3
$\begingroup$

Let $(M,g)$ be a smooth, closed Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are eigenfunctions of the Laplacian on $M$. Write $f = \phi_1 + \dots + \phi_m$.

How big can the set $\mathcal{S} := \left\{ x \in M\ :\ f(x) = \nabla f(x) = 0\right\}$ be? Is it, for example, known that $\mathrm{dim}_{\mathcal{H}}(\mathcal{S}) \leq n-2$?

Some motivation comes from the facts that:

  1. The nodal set of $f$ is known to be $(n-1)$-dimensional [Donnelley '94].
  2. If the metric is analytic, then for a single eigenfunction $\phi$, the singular set is $(n-2)$-dimensional (and more is known) [Han, Hardt, Lin, M. and T. Hoffmann-Ostenhof, Nadirashvili...].
Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ The last dimension estimate does not hold on the torus $\mathbb R^n/\mathbb Z^n$. The function $f(x)=1-\cos(2\pi x_1)$ is the sum of the first two eigenfunctions (assuming this numbering and normalization is permitted) and the set $\mathcal S$ has dimension $n-1$. Do you want to normalize your eigenfunctions? If it's important that they are the first ones, how do you treat eigenvalues with multiplicity? $\endgroup$
    – Joonas Ilmavirta
    Commented Jul 14, 2015 at 11:15
  • $\begingroup$ I am not bothered about making any stipulations about multiplicity or normalization. I guess I could really ask it for a general linear combination $f = a_1\phi_1 + \dots + a_m\phi_m$; $a_j \in \mathbb{R}$. $\endgroup$
    – Spencer
    Commented Jul 14, 2015 at 22:58

0

Reset to default

You must log in to answer this question.