Journal of Differential Geometry

The Existence of Hypersurfaces of Constant Gauss Curvature with Prescribed Boundary

Bo Guan and Joel Spruck

Abstract

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.

Article information

Source
J. Differential Geom., Volume 62, Number 2 (2002), 259-287.

Dates
First available in Project Euclid: 27 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1090950194

Digital Object Identifier
doi:10.4310/jdg/1090950194

Mathematical Reviews number (MathSciNet)
MR1988505

Zentralblatt MATH identifier
1070.58013

Citation

Guan, Bo; Spruck, Joel. The Existence of Hypersurfaces of Constant Gauss Curvature with Prescribed Boundary. J. Differential Geom. 62 (2002), no. 2, 259--287. doi:10.4310/jdg/1090950194. https://projecteuclid.org/euclid.jdg/1090950194


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