Liouville-type theorems on the complete gradient shrinking Ricci solitons

Abstract

Abstract. We prove that there does not exist non-constant positive f -harmonic function on the complete gradient shrinking Ricci solitons. We also prove the L p (p ≥ 1 or 0 < p ≤ 1) Liouville theorems on the complete gradient shrinking Ricci solitons.

A gap theorem on complete shrinking gradient Ricci solitons

“…where ∆ f R = ∆R− < ∇f, ∇R >, λ i are the eigenvalues of Ricci tensor. Then the scalar curvature must be constant (this result also can be obtained from the main result in the author and Ge's paper [10]), implies that the scalar curvature and the Ricci curvature must be zero, hence (M, g, f ) is the Gaussian soliton. Without the assumption of nonnegative Ricci curvature, Munteanu and Wang [14] proved that if |Ric| ≤ 1 100n , then (M, g, f ) is isometric to the Gaussian soliton.…”

Counting ends on shrinkers

“…on complete shrinkers, which can be further extended to the other geometric inequalities, such as Nash inequalities, Faber-Krahn inequalities and Rozenblum-Cwikel-Lieb inequalities in [43]. For more function theory on shrinkers, the interested readers are referred to [18,33,35,36,40,44,45] and references therein.…”

Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application

“…For shrinking Ricci solitons, many people studied the f -harmonic functions, that is the function u such that △u− < ∇u, ∇f >= 0. H. Ge and S. Zhang [12] showed that any positive f -harmonic function on shrinkers is constant. However, for the standard harmonic function on gradient shrinking Ricci solitons, we do not know whether any bounded harmonic function is constant.…”