Boundary-value problems on \(S^ n\) for surfaces of constant Gauss curvature. (English) Zbl 0840.53046

Guan, Bo; Spruck, Joel

Ann. Math. (2) 138, No. 3, 601-624 (1993).

The authors prove existence of smooth, locally strictly convex radial graphs over domains on \(S^n\) with prescribed Gauss curvature and boundary data. Previously known results dealt with graphs over the entire \(S^n\) and the Gauss curvature given as a function on \(S^n \times (0,\infty)\), subject to certain growth and decay conditions in radial direction. Such conditions allow to establish a priori estimates of the solution and its derivatives.
In contrast with the known cases, the authors allow the prescribed Gauss curvature to be defined only on \(S^n\), but postulate existence of subsolutions. The results have many interesting geometric applications, especially for surfaces with constant Gauss curvature. The proofs are based on a careful study of related nonlinear equations of Monge-Ampère type.

Reviewer: V.I.Oliker (Atlanta)

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58J32	Boundary value problems on manifolds
35J65	Nonlinear boundary value problems for linear elliptic equations

Keywords:

Monge-Ampère equations; radial graphs; prescribed Gauss curvature; surfaces with constant Gauss curvature