Let M be a complete non-compact Riemannian manifold whose sectional curvature is bounded between two constants-k and K. Then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in Euclidean space. The purpose of this paper is to give a justi-fication of such a statement.

We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry), which is also a generalization of Conn’s linearization theorem.

Let M be a compact Riemannian manifold with a fixed conformal structure. Then we introduce the concept of conformal volume of M in the following manner. For each branched conformal immersion q9 of M into the unit sphere S n, we consider the set of all branched conformal immersions obtained by composition of qo with the conformal automorphisms of S". We let Vc (n, qg) be the maximum volume of these branched immersions. The conformal volume of M is defined to be the infimum of V.(n, q0) where qo ranges over all branched conformal immersions of M into the unit sphere S". In this paper, we study the case when M is a compact surface and we call the conformal volume of M to be the conformal area of M. We demonstrate that this conformal invariant is non-trivial. In fact, we prove that if there exists a minimal immersion of M into S" where coordinate functions are first eigenfunctions, then the conformal area of M is given by the area of M with respect to the induced metric. This enables us to compute the conformal area for several surfaces. For example, the conformal area of RP z is 6n and the conformal area of the square torus is 27c 2. We believe that the computation of the conformal area for general surfaces will be very important in studying the geometry of compact surfaces. We demonstrate this claim by applying the concept of conformal area to two different branches of surface theory. The first application is to study the total curvature of a compact surface in R". This problem has a long history. Fenchel and Fary [9] proved that for a closed curve o in R", Slk [> 27c where k is its curvature. Then Milnor [12] r proved that 5lk]> 4~ z if cr is

A discrete conformality for polyhedral metrics on surfaces is introduced in this paper. It is shown that each polyhedral metric on a compact surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a finite dimensional variational principle.

We consider an ancient solution g(·, t) of the Ricci flow on a compact surface that exists for t 2 (−1, T ) and becomes spherical at time t = T . We prove that the metric g(·, t) is either a family of contracting spheres, which is a type I ancient solution, or a King–Rosenau solution, which is a type II ancient solution.