Critical points of invariant functions on closed orientable surfaces

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• 作者：G. Gromadzki ; J. Jezierski ; W. Marzantowicz
• 关键词：Riemann surfaces ; Finite group action ; Equivariant Lusternik–Schnirelmann category ; Riemann–Hurwitz ramification formula ; Invariant function ; Critical orbits ; Primary 55M30 ; 55M35 ; Secondary 30F10
• 刊名：Bolet篓陋n de la Sociedad Matem篓垄tica Mexicana
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：21
• 期：1
• 页码：71-88
• 全文大小：281KB
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• 作者单位：G. Gromadzki (1)
  J. Jezierski (2)
  W. Marzantowicz (3)
  
  1. Institute of Mathematics, Gdańsk University, Wita Stwosza 57, 80-952?, Gdańsk, Poland
  2. Institute of Applied Mathematics, University of Agriculture, ul. Nowoursynowska 166, 02-787?, Warsaw, Poland
  3. Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań, Umultowska 87, 61-614?, Poznań, Poland
  
• 刊物类别：Mathematics, general;
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：2296-4495

文摘

The relationship between critical points of equivariant functions and topological invariants of an equivariant action on closed manifold is an interesting problem. In this paper, we study this relationship for orientation-preserving actions of finite groups \(G\) on a closed orientable surfaces. We give an elementary, but detailed, description of the behaviour of the gradient field of an equivariant \(C^1\)-function, we present an elementary, differential, proof of the Riemann–Hurwitz formula and we construct invariant \(C^1\)-functions with the minimal number of critical orbits. These lead us to show that, with a few exceptions, the equivariant Lusternik–Schnirelmann category of a closed orientable topological surface equals the number of singular orbits of the action. Keywords Riemann surfaces Finite group action Equivariant Lusternik–Schnirelmann category Riemann–Hurwitz ramification formula Invariant function Critical orbits