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.

Evolving smooth, compact hypersurfaces in Rn+1 with normal speed equal to a positive power k of the mean curvature improves a certain ‘isoperimetric difference’ for k > n.1. As singularities may develop before the volume goes to zero, we develop a weak level-set formulation for such flows and show that the above monotonicity is still valid. This proves the isoperimetric inequality for n 6 7. Extending this to complete, simply connected 3-dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidean isoperimetric inequality on such manifolds.

Making use of the extended flux homomorphism defined in [13] on the group Symp§g of symplectomorphisms of a closed ori- ented surface §g of genus g ≥ 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently area-preserving, total holonomy. These characteristic classes are stable with respect to g and we show that they are highly non- trivial. We also prove that the second homology of the group Ham§g of Hamiltonian symplectomorphisms of §g, equipped with the discrete topology, is very large for all g ≥ 2.

We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension d for all d ≥ 4 (see Theorem 2.13). These examples result from applying CAT(0)/CAT(−1) filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps. The groups obtained have a number of interesting properties, which are established by analyzing their boundaries at infinity by a kind of Morse-theoretic technique, related to but distinct from ordinary and combinatorial Morse theory (see Section 5).

Mean curvature ows of hypersurfaces have been extensively studied and there are various dierent approaches and many beautiful results. However, relatively little is known about mean curvature ows of submanifolds of higher codimensions. This notes starts with some basic materials on submanifold geometry, and then introduces mean curvature ows in general dimensions and co-dimensions. The related techniques in the so called \blow-up" analysis are also discussed. At the end, we present some global existence and convergence results for mean curvature ows of two-dimensional surfaces in four-dimensional ambient spaces.