×

Graphs with prescribed mean curvature in the sphere. (English) Zbl 0984.35064

This paper is devoted to the following problem: is there an embedding \(Y\) from an \(n\)-dimensional Riemannian manifold \(M\) into an \((n+1)\)-dimensional Riemannian manifold \(N\) whose mean curvature is prescribed by function \({\mathcal H}\)? The authors show that for a given function \({\mathcal H}\) on the unit sphere \(S^{n+1}\), there is a unique group over the \(n\)-dimensional unit sphere whose mean curvature coincides with \({\mathcal H}\).

MSC:

35J60 Nonlinear elliptic equations
53B20 Local Riemannian geometry
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
PDFBibTeX XMLCite