of Mathematics, Kobe University, Japan)

Katrin Leschke (University of Leicester)
September 11 (Thu.), 13:30-15:00 "Integrable system methods for minimal surfaces, Part I"
September 12 (Fri.), 10:40-12:10 "Integrable system methods for minimal surfaces, Part II"
Abstract: Minimal surfaces, that is surfaces with vanishing mean curvature, are amongst the surface classes best studied and understood. One of the reasons for this is the fact that a minimal surface in 3-space is the real part of a holomorphic function into complex 3-space, and thus the classical notions and facts from Complex Analysis can be used in the study of minimal surfaces. On the other hand, harmonic maps into appropriate spaces give rise to integrable systems. In particular, integrable system methods can be used to investigate surfaces given by harmonicity. For example, constant mean curvature (CMC) surfaces have harmonic Gauss map and the associated family (for non-vanishing mean curvature) has been used to classify all CMC tori as meromorphic functions on an auxiliary Riemann surface given by the associated family, the spectral curve.

In the first talk, I will recall some of the basic facts for minimal surfaces and harmonic maps to setup notations. We will discuss the Lopez-Ros deformation and provide an explicit form of the deformation in terms of the holomorphic null curve of the minimal surface. We will also investigate periods and ends of a minimal surface in terms of the holomorphic null curve. We recall how a minimal surface gives rise to a second Willmore surface which is a twistor projection of a holomorphic curve, and introduce a generalisation of the associated family of minimal surfaces.

In the second talk, the simple factor dressing of a harmonic map will be discussed. It's application to the harmonic conformal Gauss map of a minimal surface can be computed explicitely in terms of the pole of the dressing, and two parameters in the 3-sphere. In particular, we will show that the Lopez-Ros deformation is a simple factor dressing with special choices of parameters. General simple factor dressings are given as the Lopez-Ros deformation applied to rotations in space. In particular, this gives hope that other concepts in minimal surface theory may be special cases of more general concepts of integrable system theory. If time permits we will see another instance of this; we will show that the Darboux transforms of a minimal surfaces are twistor projections of holomorphic curves. The set of Darboux transforms gives, in case when the surface is a torus, a geometric interpretation of the spectral curve. Recent results by Meeks, Perez and Ros on the uniqueness of the Riemann minimal surface indicate that the understanding of spectral data for punctured tori might lead to a further understanding in minimal surface theory.

This is joint work with Katsuhiro Moriya.
2014年9月12日(金)9:00-10:30, September 12 (Fri.)
Yuta Ogata 緒方 勇太 (Kobe University 神戸大学)
"Smyth surfaces in Lorentz-Minkowski space and their singularities "
Abstract: It is a famous result that a surface of revolution in 3-dimensional Euclidean space with constant mean curvature is a periodic Delaunay surface. We can consider a generalization of Delaunay surfaces, and treat CMC surfaces which are not those of revolution but have rotationally invariant metrics, called Smyth surfaces. There are many studies of Smyth surfaces in Riemannian spaceforms from the viewpoint of differential geometry and integrable systems. In this talk, we construct spacelike Smyth surfaces in 3-dimensional Lorentz-Minkowski space, and identify the types of singularities that Smyth surfaces have.

アブストラクト:平均曲率一定曲面(CMC曲面)でかつ回転面であるものは、「Delaunay曲面」と呼ばれ る。また、Delaunay曲面の一般化として、回転不変な計量をもつCMC曲面は「Smyth曲 面」と呼ばれ、リーマン空間形内のSmyth曲面は、微分幾何や可積分系の立場から多 くの研究が行われてきた。今回の講演では、3次元ローレンツ・ミンコフスキー空間 内の空間的Smyth型曲面の構成を行い、それらの曲面がもつ特異点の判定を行う。

リンク等 案内

大仁田 義裕: ohnita (at)
Wayne Rossman: wayne (at)
製作 のだ & ボス Last updated on 22/July/2012