Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds

By Xiang-Dong Li. 2005

328 citations recorded by the source corpus.

Let L=Δ−∇ϕ⋅∇ be a symmetric diffusion operator with an invariant measure μ(dx)=e−ϕ(x)dx on a complete non-compact Riemannian manifold M. We give the optimal conditions on “the m-dimensional Ricci curvature associated with L” so that various Liouville theorems hold for L-harmonic functions, and that the heat semigroup Pt=etL has the C0-diffusion property and is unique in L1(M,μ). As applications, we give the optimal conditions for the uniqueness of the positive L-invariant measure and the L1-uniqueness of the intrinsic Schrodinger operators on complete non-compact Riemannian manifolds. We also give a criterion for the finiteness of the total mass of the L-invariant measure and establish the Calabi–Yau volume growth theorem for the L-invariant measure on complete Riemannian manifolds on which “the m-dimensional Ricci curvature associated with L” is non-negative. This leads us to prove that if M is a complete Riemannian manifold with a finite L-invariant measure for which the associated m-dimensional Ricci curvature is non-negative, then M is compact. Moreover, we obtain an upper bound diameter estimate of such Riemannian manifolds by using the dimension of L, the total μ-volume of M and the upper bound of the μ-volume of geodesic balls of a fixed radius. Finally, using the variational formulae in Riemannian geometry, we give a new proof of the Bakry–Qian generalized Laplacian comparison theorem.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5fee9e4d80fa455add572e5df3ea838d8543e37d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Stochastic completeness for graphs with curvature dimension conditions (2015, background): “Bakry [Bak86] proved the stochastic completeness for weighted Riemannian manifolds satisfying CD(K,∞) condition for weighted Laplacians, see also Li [Li05].”
Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature (2014, background): “As an immediate consequence, we have the following variants corollaries parallel to these in [34,23].”
Comparison geometry for the Bakry-Emery Ricci tensor (2007, background): “See [17, 18] for elementary proofs of the inequality.”
Analysis of weighted Laplacian and applications to Ricci solitons (2011, unknown): “The reference [33] is a good source for many other similar results concerning the weighted Laplacian, under various curvature bounds.”
Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds (2011, background): “For the related research and improvement in this direction, see [2, 3, 5–7, 11, 14–16] and the references therein.”
A Liouville-Type Theorem for Smooth Metric Measure Spaces (2010, methodology): “See, for example [4, 7, 9, 12, 8] or [11] and the references therein.”
Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries (2011, background): “Among other things, these authors have succeeded in re-deriving, from a purely analytical perspective, several of the well-known fundamental results which, in Riemannian geometry, are obtained under the assumption that the Ricci curvature be bounded from below, see for instance [5], [9] [36], [39].”
Elliptic and parabolic equations for measures (2009, background): “Such an estimate was indeed obtained in [48], [49] for a broad class of Riemannian manifolds (concerning diffusion operators on manifolds, see also [115]).”
On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons (2013, background): “for example, Myers’ theorem, eigenvalues estimates, Li–Yau Harnack inequality, Liouville theorems, and the Cheeger–Gromoll splitting theorem [4–6,14,19,24,26,31].”
Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature (2012, methodology): “associated with the Witten Laplacian (also called symmetric diffusion operator in [ 23 ])”
Smooth metric measure spaces with non-negative curvature (2011, methodology): “We will refer the readers to [23, 32] for some of the results.”
Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds (2008, methodology): “where β = e −2Kt . It is well known that for the N -Bakry-Emery Ricci tensor, we have the Bochner formula [10, 11 ]:”