Joachimsthal surfaces with nonzero constant Gaussian curvature
Algebraic, topological and analytical techniques in differential geometry and geom...
| Full text | |
| Author(s): |
de Lima, R. F.
;
Manfio, F.
;
dos Santos, J. P.
Total Authors: 3
|
| Document type: | Journal article |
| Source: | Annali di Matematica Pura ed Applicata; v. N/A, p. 50-pg., 2022-06-20. |
| Abstract | |
We consider hypersurfaces of products M x R with constant rth mean curvature H-r >= 0 (to be called H-r-hypersurfaces), where M is an arbitrary Riemannian n-manifold. We develop a general method for constructing them, and employ it to produce many examples for a variety of manifolds M, including all simply connected space forms and the hyperbolic spaces H-F(m) (rank one symmetric spaces of noncompact type). We construct and classify complete rotational H-r(>= 0)-hypersurfaces in H-F(m) x R and in S-n x R as well. They include spheres, Delaunay-type annuli and, in the case of H-F(m) x R, entire graphs. We also construct and classify complete H-r(>= 0)-hypersurfaces of H-F(m) x R which are invariant by either parabolic isometries or hyperbolic translations. We establish a Jellett-Liebmann-type theorem by showing that a compact, connected and strictly convex H-r-hypersurface of H-n x R or S-n x R (n >= 3) is a rotational embedded sphere. Other uniqueness results for complete H-r-hypersurfaces of these ambient spaces are obtained. (AU) | |
| FAPESP's process: | 16/23746-6 - Algebraic, topological and analytical techniques in differential geometry and geometric analysis |
| Grantee: | Paolo Piccione |
| Support Opportunities: | Research Projects - Thematic Grants |