Heat diffusions on holomorphic foliations with non-degenerate singularities. (English) Zbl 07784102
A singular holomorphic foliation \(\mathscr{F}\) of a compact complex manifold by hyperbolic Riemann surfaces naturally admits leafwise heat diffusions determined by the leafwise Laplacians associated with the leafwise Poincaré metric. To each harmonic measure on \(\mathscr{F}\), one can associate an abstract heat diffusion. The author shows that when \(\mathscr{F}\) is a Brody hyperbolic foliation having only non-degenerate singularities, then each abstract heat diffusion associated with a harmonic measure on \(\mathscr{F}\) coincides (leafwise) with the leafwise heat diffusion. It follows, in this case, that abstract heat diffusions are independent of the harmonic measures with which they are associated.
Reviewer: Gautam Bharali (Bangalore)
MSC:
| 32M25 | Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions |
| 32S65 | Singularities of holomorphic vector fields and foliations |
| 37F75 | Dynamical aspects of holomorphic foliations and vector fields |
Keywords:
singular holomorphic foliation; leafwise Poincaré metric; heat diffusions; harmonic measureReferences:
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