On Hessian Limit Directions along Gradient Trajectories

Abstract

AbstractGiven a non-oscillating gradient trajectory |γ|of a real analytic function f, we show that the limit v of the secants at the limit point 0of |γ|along the trajectory |γ| is an eigenvector of the limit of the direction of the Hessian matrix Hess(f) at 0along |γ|. 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.