The 9th DIFFERENTIAL GEOMETRY DAY

at LUND

Wednesday the 15th of May 2013


ABSTRACTS

David Brander (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. Romain Gicquaud (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. Pascal Romon (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.