Complex Gradient Systems

  • Giuseppe Tomassini
  • Sergio Venturini
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 16)


Let \(\widetilde{M}\) be a complex manifold of complex dimension n+k. We say that the functions u 1,…,u k and the vector fields ξ 1,…,ξ k on \(\widetilde{M}\) form a complex gradient system if ξ 1,…,ξ k , 1,…, k are linearly independent at each point \(p\in \widetilde{M}\) and generate an integrable distribution of \(T \widetilde{M}\) of dimension 2k and du α (ξ β )=0, d c u α (ξ β )=δ αβ for α,β=1,…,k. We prove a Cauchy theorem for such complex gradient systems with initial data along a CR-submanifold of type (n,k). We also give a complete local characterization for the complex gradient systems which are holomorphic and abelian, which means that the vector fields \(\xi _{\alpha }^{c}=\xi _{\alpha }-iJ \xi _{\alpha }\), α=1,…,k, are holomorphic and satisfy \([\xi _{\alpha }^{c},\bar{\xi _{\beta }^{c}} ]=0\) for each α,β=1,…,k.


Vector Field Complex Manifold Real Hypersurface Symplectic Action Left Representation 
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Copyright information

© Springer-Verlag Italia 2012

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Dipartimento Di MatematicaUniversità di BolognaBolognaItaly

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