  A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a
natural partition into Neumann domains (a.k.a. a Morse--Smale complex). This
partition is generated by gradient flow lines of the eigenfunction, which bound
the so-called Neumann domains. We prove that the Neumann Laplacian defined on a
Neumann domain is self-adjoint and has a purely discrete spectrum. In addition,
we prove that the restriction of an eigenfunction to any one of its Neumann
domains is an eigenfunction of the Neumann Laplacian. By comparison, similar
statements about the Dirichlet Laplacian on a nodal domain of an eigenfunction
are basic and well-known. The difficulty here is that the boundary of a Neumann
domain may have cusps and cracks, so standard results about Sobolev spaces are
not available.
  Another very useful common fact is that the restricted eigenfunction on a
nodal domain is the first eigenfunction of the Dirichlet Laplacian. This is no
longer true for a Neumann domain. Our results enable the investigation of the
resulting spectral position problem for Neumann domains, which is much more
involved than its nodal analogue.
