Existence and Liouville theorems for V -harmonic maps from complete manifolds

Abstract

Abstract. We establish existence and uniqueness theorems for V -harmonic maps from complete noncompact manifolds. This class of maps includes Hermitian harmonic maps, Weyl harmonic maps, affine harmonic maps and Finsler maps from a Finsler manifold into a Riemannian manifold. We also obtain a Liouville type theorem for V -harmonic maps. In addition, we prove a V-Laplacian comparison theorem under the BakryEmery Ricci condition.

Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds

“…where ∇ and ∆ are respectively the Levi-Civita connection and the Laplace-Beltrami operator with respect to g, V is a smooth vector field on M . In [1] and [6], the authors introduced two curvatures…”

Gradient estimates of Hamilton–Souplet–Zhang type for a general heat equation on Riemannian manifolds

“…V is a smooth vector field on M . A natural generalization of Bakry-Émery curvature and N -Bakry-Émery curvature are the following two tensors ( [3,7])…”

Li–Yau–Hamilton estimates and Bakry–Emery–Ricci curvature

“…An important generalization is a diffusion operator (1.1) ∆ V := ∆ + V, ∇ on a Riemannian manifold (M, g) of dimension m, where ∇ and ∆ are respectively the Levi-Civita connection and Beltrami-Laplace operator of g, and where V is a smooth vector field on M. This operator is also a special case of V -harmonic map introduced in [11]. As in [4,10], we introduce Bakey-Emery Ricci tensor fields is exactly the Ricci soliton equation, which is one-to-one corresponding to a selfsimilar solution of Ricci flow (see, [13] It is easy to see that the scalar curvature of g cs is 4/(1+x 2 +y 2 ) and hence the cigar soliton is not Ricci-flat. An important result about the cigar soliton is that it is rotationally symmetric, has positive Gaussian curvature, is asymptotic to a cyclinder near infinity, and, up to homothety, is the unique rotationally symmetric gradient Ricci soliton of positive curvature on R 2 .…”

“…In our geometric proof we require the curvature condition Ric n,m f ≥ −K in order to use the Bakry-Qian's Laplacian comparison theorem without any additional requirement on the potential function f . If we use the curvature condition Ric f ≥ −K in our geometric proof, then some conditions on f would be required (see [10,37]). A probabilistic proof of Li [25] shows a similar estimate…”