JOURNAL BROWSE
Search
Advanced SearchSearch Tips
ON THE STRUCTURE OF MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF NON-NEGATIVE CURVATURE
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
ON THE STRUCTURE OF MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF NON-NEGATIVE CURVATURE
Yun, Gab-Jin; Kim, Dong-Ho;
  PDF(new window)
 Abstract
Let M be a complete oriented non-compact minimally immersed submanifold in a complete Riemannian manifold N of nonnegative curvature. We prove that if M is super-stable, then there are no non-trivial L harmonic one forms on M. This is a generalization of the main result in [8].
 Keywords
minimal submanifold;super-stable minimal submanifold;L2 harmonic form;
 Language
English
 Cited by
 References
1.
H.-D. Cao, Y Shen, and S. Zhu, The structure of stable minimal hypersurfaces in $R^{n+1}$, Math. Res. Lett. 4 (1997), no. 5, 637-644 crossref(new window)

2.
J. Dodziuk, $L^2$ harmonic forms on complete manifolds, in Seminar on Differential Geometry, Princeton University Press, Princeton N. J., 1982, 291-302

3.
D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199-211 crossref(new window)

4.
H. B. Lawson, Lectures on Minimal Submanifolds. Vol. I, Publish or Perish, Inc., Wilmington, Del., 1980

5.
P.-F. Leung, An estimate on the Ricci curvature of a submanifold and some applications, Proc. Amer. Math. Soc. 114 (1992), no. 4, 1051-1061 crossref(new window)

6.
P. Li, Curvature and function theory on Riemannian manifolds, Surveys in differential geometry, 375-432, Surv. Differ. Geom., VII, Int. Press, Somerville, MA, 2000

7.
P. Li, Differential geometry via harmonic functions, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 293-302, Higher Ed. Press, Beijing, 2002

8.
R. Miyaoka, $L^2$ harmonic 1-forms on a complete stable minimal hypersurface, Geometry and global analysis (Sendai, 1993), 289-293, Tohoku Univ., Sendai, 1993

9.
K. Seo, Rigidity of minimal submanifolds with flat normal bundle, Commun. Korean Math. Soc. 23 (2008), no. 3, 421-426 crossref(new window)

10.
J. Spruck, Remarks on the stability of minimal submanifolds of $R^n$, Math. Z. 144 (1975), no. 2, 169-174 crossref(new window)

11.
S. Tanno, $L^2$ harmonic forms and stability of minimal hypersurfaces, J. Math. Soc. Japan 48 (1996), no. 4, 761-768 crossref(new window)

12.
Q. Wang, On minimal submanifolds in an Euclidean space, Math. Nachr. 261/262 (2003), 176-180 crossref(new window)

13.
S.-T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659-670 crossref(new window)

14.
G. Yun, Total scalar curvature and $L^2$ harmonic 1-forms on a minimal hypersurface in Euclidean space, Geom. Dedicata 89 (2002), 135-141