We survey our recent new results on the geometry of Teichmuller and moduli spaces of Riemann surfaces and Calabi-Yau manifolds.

Scuola Normale Superiore, Pisa, 1984, tous droits rservs. Laccs aux archives de la revue Annali della Scuola Normale Superiore di Pisa, Classe di Scienze(http://www. sns. it/it/edizioni/riviste/annaliscienze/) implique laccord avec les conditions gnrales dutilisation (http://www. numdam. org/conditions). Toute utilisation commerciale ou impression systmatique est constitutive dune infraction pnale. Toute copie ou impression de ce fichier doit contenir la prsente mention de copyright.

The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is a isometric to a flat polyhedron.

Suppose L is a lamination of a Riemannian manifold by hypersurfaces with the same constant mean curvature H. We prove that every limit leaf of L is stable for the Jacobi operator. A simple but important consequence of this result is that the set of stable leaves of L has the structure of a lamination.

A Schubert class  in the Grassmannian G(k, n) is rigid if the only proper subvarieties representing that class are Schubert varieties . We prove that a Schubert class  is rigid if and only if it is defined by a partition  satisfying a simple numerical criterion. If  fails this criterion, then there is a corresponding hyperplane class in another Grassmannian G(k0, n0) such that the deformations of the hyperplane in G(k0, n0) yield non-trivial deformations of the Schubert variety . We also prove that, if a partition  contains a sub-partition defining a rigid and singular Schubert class in another Grassmannian G(k0, n0), then there does not exist a smooth subvariety of G(k, n) representing .