 
Summary: GRADIENT ESTIMATE AND HARNACK INEQUALITY
ON NONCOMPACT RIEMANNIAN MANIFOLDS
MARC ARNAUDON, ANTON THALMAIER, AND FENGYU WANG
Abstract. A gradiententropy inequality is established for elliptic diffusion semi
groups on arbitrary complete Riemannian manifolds. As applications, a global Har
nack inequality with power and a heat kernel estimate are derived without curvature
conditions.
1. The main result
Let M be a noncompact complete connected Riemannian manifold, and Pt be the
Dirichlet diffusion semigroup generated by L = + V for some C2
function V . We
intend to establish reasonable gradient estimates and Harnack type inequalities for Pt.
In case that Ric  HessV is bounded below, a dimensionfree Harnack inequality was
established in [15], which according to [17], is indeed equivalent to the corresponding
curvature condition. See e.g. [2] for equivalent statements on heat kernel functional
inequalities; see also [8, 3, 9] for a parabolic Harnack inequality using the dimension
curvature condition by shifting time, which goes back to the classical local parabolic
Harnack inequality of Moser [10].
Recently, some sharp gradient estimates have been derived in [13, 19] for the Dirichlet
semigroup on relatively compact domains. More precisely, for V = 0 and a relatively
