• Journal of the Chungcheong Mathematical Society
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• v.34 no.4
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• pp.369-378
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• 2021

In this paper, we give an upper bound for the fundamental tone of stable constant mean curvature hypersurfaces in hyperbolic space. Let M be an n-dimensional complete non-compact constant mean curvature hypersurface with finite L²-norm of the traceless second fundamental form. If M is weakly stable, then λ₁(M) is bounded above by n² + O(n^2+s) for arbitrary s > 0.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1245-1252
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• 2011

Consider the $L^2$-adjoint $s_g^{'*}$ of the linearization of the scalar curvature $s_g$. If ker $s_g^{'*}{\neq}0$ on an n-dimensional compact manifold, it is well known that the scalar curvature $s_g$ is a non-negative constant. In this paper, we study the structure of the level set ${\varphi}^{-1}$(0) and find the behavior of Ricci tensor when ker $s_g^{'*}{\neq}0$ with $s_g$ > 0. Also for a nontrivial solution (g, f) of $z=s_g^{'*}(f)$ on an n-dimensional compact manifold, we analyze the structure of the regular level set $f^{-1}$(-1). These results give a good understanding of the given manifolds.

• Kyungpook Mathematical Journal
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• v.60 no.1
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• pp.185-195
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• 2020

This paper aims to study the 𝒲^⁎-curvature tensor on relativistic space-times. The energy-momentum tensor T of a space-time having a semi-symmetric 𝒲^⋆-curvature tensor is semi-symmetric, whereas the whereas the energy-momentum tensor T of a space-time having a divergence free 𝒲^⋆-curvature tensor is of Codazzi type. A space-time having a traceless 𝒲^⁎-curvature tensor is Einstein. A 𝒲^⁎-curvature flat space-time is Einstein. Perfect fluid space-times which admits 𝒲^⋆-curvature tensor are considered.