Eigenvalue maximization for surfaces of revolution with prescribed   boundary
  Fix two parallel circles in $\mathbb{R}^3$ centered about a common axis. Among surfaces of revolution immersed in $\mathbb{R}^3$ whose boundary is given by these circles, there is one which maximizes the first Dirichlet eigenvalue. If the circles are sufficiently close together, then this surface is unique. 
Sinan Ariturk
math.AP math.DG