• Bulletin of the Korean Mathematical Society
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• v.46 no.6
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• pp.1213-1219
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• 2009

Let M$^n$ be a complete oriented non-compact minimally immersed submanifold in a complete Riemannian manifold N$^{n+p}$ of nonnegative 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].

• Communications of the Korean Mathematical Society
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• v.23 no.3
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• pp.421-426
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• 2008

Let $M^n$ be a complete immersed super stable minimal submanifold in $\mathbb{R}^{n+p}$ with fiat normal bundle. We prove that if M has finite total $L^2$ norm of its second fundamental form, then M is an affine n-plane. We also prove that any complete immersed super stable minimal submanifold with flat normal bundle has only one end.

• Korean Journal of Mathematics
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• v.14 no.1
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• pp.13-33
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• 2006

In this article, we study the relations of horizontally conformal maps and harmonic morphisms with the stability of minimal fibers. Let ${\varphi}:(M^n,g){\rightarrow}(N^m,h)$ be a horizontally conformal submersion. There is a tensor T measuring minimality or totally geodesics of fibers of ${\varphi}$. We prove that if T is parallel and the horizontal distribution is integrable, then any minimal fiber of ${\varphi}$ is volume-stable. As a corollary, we obtain that any fiber of a submersive harmonic morphism whose fibers are totally geodesics and the horizontal distribution is integrable is volume-stable. As a consequence, we obtain if ${\varphi}:(M^n,g){\rightarrow}(N^2,h)$ is a submersive harmonic morphism of minimal fibers from a compact Riemannian manifold M into a surface N, T is parallel and the horizontal distribution is integrable, then ${\varphi}$ is energy-stable.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.493-511
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• 2008

In this article, we study the relations of horizontally homothetic harmonic morphisms with the stability of totally geodesic submanifolds. Let $\varphi:(M^n,g)\rightarrow(N^m,h)$ be a horizontally homothetic harmonic morphism from a Riemannian manifold into a Riemannian manifold of non-positive sectional curvature and let T be the tensor measuring minimality or totally geodesics of fibers of $\varphi$. We prove that if T is parallel and the horizontal distribution is integrable, then for any totally geodesic submanifold P in N, the inverse set, $\varphi^{-1}$(P), is volume-stable in M. In case that P is a totally geodesic hypersurface the condition on the curvature can be weakened to Ricci curvature.