MSI Weekly Bulletin - Week starting Monday 19 March, 2007

We all learned that, given a short exact sequence of chain complexes, there is an induced long exact sequence on homology. Now suppose we turn this around: given a long exact sequence, we can ask if it can be realized as the long exact sequence associated to a short exact sequence of chain complexes. The answer is No. We will give a necessary criterion.

We consider second-order linear elliptic operators of nondivergence type which is intrinsically defined on Riemannian manifolds. Cabre proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature is nonnegative. We improve Cabre's result and, as a consequence, we give another proof to Harnack inequality for Yau for positive harmonic functions on Riemannian manifolds with nonnegative Ricci curvature using the nondivergence structure of the Laplace operator.