Liouville-type theorems for biharmonic maps between Riemannian manifolds

2010 ◽  
Vol 3 (1) ◽  
Author(s):  
Paul Baird ◽  
Ali Fardoun ◽  
Seddik Ouakkas
Keyword(s):  
Riemannian Manifolds ◽  
Biharmonic Maps ◽  
Liouville Type ◽  
Liouville Type Theorems

scholarly journals F-biharmonic maps into general Riemannian manifolds

2019 ◽  
Vol 17 (1) ◽  
pp. 1249-1259
Author(s):  
Rong Mi
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Biharmonic Maps ◽  
Type Result ◽  
Integral Conditions ◽  
Liouville Type ◽  
Nonexistence Result ◽  
Biharmonic Map ◽  
Euclidean Type

Abstract Let ψ:(M, g) → (N, h) be a map between Riemannian manifolds (M, g) and (N, h). We introduce the notion of the F-bienergy functional $$\begin{array}{} \displaystyle E_{F,2}(\psi)=\int\limits_{M}F\left(\frac{|\tau(\psi)|^{2}}{2}\right)\text{d}V_{g}, \end{array}$$ where F : [0, ∞) → [0, ∞) be C3 function such that F′ > 0 on (0, ∞), τ(ψ) is the tension field of ψ. Critical points of τF,2 are called F-biharmonic maps. In this paper, we prove a nonexistence result for F-biharmonic maps from a complete non-compact Riemannian manifold of dimension m = dimM ≥ 3 with infinite volume that admit an Euclidean type Sobolev inequality into general Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the Lp-norm (p > 1) of the tension field is bounded and the m-energy of the maps is sufficiently small, then every F-biharmonic map must be harmonic. We also get a Liouville-type result under proper integral conditions which generalize the result of [Branding V., Luo Y., A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds, 2018, arXiv: 1806.11441v2].

Liouville Type Theorems in the Theory of Mappings of Complete Riemannian Manifolds

2017 ◽  
Vol 221 (6) ◽  
pp. 737-744
Author(s):  
I. A. Aleksandrova ◽  
J. Mikeš ◽  
S. E. Stepanov ◽  
I. I. Tsyganok
Keyword(s):  
Riemannian Manifolds ◽  
Liouville Type ◽  
Liouville Type Theorems ◽  
Complete Riemannian Manifolds
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