Open Access
October, 2002 The Existence of Hypersurfaces of Constant Gauss Curvature with Prescribed Boundary
Bo Guan, Joel Spruck
J. Differential Geom. 62(2): 259-287 (October, 2002). DOI: 10.4310/jdg/1090950194


We are concerned with the problem of finding hypersurfaces of constant Gauss curvature (K-hypersurfaces) with prescribed boundary Γ in Rn+1, using the theory of Monge-Ampère equations. We prove that if Γ bounds a suitable locally convex hypersurface Σ, then Γ bounds a locally convex K-hypersurface. The major difficulty lies in the lack of a global coordinate system to reduce the problem to solving a fixed Dirichlet problem of Monge-Ampère type. In order to overcome this difficulty we introduced a Perron method to deform (lift) Σ to a solution. The success of this method is due to some important properties of locally convex hypersurfaces, which are of independent interest. The regularity of the resulting hypersurfaces is also studied and some interesting applications are given.


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Bo Guan. Joel Spruck. "The Existence of Hypersurfaces of Constant Gauss Curvature with Prescribed Boundary." J. Differential Geom. 62 (2) 259 - 287, October, 2002.


Published: October, 2002
First available in Project Euclid: 27 July 2004

zbMATH: 1070.58013
MathSciNet: MR1988505
Digital Object Identifier: 10.4310/jdg/1090950194

Rights: Copyright © 2002 Lehigh University

Vol.62 • No. 2 • October, 2002
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