DOI QR코드

DOI QR Code

A LIOUVILLE TYPE THEOREM FOR HARMONIC MORPHISMS

  • Jung, Seoung-Dal (DEPARTMENT OF MATHEMATICS CHEJU NATIONAL UNIVERSITY) ;
  • Liu, Huili (DEPARTMENT OF MATHEMATICS NORTHEASTERN UNIVERSITY) ;
  • Moon, Dong-Joo (DEPARTMENT OF MATHEMATICS CHEJU NATIONAL UNIVERSITY)
  • 발행 : 2007.07.30

초록

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of nonpositive scalar curvature. Let ${\mu}0$ be the least eigenvalue of the Laplacian acting on $L^2-functions$ on M. We show that if $Ric^M{\ge}-{\mu}0$ at all $x{\in}M$ and either $Ric^M>-{\mu}0$ at some point x0 or Vol(M) is infinite, then every harmonic morphism ${\phi}:M{\to}N$ of finite energy is constant.

키워드

참고문헌

  1. P. Berard, A note on Bochner type theorems for complete manifolds, Manuscripta Math. 69 (1990), no. 3, 261-266 https://doi.org/10.1007/BF02567924
  2. G. Choi and G. Yun, A theorem of Liouville type for harmonic morphisms, Geom. Dedicata 84 (2001), 179-182 https://doi.org/10.1023/A:1010329618346
  3. J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1-68 https://doi.org/10.1112/blms/10.1.1
  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 https://doi.org/10.1007/BF02568161
  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 https://doi.org/10.1002/cpa.3160280203
  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 https://doi.org/10.1512/iumj.1976.25.25051

피인용 문헌

  1. LIOUVILLE TYPE THEOREM FOR p-HARMONIC MAPS II vol.29, pp.1, 2014, https://doi.org/10.4134/CKMS.2014.29.1.155
  2. Liouville type theorems for p-harmonic maps vol.342, pp.1, 2008, https://doi.org/10.1016/j.jmaa.2007.12.018