Giona Veronelli (Université Paris 13) will speak in the geometry seminar on Tuesday 12 February, at 1.30pm in the Salle de Profs. Giona’s title is *Distance-like functions on Riemannian manifolds* and his abstract is below.

Let M be a complete non-compact Riemannian manifold. The behaviour of the distance function r(x) (from a fixed reference point) can reflect how different M is from a Euclidean space.

In general r(x) is 1-lipschitz on M, but only a.e. differentiable. However a well-known result by Greene and Wu shows the existence of a function H(x) on M which is smooth, distance-like (i.e. r(x)/C < H(x) < Cr(x) outside a compact set), and whose gradient is bounded. It is natural to generalize this result, giving geometric assumptions implying the existence of a distance-like function with controlled higher order derivatives. In this talk we will show some classical results and some more recent answers to this problem.

Then we will discuss how distance-like functions can be used to prove on non-compact manifolds classical analytic tools available in the Euclidean setting, such as the density of smooth compactly supported functions in Sobolev spaces, or Sobolev and Calderon-Zygmund inequalities.