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Three dimensional affine hyperspheres generated by 2-dimensional partial differential equations

Vrancken, Luc

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

It is well known that locally strongly convex affine hyperspheres can be determined as solutions of differential euqations of Monge-Ampere type. In this paper we study in partivular the 3-dimensional case and we assume that the hypersphere admits a Killing vector field (with respect to the affine metric) whose integral curves are geodesics with respect to both the induced affine connection and the Levi Civita connection of the affine metric. We show that besides the already known examples, such hyperspheres can be constructed starting from the 2-dimensional Laplace equation, the 2-dimensional sine-Gordon equation or the 2-dimensional cosh-Gordon equation.