Real Algebraic and Analytic Geometry |

Gradient flow of a harmonic function in

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Submission: 2005, October 20.

*Abstract:
In the paper I study the gradient field of a harmonic function $f$
in R^{3} in a neighbourhood of a critical point $0$. I show that
the flow of $\nabla f$, as a mapping between level sets of $f$, is
a stratified mapping -- that gives, in our case, an answer to the
problem of stratifying the space of orbits of the field $\nabla f$
posed by R. Thom. I also show that the trajectories of $\nabla f$
having $0$ as a limit point satisfy the finiteness conjecture and
have generalized \mbox{tangents at 0}.*

Mathematics Subject Classification (2000): 37C10, 32B20, 37B35.

Keywords and Phrases: stratification, gradient, gradient conjecture, finiteness conjecture.

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