Real Algebraic and Analytic Geometry
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175. Paweł Goldstein:
Gradient flow of a harmonic function in R3.

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Submission: 2005, October 20.

Abstract:
In the paper I study the gradient field of a harmonic function \$f\$ in R3 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.