Let M be a compact Riemann surface with smooth boundary dM. Let
be the Laplacian with Dirichlet boundary condition. If
are the non-zero eigenvalues of
, then the zeta function
is holomorphic at
is the functional determinant. This is a non-local spectral invariant. A metric g on M is said to be uniform if (i)
and the metric g has constant curvature or (ii) dM
and g is flat. The authors show that the uniform metric minimizes ths inducing normal variation. Then structures are preserved by invariant, isometric infinitesimal variations.