extension of the projectivized gradient of a harmonic function

Question

Let $(M,g)$ be a riemannian manifold, $\Delta$ the associated Laplacian, and $\{ f_i \}$ the real-valued eigenfunctions of $\Delta$. Then, $\nabla f_i \in \Gamma ^{\infty } (\mathrm{T} M) $ is defined everywhere, but may have zeros.

Is it true that, for every $i$, $\pi ( \frac{\nabla f_i (\cdot )}{|\nabla f_i (\cdot )|} ) \in \Gamma ^{\infty } (\mathrm{PT} M) $ ? Here, $\mathrm{PT} M$ is the projectivized tangent bundle of $M$ (I hope it is the right notation) and $\pi$ is the projection on the projective space.

In other words, does the gradient define a direction field at every point of the manifold, even where the gradient is $0$, and does this direction field depend smoothly on the point? Is there a reference?

edit: in Steve Zelditch's paper "Eigenfunctions and Nodal sets" one can find that for a generic metric, all eigenfunctions are Morse, so the projectivized gradient does admit an extension to critical points. on the other hand, the flat metric on the torus, which is not generic from this point of view, satisfies the conclusion, and so does the euclidean metric on $S^3$.

thanks, nikos

Answer 1

This problem looks like how to extend the normal bundle of minimal surface over branch point. It is solve in Gulliver, Osserman and Royden. Prooving that branch point are of the form $az^k+o(z^k)$ in a appropriate choice of coordinate. I guess that here if you complexcify thing you can probably prove something similar. The fact that the metric should be generic for the eigenfunction to be a Morse function is clearly necessary and make me think to the bumpy metric theorem of White(The space of minimal submanifolds for varying Riemannian metrics).

Answer 2

in fact, as can show the example $\sin x \sin y$ on the torus $\mathbb{T} ^2$ there can be a singularity of codimension $2$ where the projectivized gradient cannot be extended.