Isometric Immersions and Embeddings of Nonnegatively Curved Hypersurfaces in Hyperbolic Space
Currier, Robert John
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https://hdl.handle.net/2142/71224
Description
Title
Isometric Immersions and Embeddings of Nonnegatively Curved Hypersurfaces in Hyperbolic Space
Author(s)
Currier, Robert John
Issue Date
1984
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
Sacksteder showed that immersions into Euclidean space of complete hypersurfaces with positive semi-definite second fundamental form and some positive sectional curvature are embeddings, and that the hypersurface splits metrically as the product of flat Euclidean space and a manifold diffeomorphic to a sphere or to Euclidean space having strictly positive sectional curvature at some point.
My result is an analogue of Sacksteder's theorem, in which the hypersurface is immersed into hyperbolic space. Plausible versions of the curvature condition on the hypersurface include: (1) the second fundamental form is positive semi-definite, (2) the second fundamental form has all eigenvalues greater than or equal to one, and (3) the hypersurface has intrinsic nonnegative curvature. The corresponding first result is false; there are examples of complete hypersurfaces with positive definite second fundamental form which are not embedded.
The corresponding second result is true. Using slices of the hypersurface by horospheres, I showed that in this case the hypersurface must be embedded, and is diffeomorphic to a sphere, or is a horosphere.
The third case is the most interesting. If the immersed hypersurface has two intrinsic ends, then a result of Toponogov guarantees that the manifold splits metrically as R x M'. Here my result is that a complete manifold of this form must be embedded as an equidistant hypersurface, i.e. the set of points at a constant distance from a goedesic.
We introduce a slicing procedure based on the intersection of the manifold with translates along the normal geodesic through a point of the manifold. This slicing procedure either shows that the immersion is an embedding, or it results in an embedded noncompact domain that is subject to further study.
To examine the structure at infinity of this domain we introduce a smoothness hypothesis: roughly, that the hypersurface is of Class C('4) at certain points at infinity. This hypothesis implies that each point at infinity of the domain found by the slicing procedure is isolated. Consequently, an embedded, nonnegatively curved, hypersurface with a single point at infinity is globally supported by a horosphere.
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