Bounds on the multiplicity of eigenvalues for fixed membranes
  For a membrane in the plane the multiplicity of the $k$-th eigenvalue is known to be not greater than $2k-1$. Here we prove that it is actually not greater than $2k-3$, for $k\ge 3$. 
Thomas Hoffman-Ostenhof, Peter W. Michor, Nikolai Nadirashvili