• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.769-780
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• 2016

In this paper, we first prove some Liouville type theorems for elliptic inequalities on weighted manifolds which support a weighted Sobolev-type inequality. Secondly, applying the Liouville type theorems to self-shrinkers, we obtain some global rigidity theorems.

• Bulletin of the Korean Mathematical Society
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• v.46 no.2
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• pp.321-329
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• 2009

In this paper, we have considered linear Weingarten hypersurfaces in a sphere and obtained some rigidity theorems. The purpose of this paper is to give some extension of the results due to Cheng-Yau [3] and Li [7].

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1053-1068
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• 2021

We investigate the spacelike hypersurface Mⁿ with constant mean curvature (CMC) in a Lorentz Einstein manifold Lⁿ⁺¹₁, which is supposed to obey some appropriate curvature constraints. Applying a suitable Simons type formula jointly with the well known generalized maximum principle of Omori-Yau, we obtain some rigidity classification theorems and pinching theorems of hypersurfaces.

• Communications of the Korean Mathematical Society
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• v.12 no.2
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• pp.365-371
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• 1997

We prove some relation of volume with the covering radius of Riemannian manifolds and reprove well-known sphere rigidity theorems by using it.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.97-103
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• 2013

As a suitable application of the well known generalized maximum principle of Omori-Yau, we obtain rigidity results concerning to a complete hypersurface immersed with bounded mean curvature in the $(n+1)$-dimensional hyperbolic space $\mathbb{H}^{n+1}$. In our approach, we explore the existence of a natural duality between $\mathbb{H}^{n+1}$ and the half $\mathcal{H}^{n+1}$ of the de Sitter space $\mathbb{S}_1^{n+1}$, which models the so-called steady state space.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1457-1469
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• 2016

In this paper, we study a class of Finsler metric. First, we find some rigidity results of the dually flat (${\alpha}$, ${\beta}$)-metric where the underline Riemannian metric ${\alpha}$ satisfies nonnegative curvature properties. We give a new geometric approach of the Monge-$Amp{\acute{e}}re$ type equation on $R^n$ by using those results. We also get the non-existence of the compact globally dually flat Riemannian manifold.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.1049-1060
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• 2020

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension two surfaces, which is different from the conditions of Risa and Sinestrari in [26] and we also remove the condition that the second fundamental form is uniformly bounded when t ∈ (-∞, -1).