The 9th DIFFERENTIAL GEOMETRY DAY
Wednesday the 15th of May 2013
(Danish Technical University, DE)
Natural deformations of minimal surfaces to non-minimal CMC surfaces
The Weierstrass representation associates to any minimal surface a holomorphic
1-form and a meromorphic function on a simply connected Riemann surface; and
the converse. The surface is obtained by integration and taking the real part.
Analogously, for non-minimal CMC surfaces there is the generalized Weierstrass
representation (or DPW method), which associates the same type of data. Here
the integration takes place in a loop group and the analogue of taking the
real part is performing an Iwasawa decomposition. Because the loop group
decomposition is non-trivial, the relation between the geometry and the
holomorphic data is not simple, making it more difficult than in the minimal
case to use this method.
In recent work with J. Dorfmeister we show how to use these two representations
to give canonical deformations between minimal and non-minimal CMC surfaces,
once a basepoint is fixed. Symmetries associated with the basepoint are
preserved, and hence for these types of surfaces there is a canonical choice
of basepoint. This gives a new way to construct CMC surfaces, canonically
associated to known minimal surfaces, with explicit formulae for the Weierstrass data.
(University of Tours, FR)
On the geometry at infinity of asymptotically hyperbolic manifolds
Since the pioneer work of Charles Fefferman and Robin Graham, there has
been a great interest in the study of asymptotically hyperbolic manifolds.
The standard definition of these manifolds is an extension of the ball model
of the hyperbolic space where one allows the conformal infinity to be much
more general than the round sphere. These manifolds also show up in general
relativity where they form a natural class of Cauchy surface. As a consequence,
it is natural to study the geometry of these manifolds. In this talk I will
concentrate mostly on rigidity and almost rigidity results under assumptions
on scalar curvature and Ricci curvature.
Paul Andi Nagy
(University of Murcia, ES)
Geometric structures from twists
A brief overview of some of the Kaehler geometry aspects of
symplectic couplings will be given. Then, I will explain how to
parametrise explicitely admissible complex structure on ruled
manifolds when the symplectic form is fixed.
Various applications, including the construction of holomorphic
harmonic morphisms will be described.
(Université Paris-Est, Marne-la-Vallée, FR)
Symplectic and metric structures on the tangent bundle
When M is a differential manifold, its tangent bundle TM is a also differential
manifold, of twice the dimension of M. What kind of structure can TM be endowed
with? For example, if M is riemannian, is there a "natural" metric on TM?
We consider three different but related type of structures: metric (riemannian
or pseudo-riemannian), symplectic and complex or almost-complex (even paracomplex),
and see that such structures exist on TM, possibly depending on analogous structures
on M (and sometimes not). That also has an influence on the geometry of submanifolds
(especially lagrangian submanifolds) of TM.