## The 6th DIFFERENTIAL GEOMETRY DAY

## at LUND

## Wednesday the 12th of May 2010

## ABSTRACTS

Anna Sakovich(Stockholm)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.The constraint equations on asymptotically hyperbolic manifoldsMartin Svensson(Odense)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.Harmonic maps, unitons and filtrationsRadu Slobodeanu(Bucarest)High power energy functionals in applicationsStefano Montaldo(Cagliari)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.Invariant surfaces in a three dimensional manifold