SCHWARZIAN DERIVATIVES AND FLOWS OF SURFACES
SCHWARZIAN DERIVATIVES AND FLOWS OF SURFACES
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DOI：
10.1090/conm/308/05311
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发表时间：
2001-11
期刊：
影响因子：
--
通讯作者：
F. Burstall;F. Pedit;U. Pinkall
中科院分区：
文献类型：
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作者：
F. Burstall;F. Pedit;U. Pinkall
本文对如何在n-空间中的共形浸入环面空间上构造可积流族作了一些解释。这些流首先出现在数学物理中--Novikov-Veselov和Davey-Stewartson层次--作为Dirac算子的核维保持变形。后来，使用表面的旋量表示，同样的流动被解释为3-和4-空间中保持Willmore能量的表面变形。这最后一个属性表明，正确的几何设置为这个理论是Moebius不变曲面几何。我们发展这一观点在第一部分的文件中，我们推导出的基本不变量-的Schwarzian导数，Hopf微分和正常的连接-共形浸入到n-空间连同其可积性方程。为了证明我们的方法的有效性，我们讨论和证明了各种旧的和新的结果从共形曲面理论。本文的第二部分利用Moebius不变几何变形导出了共形浸入环面上的Novikov-Veselov流和Davey-Stewartson流。我们指出的类比类似的推导KdV层次的流Schwarzian的亚纯函数。特殊的表面类，如Willmore表面和等温表面，被保存的流动。
This paper goes some way in explaining how to construct an integrable hierarchy of flows on the space of conformally immersed tori in n-space. These flows have first occured in mathematical physics -- the Novikov-Veselov and Davey-Stewartson hierarchies -- as kernel dimension preserving deformations of the Dirac operator. Later, using spinorial representations of surfaces, the same flows were interpreted as deformations of surfaces in 3- and 4-space preserving the Willmore energy. This last property suggest that the correct geometric setting for this theory is Moebius invariant surface geometry. We develop this view point in the first part of the paper where we derive the fundamental invariants -- the Schwarzian derivative, the Hopf differential and a normal connection -- of a conformal immersion into n-space together with their integrability equations. To demonstrate the effectivness of our approach we discuss and prove a variety of old and new results from conformal surface theory. In the the second part of the paper we derive the Novikov-Veselov and Davey-Stewartson flows on conformally immersed tori by Moebius invariant geometric deformations. We point out the analogy to a similar derivation of the KdV hierarchy as flows on Schwarzian's of meromorphic functions. Special surface classes, e.g. Willmore surfaces and isothermic surfaces, are preserved by the flows.