Non existence of quasi-harmonic spheres

In this note we prove that the maximum principle of Jäger-Kaul for harmonic maps holds for a more general class of maps, V -harmonic maps. This includes Hermitian harmonic maps (Jost and Yau, Acta Math 170:221-254, 1993), Weyl harmonic maps (Kokarev, Proc Lond Math Soc 99:168-194, 2009), affine harmonic maps Jost and Simsir (Analysis (Munich) 29: [185][186][187][188][189][190][191][192][193][194][195][196][197] 2009), and Finsler maps from a Finsler manifold into a Riemannian manifold. With this maximum principle we establish the existence of V -harmonic maps into regular balls.

show abstract

“…which does not depend on T , a contradiction to (26). Therefore T * = ∞, which, together with Substep 1, proves the Claim.…”

Section: Existence Resultsmentioning

confidence: 46%

“…Harmonic maps from a Riemannian measure space are also f -harmonic. For a recent study of f -harmonic maps, see [25,26,29,30,42]. In general, however, Eq.…”

Section: Introductionmentioning

confidence: 96%

A Maximum Principle for Generalizations of Harmonic Maps in Hermitian, Affine, Weyl, and Finsler Geometry

2014

View full text Add to dashboard Cite

show abstract

“…To prove Theorem 1.1 we will use Moser iteration instead of using the monotonicity inequality for quasi-harmonic spheres and the John-Nirenberg inequality for BMO spaces as in [7]. As such, the methods in the current paper are comparatively elementary.…”

Section: Introductionmentioning

confidence: 97%

Nonexistence of quasi-harmonic spheres with large energy

Yang

2011

manuscripta math.

Self Cite

View full text Add to dashboard Cite

The absence of quasi-harmonic spheres is necessary for long time existence and convergence of harmonic map heat flows. Let (N , h) be a complete noncompact Riemannian manifold. Assume the universal covering of (N , h) admits a nonnegative strictly convex function with polynomial growth. Then there is no non-constant quasi-harmonic sphere u : R n → N such that lim r →∞ r n e − r 2 4 |x|≤r e − |x| 2 4 |∇u| 2 dx = 0.This generalizes a result of the first author and X. Zhu (Calc. Var., 2009). Our method is essentially the Moser iteration and thus comparatively elementary.

show abstract

“…A self-similar solution of the harmonic heat flow is an f -harmonic map with a special f , see [27]. For recent studies of f -harmonic maps, see [27,24,25,26,35].…”

Section: Introductionmentioning

confidence: 99%

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

2012

View full text Add to dashboard Cite

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.

show abstract

Non existence of quasi-harmonic spheres

Cited by 15 publications

References 13 publications

A Maximum Principle for Generalizations of Harmonic Maps in Hermitian, Affine, Weyl, and Finsler Geometry

A Maximum Principle for Generalizations of Harmonic Maps in Hermitian, Affine, Weyl, and Finsler Geometry

Nonexistence of quasi-harmonic spheres with large energy

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

Contact Info

Product

Resources

About