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On Hessian Limit Directions along Gradient Trajectories |
16 pages
Published:2012-12-29
- Vincent Grandjean,
Abstract
Given a non-oscillating gradient trajectory $|\gamma|$ of a real analytic function $f$,
we show that the limit $\nu$ of the secants at the limit point
$\mathbf{0}$
of $|\gamma|$ along the trajectory
$|\gamma|$ is an eigen-vector of the limit of the direction of the
Hessian matrix $\operatorname{Hess} (f)$ at $\mathbf{0}$
along $|\gamma|$. The same holds true at infinity if the function is globally sub-analytic. We also deduce
some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is
of metric nature and still holds in a general Riemannian analytic setting.
| Keywords: | gradient trajectories, non-oscillation, limit of Hessian directions, limit of secants, trajectories at infinity |
| MSC Classifications: |
34A26 - Geometric methods in differential equations 34C08 - Connections with real algebraic geometry (fewnomials, desingularization, zeros of Abelian integrals, etc.) 32Bxx - Local analytic geometry [See also 13-XX and 14-XX] 32Sxx - Singularities [See also 58Kxx] |