Let M be a complex manifold of complex dimension n+k. We say that the functions u1, . . . ,uk and the vector fields x1, . . .,xk on M form a complex gradient system if x1, . . . ,xk,Jx1, . . . ,Jxk are linearly independent at each point p ∈ eM and generate an integrable distribution of TM of dimension 2k and dua (xb ) = 0, dcua (xb ) = dab for a,b = 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 xca = xa −iJxa , a = 1, . . . ,k are holomorphic and satisfy xca ,xcb = 0 for each a,b = 1, . . . ,k.

G. Tomassini, S. Venturini (2012). Complex Gradient Systems. MILANO : Springer [10.1007/978-88-470-1947-8].

Complex Gradient Systems

VENTURINI, SERGIO
2012

Abstract

Let M be a complex manifold of complex dimension n+k. We say that the functions u1, . . . ,uk and the vector fields x1, . . .,xk on M form a complex gradient system if x1, . . . ,xk,Jx1, . . . ,Jxk are linearly independent at each point p ∈ eM and generate an integrable distribution of TM of dimension 2k and dua (xb ) = 0, dcua (xb ) = dab for a,b = 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 xca = xa −iJxa , a = 1, . . . ,k are holomorphic and satisfy xca ,xcb = 0 for each a,b = 1, . . . ,k.
2012
The Mathematical Legacy of Leon Ehrenpreis
309
326
G. Tomassini, S. Venturini (2012). Complex Gradient Systems. MILANO : Springer [10.1007/978-88-470-1947-8].
G. Tomassini; S. Venturini
File in questo prodotto:
Eventuali allegati, non sono esposti

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/122807
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 1
  • ???jsp.display-item.citation.isi??? ND
social impact