This monograph is a self-contained introduction to the geometry
of Riemann Surfaces of constant curvature 1 and their length and
eigenvalue spectra. It focuses on two subjects: the geometric
theory of compact Riemann surfaces of genus greater than one, and
the relationship of the Laplace operator with the geometry of such
surfaces. Research workers and graduate students interested in
compact Riemann surfaces will find here a number of useful tools
and insights to apply to their investigations.
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