On Hessian Limit Directions along Gradient Trajectories

Grandjean, Vincent

Canadian journal of mathematics, Tome 65 (2013) no. 4, pp. 808-822

Résumé

Given a non-oscillating gradient trajectory $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ of a real analytic function $f$ , we show that the limit $v$ of the secants at the limit point $0$ of $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ along the trajectory $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ is an eigenvector of the limit of the direction of the Hessian matrix Hess $\left( f \right)$ at $0$ along $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ . 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.

Détail

DOI : 10.4153/CJM-2012-021-6

Mots-clés : 34A26, 34C08, 32Bxx, 32Sxx, gradient trajectories, non-oscillation, limit of Hessian directions, imit of secants, trajectoriesat infinity