• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.709-723
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• 2022

In this note, we revise some vanishing and finiteness results on hypersurfaces with finite index in ℝⁿ⁺¹. When the hypersurface is stable minimal, we show that there is no nontrivial L^2p harmonic 1-form for some p. The our range of p is better than those in [7]. With the same range of p, we also give finiteness results on minimal hypersurfaces with finite index.

• 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.

• Bulletin of the Korean Chemical Society
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• v.21 no.10
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• pp.953-954
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• 2000

The equilibrium geometrical structures of silver atom clusters at their electronic ground states have been theo-retically determined by using the nonrelativistic semiempirical INDO/1 method. The clusters investigated are Agn, Agn+, and Agn- (n = 7 , 8, 9). In order to find the most stable structure, i.e., the global minimum in energy hypersurface, geometry optimization and energy calculation processes have been repeatedly performed for all the possible graphical models by changing the bond parameters (resonance integral values). The heptamers are pentagonal bipyramidal-Ag7(D5h), Ag7+ (D5h), Ag7- (D5h); the octamers are pentagonal bipyramidal with one atom capped-Ag8(D2d), Ag8+ (Cs), Ag8- (D2d); the nonamers are pentagonal bipyramidal with two atoms capped -Ag9(C2v), Ag9+ (C2v), Ag9- (C2v). Our structures are in good agreement with those by ab initio calculations ex-cept for the anionic Ag9- cluster. And it is noted that the INDO/1 method can accurately predict the Ag cluster geometries when a proper set of bond parameters is used.