  We prove conjectures of Rene Thom and Vladimir Arnold for C^2 solutions to
the degenerate elliptic equation that is the level set equation for motion by
mean curvature.
  We believe these results are the first instances of a general principle:
Solutions of many degenerate equations behave as if they are analytic, even
when they are not. If so, this would explain various conjectured phenomena.
