I am having to perform oscillatory integrations like $e^{iS}$ using Picard-Lefschetz theory. One can write this as $e^{h+is}$ where $h(x,y)=-{\rm Im}(S(x,y))$ is the Morse function. To perform these kinds of integrals one deforms the integration contour along Lefschetz thimbles on which $is$ freezes to a constant and and the Morse function has a large negative value towards the end points of the contour. Then the integration can be well approximated by the steepest descent method. Now, I need to figure out the angle this steepest descent contour makes at the saddle point. In this paper (open access) it is stated that "the direction of the steepest descent path at the saddle point is given by the direction $\theta$ of the eigenvector of the Hessian matrix for the Morse function, associated with the negative eigenvalue" (see, page 13, left column, after eq. no. 65).
I want to know how to prove this: "the direction of the steepest descent path at the saddle point is given by the direction $\theta$ of the eigenvector of the Hessian matrix for the Morse function, associated with the negative eigenvalue"
