On stable quasi-harmonic maps and Liouville type theorems
Authors:
Deliang Hsu and Chunqin Zhou
Journal:
Proc. Amer. Math. Soc. 130 (2002), 3415-3422
MSC (2000):
Primary 58G30, 35B05
DOI:
https://doi.org/10.1090/S0002-9939-02-06499-7
Published electronically:
May 8, 2002
MathSciNet review:
1913022
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We consider Liouville type problems of stable quasi-harmonic maps, by ``stable'' we mean that the second variation of quasi-energy functional 
is nonnegative, and we prove that the stable quasi-harmonic maps must be constant under some geometry conditions.
- 1. Shiu Yuen Cheng, Liouville theorem for harmonic maps, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 147–151. MR 573431
- 2. Wei Yue Ding and Fang-Hua Lin, A generalization of Eells-Sampson’s theorem, J. Partial Differential Equations 5 (1992), no. 4, 13–22. MR 1192714
- 3. J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, https://doi.org/10.1112/blms/10.1.1
- 4. FangHua Lin and ChangYou Wang, Harmonic and quasi-harmonic spheres, Comm. Anal. Geom. 7 (1999), no. 2, 397–429. MR 1685578, https://doi.org/10.4310/CAG.1999.v7.n2.a9
- 5. Ralph Howard, The nonexistence of stable submanifolds, varifolds, and harmonic maps in sufficiently pinched simply connected Riemannian manifolds, Michigan Math. J. 32 (1985), no. 3, 321–334. MR 803835, https://doi.org/10.1307/mmj/1029003241
- 6. Ralph Howard and S. Walter Wei, Nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of Euclidean space, Trans. Amer. Math. Soc. 294 (1986), no. 1, 319–331. MR 819950, https://doi.org/10.1090/S0002-9947-1986-0819950-4
- 7.
WEI S. W., Liouville theorem for stable harmonic maps into either strongly unstable, or
-pinched manifolds. Proc. of Symp. in Pare Math. Vol 44, 406-412(1986).
- 8. S. Walter Wei, Liouville theorems and regularity of minimizing harmonic maps into super-strongly unstable manifolds, Geometry and nonlinear partial differential equations (Fayetteville, AR, 1990) Contemp. Math., vol. 127, Amer. Math. Soc., Providence, RI, 1992, pp. 131–154. MR 1155415, https://doi.org/10.1090/conm/127/1155415
- 9. Y. L. Xin, Some results on stable harmonic maps, Duke Math. J. 47 (1980), no. 3, 609–613. MR 587168
- 10. HSU D. L. & ZHOU C Q., On the finiteness of energy of quasi-harmonic sphere, preprint.
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58G30, 35B05
Retrieve articles in all journals with MSC (2000): 58G30, 35B05
Additional Information
Deliang Hsu
Affiliation:
Department of Applied Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
Email:
hsudl@online.sh.cn
Chunqin Zhou
Affiliation:
Department of Applied Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
DOI:
https://doi.org/10.1090/S0002-9939-02-06499-7
Keywords:
Quasi-harmonic map,
stableness,
Liouville type theorems
Received by editor(s):
April 19, 2000
Received by editor(s) in revised form:
June 25, 2001
Published electronically:
May 8, 2002
Additional Notes:
The first author was supported by NSF of Shanghai Jiao Tong University
Communicated by:
Bennett Chow
Article copyright:
© Copyright 2002
American Mathematical Society
