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On the structure of minimal submanifolds in a Riemannian manifold of non-negative curvature

김동호|윤갑진
대한수학회 (The Korean Mathematical Society)|2009.11|대한수학회보 (Bulletin of the KMS)|제46권 제6호|pp. 1213-1219

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참고문헌

키워드

minimal submanifoldsuper-stable minimal submanifoldL^2 harmonic form
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초록

Let M^{n} be a complete oriented non-compact minimally immersed submanifold in a complete Riemannian manifold N^{n+p} of non-negative curvature. We prove that if M is super-stable, then there are no non-trivial L^2 harmonic one forms on M. This is a generalization of the main result in [8].

Abstract

Let M^{n} be a complete oriented non-compact minimally immersed submanifold in a complete Riemannian manifold N^{n+p} of non-negative curvature. We prove that if M is super-stable, then there are no non-trivial L^2 harmonic one forms on M. This is a generalization of the main result in [8].

참고문헌 (14)

  1. H.-D. Cao, "The structure of stable minimal hypersurfaces in Rn+1", Math. Res. Lett, 4(5), pp.637~644 (1997)
  2. J. Dodziuk, "L2 harmonic forms on complete manifolds, in Seminar on Differential Geometry", pp.291~302 (1982), Princeton University Press
  3. D. Fischer-Colbrie, "The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature", Comm. Pure Appl. Math, 33(2), pp.199~211 (1980)
  4. H. B. Lawson, "Lectures on Minimal Submanifolds" (1980), Publish or Perish, Inc
  5. P.-F. Leung, "An estimate on the Ricci curvature of a submanifold and some applications", Proc. Amer. Math. Soc, 114(4), pp.1051~1061 (1992)
  6. P. Li, "Curvature and function theory on Riemannian manifolds, Surveys in differential geometry", Surv. Differ. Geom., VII, Int. Press, 375~432 (2000)
  7. P. Li, "Differential geometry via harmonic functions", Proceedings of the International Congress of Mathematicians, Vol. II, pp.293~302 (2002), Higher Ed. Press
  8. R. Miyaoka, "L2 harmonic 1-forms on a complete stable minimal hypersurface, Geometry and global analysis", Tohoku Univ, 289~293 (1993)
  9. 서검교, "Rigidity of minimal submanifolds with flat normal bundle", 대한수학회논문집, 23(3), pp.421~426 (2008), 대한수학회
  10. J. Spruck, "Remarks on the stability of minimal submanifolds of Rn", Math. Z, 144(2), pp.169~174 (1975)
  11. S. Tanno, "L2 harmonic forms and stability of minimal hypersurfaces", J. Math. Soc. Japan, 48(4), pp.761~768 (1996)
  12. Q. Wang, "On minimal submanifolds in an Euclidean space", Math. Nachr, 261, pp.176~180 (2003)
  13. S.-T. Yau, "Some function-theoretic properties of complete Riemannian manifold and their applications to geometry", Indiana Univ. Math. J, 25(7), pp.659~670 (1976)
  14. G. Yun, "Total scalar curvature and L2 harmonic 1-forms on a minimal hypersurface in Euclidean space", Geom. Dedicata, 89, pp.135~141 (2002)

관련 논문AI추천

Structure of stable minimal hypersurfaces in a Riemannian manifold of nonnegative Ricci curvature김정진, 윤갑진대한수학회 · 대한수학회보2013 · 제50권 제4호Rigidity of minimal submanifolds with flat normal bundle서검교대한수학회 · 대한수학회논문집2008 · 제23권 제3호Rigidity of immersed submanifolds in a hyperbolic spaceNguyen Thac Dung대한수학회 · 대한수학회보2016 · 제53권 제6호STABLE MINIMAL HYPERSURFACES WITH WEIGHTED POINCARE INEQUALITY IN A RIEMANNIAN MANIFOLDNguyen Dinh Sang, Nguyen Thi Thanh대한수학회 · 대한수학회논문집2014 · 제29권 제1호Complete noncompact submanifolds of manifolds with negative curvatureJing Mao, Ya Gao, Yanling Gao, Zhiqi Xie대한수학회 · 대한수학회지2024 · 제61권 제1호The rigidity of minimal submanifolds in a locally symmetric spaceShunjuan Cao대한수학회 · 대한수학회보2013 · 제50권 제1호

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