Hyperbolic space, the geometry which has constant negative curvature,
is one of the central examples in Riemannian geometry and has deep
connections to the theory of 3-manifolds. Complex hyperbolic space is
its "complexification", and is a homogeneous geometry of variable
negative curvature. In both cases, it is very interesting to try to
understand discrete groups of isometries of these geometries, that is,
discrete subgroups of the Lie groups SO(1, n) and SU(1,
n). Equivalently, we want to understand manifolds with metrics
locally isometric to these hyperbolic spaces. When the quotient has finite
volume (i.e. when the group is a lattice in the associated Lie group),
they are very rigid; Mostow showed that if two such quotients of dim >
2 have the same fundamental group then they are isometric. On the
other hand, when the quotient has infinite volume they frequently are
quite flexible. Moreover, in this case, it is interesting to
understand the dynamics of the action of the group on the sphere at
infinity, contrasting the regions where the action is chaotic with
those where the action is discrete.
I will try to cover the following topics:
Weeks 1-2: Overview of real and complex hyperbolic spaces,
their isometry groups, and basic examples of lattices. For
concreteness, I will focus of real dimensions 2 and 3, and complex
dimensions 1 and 2. In the complex case, this will mostly follow:
D.B.A. Epstein, Complex Hyperbolic Geometry, in Analytical and
geometric aspects of hyperbolic space (Coventry/Durham, 1984),
93--111, Cambridge Univ. Press, 1987
Weeks 3-4: Thurston gave a very elegant construction of
certain lattices in real and complex hyperbolic space based on looking
at configuration spaces of polygons and polyhedra, respectively. (In
the complex case, these are the same as the Deligne-Mostow examples.)
I will explain these ideas following:
as well as papers of Yamashita, Nishi, and Kojima which describe the
simpler real hyperbolic case.
Weeks 4-5: This course will focus more on flexibility than
rigidity. I'll start with Thurston's Hyperbolic Dehn Surgery
Theorem. This is something special to dimension 3 whereby a
non-cocompact finite-volume hyperbolic 3-manifold can be perturbed to
an infinite family of compact hyperbolic 3-manifolds. This
will follow the appendix to
Boileau and Porti, Geometrization of 3-orbifolds of cyclic type, Asterisque 272.
Weeks 6-10: The rest of this course will be devoted to
understanding Rich Schwartz's work on the interplay between real and
complex hyperbolic geometry. In particular, he shows that there are
many complex hyperbolic 4-manifolds whose boundary at infinitely is a
compact real hyperbolic 3-manifold. This gives examples of compact
hyperbolic 3-manifolds which admit a spherical CR-structure, which is
a particularly rigid type of contract structure. I will follow his
monograph and the condensed version thereof:
Familiarity with smooth manifolds, elementary Riemannian geometry, and
a little knowledge of the hyperbolic plane (which you can obtain from reading
chapter 2 of Thurston's book listed below).
Grading
The course grade will be determined by a final paper. This will be a
6-8 page paper written on a topic in related to the content of this
class. The topic will be chosen in consolation with me in the 5th or
6th week of class, and will be due on Friday, June 4. Here are some
ideas, with sources:
Mostow Rigidity (mentioned above): Munkholm, "Gromov's proof of
Mostow following Thurston", Springer Lecture Notes in Math, number
788. See aslo Thurston's original notes (available at msri.org) and
Ratcliffe's book.
Deformation and classification of infinite volume hyperbolic
3-manifolds: Yair Minsky, "Combinatorial and Geometrical Aspects of
Hyperbolic 3-Manifolds", arXiv.org:math.GT/0205173. A good book on
this is Matsuzaki and Taniguchi, "Hyperbolic Manifolds and Kleinian
Groups".
Existence of non-arithmetic lattices for all real hyperbolic
spaces. See original paper by Gromov and Piatetski-Shapiro in IHES,
1988.
The general construction of arithmetic lattices for real
hyperbolic 3-space: Maclachlan and Reid, "The Arithmetic of Hyperbolic
3-manifolds", Springer book.
Construction of hyperbolic structures on 3-manifolds in practice.
Weeks, Computation of Hyperbolic Structures in Knot
Theory. arXiv.org:math.GT/0309407.
Aspects of complex hyperbolic space from the point of view of
complex and Kahler geometry. E.g. the Kobayashi metric, holomorphic
curvature, Kahler hyperbolicity, etc. This links in with Ma 157b.
Scheduling
There will be no class the week of May 3rd. To make up for this,
class will continue through June 3rd. Alternatively, extra classes will be
scheduled.
Further references
In addition to the sources listed above, there is
W. Goldman, Complex Hyperbolic Geometry Oxford University
Press, 1999.
For a good intro to real hyperbolic geometry, see Chapter 2 of
W. Thurston, Three-dimensional geometry and topology, vol 1,
Princeton University Press, 1997.