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The geometry and spectrum of the one holed torus. (English) Zbl 0649.53028
The authors examine the space of Riemann surfaces of signature (1,1) with metric of curvature -1 and geodesic boundary. They solve explicitly the moduli problem in this case and show furthermore that two surfaces of this type having the same length spectrum (this referring to smooth closed geodesics including the boundary) are isometric. They announce the same type of result for genus two surfaces without boundary. The problem whether the analogous assertion holds for the spectrum of the Laplacian with respect to the Neumann or Dirichlet conditions is open.
Reviewer: M.Burger

53C22 Geodesics in global differential geometry
30F20 Classification theory of Riemann surfaces
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
11F06 Structure of modular groups and generalizations; arithmetic groups
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