Differential Geometry

We consider compact complex surfaces with Hermitian metrics which are Einstein but not Kaehler. It is shown that the manifold must be CP2 blown up at 1,2, or 3 points, and the isometry group of the metric must contain a 2-torus. Thus the Page metric on CP2#(-CP2) is almost the only metric of this type.

We provide a simple way to obtain the meromorphic extension of Eisenstein series and Scattering matrices under conditions which generalize the case of discrete groups acting convex cocompactly on hyperbolic spaces.

In a previous preprint we defined an energy associated to every embedding of a surface into R n or S n . This energy is invariant under Moebius tranformations and the "round" sphere is its only absolute minimum. Here we sketch a proof of the compactness property for a variant of it. The details will appear elsewhere.

In this paper it is proved that if a finitely presented group acts properly discontinuously, cocompactly and by isometries on a simply connected Riemannian manifold, then the two Dehn functions, of the group and the manifold, respectively, are equivalent.

We show that J. Lott's equivariant higher analytic torsion for compact group actions depends only on the equivariant Euler characteristic.

Assume that the circle group acts holomorphically on a compact Kähler manifold with isolated fixed points and that the action can be lifted holomorphically to a holomorphic Hermitian vector bundle. We give a heat kernel proof of the equivariant holomorphic Morse inequalities. We use some techniques developed by Bismut and Lebeau. These inequalities, first obtained by Witten using a different argument, produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomologies in terms of the data of the fixed points.

We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying directions in the Lie algebra of the torus. We apply the results to invariant line bundles over toric manifolds. If the group is non-Abelian, there is in addition an action of the Weyl group on the fixed-point set of its maximal torus. The sum over the fixed points can be rearranged into sums over the Weyl group (having incorporated the character of the isotropy representation on the fiber) and over its orbits.

We prove the equivariant holomorphic Morse inequalities for a holomorphic torus action on a holomorphic vector bundle over a compact Kahler manifold when the fixed-point set is not necessarily discrete. Such inequalities bound the twisted Dolbeault cohomologies of the Kahler manifold in terms of those of the fixed-point set. We apply the inequalities to obtain relations of Hodge numbers of the connected components of the fixed-point set and the whole manifold. We also investigate the consequences in geometric quantization, especially in the context of symplectic cutting.

We compute the equivariant K -theory K ∗ G (G) for a simply connected Lie group G (acting on itself by conjugation). We prove that K ∗ G (G) is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a non-simply connected Lie group G , namely PSU(3), and compute the corresponding equivariant K -theory.

We establish an equivariant generalization of the Novikov inequalities which allow to estimate the topology of the set of critical points of a closed basic invariant form by means of twisted equivariant cohomology of the manifold. We apply these inequalities to study cohomology of the fixed points set of a symplectic torus action. We show that in this case our inequalities are perfect, i.e. they are in fact equalities.