We survey briefly the definition of the Rozansky-Witten invariants, and review their relevance to the study of compact hyperkähler manifolds. We consider how various generalisations of the invariants might prove useful for the study of non-compact hyperkähler manifolds, of quaternionic-Kähler manifolds, and of relations between hyperkähler manifolds and Lie algebras. The paper concludes with a ...

We establish two-sided bounds for the complexity of two infinite series of closed orientable 3-dimensional hyperbolic manifolds, the Löbell manifolds and the Fibonacci manifolds.

We obtain an embedding theorem for compact strongly pseudoconvex CR manifolds which are bounadries of some complete Hermitian manifolds. We use this to compactify some negatively curved Kähler manifolds with compact strongly pseudoconvex boundary. An embedding theorem for Sasakian manifolds is also derived.

Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron’s method. Th...

H-minimal Lagrangian submanifolds in general Kähler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in Kähler manifolds to construct minimal Lagrangian torus in certain Kähler-Einstein manifolds with negative first Chern class.

Among the eigenvalue problems of the Laplacian, the biharmonic operator eigenvalue problems are interesting projects because these problems root in physics and geometric analysis. The buckling problem is one of the most important problems in physics, and many studies have been done by the researchers about the solution and the estimate of its eigenvalue. In this paper, first, we obtain the evol...

This paper adresses the following problem: Given a closed orientable threemanifold M , are there at most finitely many closed orientable three-manifolds 1-dominated by M? We solve this question for the class of closed orientable graph manifolds. More presisely the main result of this paper asserts that any closed orientable graph manifold 1dominates at most finitely many orientable closed three...

It is often advantageous in image processing and computer vision to consider images as surfaces imbedded in higher dimensional manifolds. It is therefore important to consider the theoretical and applied aspects of proper sampling of manifolds. We present a new sampling theorem for surfaces and higher dimensional manifolds. The core of the proof resides in triangulation results for manifolds wi...

We compute the L-cohomology spaces of some negatively curved manifolds. We deal with two cases: manifolds with finite volume and sufficiently pinched negative curvature, and conformally compact manifolds.