Abstract view
On Hessian Limit Directions along Gradient Trajectories


Published:20121229
Printed: Aug 2013
Vincent Grandjean,
partamento de Matemática, UFC, Av. Humberto Monte s/n, Campus do Pici Bloco 914, CEP 60.455760, FortalezaCE, Brasil
Abstract
Given a nonoscillating 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 eigenvector 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 subanalytic. 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.