Uniformly elliptic operators on Riemannian manifolds.

*(English)*Zbl 0735.58032Given a Riemannian manifold \((M,g)\), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to \(g\). Typical examples of such operators are the Laplace operators of Riemannian structures which are quasi-isometric to \(g\). We first prove some Poincaré and Sobolev inequalities on geodesic balls. Then we use Moser’s iteration to obtain Harnack inequalities. Gaussian estimates, uniqueness theorems, and other applications are also discussed. These results involve local or global lower bound hypotheses on the Ricci curvature of \(g\). Some of them are new even when applied to the Laplace operator of \((M,g)\).

Reviewer: L.Saloff-Coste

##### MSC:

58J35 | Heat and other parabolic equation methods for PDEs on manifolds |

47F05 | General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) |