The 6th DIFFERENTIAL GEOMETRY DAY
Wednesday the 12th of May 2010
Anna Sakovich (Stockholm)
The constraint equations on asymptotically hyperbolic manifolds
The Einstein field equations of General Relativity can be formulated as an
initial value problem, which is well posed provided that the initial data on
the Cauchy hypersurface is chosen to be a solution to the constraint equations.
The constraint equations have been studied in different contexts, e.g. on
compact manifolds. Nevertheless, there are physical reasons for choosing
the Cauchy hypersurface to be either an asymptotically Euclidean or an
asymptotically hyperbolic manifold, the later option being more suitable for
numerical studies of the Einstein equations for asymptotically flat spacetimes.
In this talk we will first discuss the geometric origins of the constraint
equations and their conformal reformulation, and also review known solutions.
Then we will describe the class of asymptotically hyperbolic manifolds and
present a study of the constraint equations on these spaces.
Martin Svensson (Odense)
Harmonic maps, unitons and filtrations
Harmonic maps have been intensively researched in differential geometry for
several decades. Since the work of Uhlenbeck, it is known that harmonic
maps from a Riemann surface into U(n) can be generated from holomorphic
data by a transformation called - adding a uniton - when the map is of
finite uniton number, this procedure gives all harmonic maps. This leads
to the notion of extended solutions: holomorphic maps into loop groups
satisfying a first order ordinary differential equation, thus generalizing
the classical twistor constructions of Calabi. In this talk, I will discuss
some different types of unitons, and show how one may produce these using
the Grassmannian model of Segal. I will also discuss how one may factorize
extended solutions which correspond to harmonic maps into SO(n) or Sp(n).
Finally, I will give some examples of both real and symplectic harmonic
maps, and their unitons.
Radu Slobodeanu (Bucarest)
High power energy functionals in applications
Stefano Montaldo (Cagliari)
Invariant surfaces in a three dimensional manifold
The theory of surfaces in three dimensional manifolds is having, in the
last decades, a new golden age evidenced by the great number of papers on
the subject. An important geometric class of surfaces in a three dimensional
manifold is that of invariant surfaces, that is surfaces which are invariant
under the action of a one-parameter group of isometries of the ambient space.
In this lectures we shall present the problem of classifying invariant surfaces,
according to the value of their Gaussian, mean o extrinsic curvature,
in many remarkable three dimensional spaces.