Archiv der Mathematik

, Volume 79, Issue 3, pp 223-231

First online:

Extremal properties of the principal Dirichlet eigenvalue for regular polygons in the hyperbolic plane

  • L. KarpAffiliated withDepartment of Mathematics, CUNY Graduate Center and Lehman College, 365 Fifth Avenue, New York, NY 10016, USA,¶ e-mail:
  • , N. PeyerimhoffAffiliated withFakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstr. 150, Gebäude NA 5/32, D-44780 Bochum, Germany,¶ e-mail:

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We prove that amongst all hyperbolic triangles of equal perimeter or quadrilaterals in a given geodesic ball the regular polygon is the unique minimum for the first Dirichlet eigenvalue. Moreover, we give a geometric description of the set of all hyperbolic triangles with a fixed base and prescribed area.