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On Hessian Limit Directions along Gradient Trajectories

Published:2012-12-29
Printed: Aug 2013
• Vincent Grandjean,
partamento de Matemática, UFC, Av. Humberto Monte s/n, Campus do Pici Bloco 914, CEP 60.455-760, Fortaleza-CE, Brasil
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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]