The Li-Yau inequality and applications under a curvature-dimension   condition
  We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequents bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method. 
Dominique Bakry (IMT), Fran\c{c}ois Bolley (LPMA), Ivan Gentil (ICJ)
math.DG math.AP math.FA math.PR