• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.281-290
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• 2024

We consider the total scalar curvature functional, and show that if the second variation in the transverse traceless tensor direction is negative, then the metric is Einstein. We also find the relation between the second variation and the Lichnerowicz Laplacian.

• Communications of the Korean Mathematical Society
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• v.12 no.1
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• pp.101-108
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• 1997

We let (M,g) be a noncompact complete Riemannina manifold of dimension $n \geq 3$ with finite volume and positive scalar curvature. We show the existence of a conformal metric with constant positive scalar curvature on (M,g) by gluing solutions of Yamabe equation on each compact subsets $K_i$ with $\cup K_i = M$ .

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.867-871
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• 2013

It has been conjectured that, on a compact 3-dimensional orientable manifold, a critical point of the total scalar curvature restricted to the space of constant scalar curvature metrics of unit volume is Einstein. In this paper we prove this conjecture under a condition that ker $s^{\prime}^*_g{\neq}0$, which generalizes the previous partial results.

• The Pure and Applied Mathematics
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• v.20 no.3
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• pp.199-206
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• 2013

We present a 4-dimensional nil-manifold as the first example of a closed non-K$\ddot{a}$hlerian symplectic manifold with the following property: a function is the scalar curvature of some almost K$\ddot{a}$hler metric iff it is negative somewhere. This is motivated by the Kazdan-Warner's work on classifying smooth closed manifolds according to the possible scalar curvature functions.

• Honam Mathematical Journal
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• v.29 no.4
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• pp.641-647
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• 2007

In this paper, we consider the problem of achieving constant scalar curvature on warped product manifolds according to fiber manifolds with constant scalar curvature.

• Honam Mathematical Journal
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• v.28 no.2
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• pp.233-240
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• 2006

In this paper, we consider the problem of achieving a prescribed scalar curvature on warped product manifolds according to fiber manifolds with constant scalar curvature.

• Journal of the Korean Mathematical Society
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• v.38 no.1
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• pp.135-146
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• 2001

Let M(sup)n be a space-ike submanifold in a de Sitter space M(sub)p(sup)n+p (c) with constant scalar curvature. We firstly extend Cheng-Yau's Technique to higher codimensional cases. Then we study the rigidity problem for M(sup)n with parallel normalized mean curvature vector field.

• Bulletin of the Korean Mathematical Society
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• v.48 no.1
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• pp.129-140
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• 2011

The aim of this paper is to study the structure of the fundamental group of a closed oriented Riemannian manifold with positive scalar curvature. To be more precise, let M be a closed oriented Riemannian manifold of dimension n (4 $\leq$ n $\leq$ 7) with positive scalar curvature and non-trivial first Betti number, and let be $\alpha$ non-trivial codimension one homology class in $H_{n-1}$(M;$\mathbb{R}$). Then it is known as in [8] that there exists a closed embedded hypersurface $N_{\alpha}$ of M representing $\alpha$ of minimum volume, compared with all other closed hypersurfaces in the homology class. Our main result is to show that the fundamental group ${\pi}_1(N_{\alpha})$ is always virtually free. In particular, this gives rise to a new obstruction to the existence of a metric of positive scalar curvature.

• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.749-761
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• 2018

Here, we calculate the evolution equation of the reduced hh-curvature and the Ricci scalar along the Finslerian Ricci flow. We prove that Finsler Ricci flow preserves positivity of the reduced hh-curvature on finite time. Next, it is shown that evolution of Ricci scalar is a parabolic-type equation and moreover if the initial Finsler metric is of positive flag curvature, then the flag curvature, as well as the Ricci scalar, remain positive as long as the solution exists. Finally, we present a lower bound for Ricci scalar along Ricci flow.

• Communications of the Korean Mathematical Society
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• v.25 no.2
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• pp.245-249
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• 2010

We show that on a compact locally conformal K$\ddot{a}$hler manifold $M^{2n}$ (dim $M^{2n}\;=\;2n\;{\geq}\;4$), $M^{2n}$ is K$\ddot{a}$hler if and only if its conformal scalar curvature k is not smaller than the scalar curvature s of $M^{2n}$ everywhere. As a consequence, if a compact locally conformal K$\ddot{a}$hler manifold $M^{2n}$ is both conformally flat and scalar flat, then $M^{2n}$ is K$\ddot{a}$hler. In contrast with the compact case, we show that there exists a locally conformal K$\ddot{a}$hler manifold with k equal to s, which is not K$\ddot{a}$hler.