A LIOUVILLE TYPE THEOREM FOR HARMONIC MORPHISMS

Title & Authors
A LIOUVILLE TYPE THEOREM FOR HARMONIC MORPHISMS
Jung, Seoung-Dal; Liu, Huili; Moon, Dong-Joo;

Abstract
Let M be a complete Riemannian manifold and let N be a Riemannian manifold of nonpositive scalar curvature. Let $\small{{\mu}0}$ be the least eigenvalue of the Laplacian acting on $\small{L^2-functions}$ on M. We show that if $\small{Ric^M{\ge}-{\mu}0}$ at all $\small{x{\in}M}$ and either $Ric^M>-{\mu}0$ at some point x0 or Vol(M) is infinite, then every harmonic morphism $\small{{\phi}:M{\to}N}$ of finite energy is constant.
Keywords
harmonic map;harmonic morphism;
Language
English
Cited by
1.
LIOUVILLE TYPE THEOREM FOR p-HARMONIC MAPS II,;

대한수학회논문집, 2014. vol.29. 1, pp.155-161
1.
Liouville type theorems for p-harmonic maps, Journal of Mathematical Analysis and Applications, 2008, 342, 1, 354
2.
LIOUVILLE TYPE THEOREM FOR p-HARMONIC MAPS II, Communications of the Korean Mathematical Society, 2014, 29, 1, 155
References
1.
P. Berard, A note on Bochner type theorems for complete manifolds, Manuscripta Math. 69 (1990), no. 3, 261-266

2.
G. Choi and G. Yun, A theorem of Liouville type for harmonic morphisms, Geom. Dedicata 84 (2001), 179-182

3.
J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1-68

4.
B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 2, 107-144

5.
S. D. Jung, Harmonic maps of complete Riemannian manifolds, Nihonkai Math. J. 8 (1997), no. 2, 147-154

6.
A. Kasue and T. Washio, Growth of equivariant harmonic maps and harmonic morphisms, Osaka J. Math. 27 (1990), no. 4, 899-928

7.
N. Nakauchi, A Liouville type theorem for p-harmonic maps, Osaka J. Math. 35 (1998), no. 2, 303-312

8.
R. Schoen and S. T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds of nonnegative Ricci curvature, Comm. Math. Helv. 51 (1976), no. 3, 333-341

9.
H. Wu, The Bochner technique in differential geometry, Math. Rep. 3 (1988), no. 2, 289-538

10.
S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), no. 7, 201-228

11.
S. T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), 659-670