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

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.

### 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
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