• 호남수학학술지
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• 제35권4호
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• pp.647-655
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• 2013

We obtain $C^{\infty}$-continuous paths of explicit Riemannian metrics $g_t$, $0{\leq}t$ < ${\varepsilon}$, whose scalar curvatures $s(g_t)$ decrease, where $g_0$ is a flat metric, i.e. a metric with vanishing curvature. Most of them can exist on tori of dimension ${\geq}3$. Some of them yield scalar curvature decrease on a ball in the Euclidean space.

• 대한수학회보
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• 제54권6호
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• pp.2119-2139
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• 2017

Here, certain Ricci flow for Finsler n-manifolds is considered and deformation of Cartan hh-curvature, as well as Ricci tensor and scalar curvature, are derived for spaces of scalar flag curvature. As an application, it is shown that on a family of Finsler manifolds of constant flag curvature, the scalar curvature satisfies the so-called heat-type equation. Hence on a compact Finsler manifold of constant flag curvature of initial non-negative scalar curvature, the scalar curvature remains non-negative by Ricci flow and blows up in a short time.

• 대한수학회보
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• 제61권1호
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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.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제20권3호
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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.

• 대한수학회보
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• 제35권2호
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• pp.195-201
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• 1998

We let (M,g) be a noncompact complete Riemannian manifold of dimension n $\geq$ 3 with scalar curvatue S, which is close to -1. We show the existence of a conformal metric $\bar{g}$, near to g, whose scalar curvature $\bar{S}$ = -1 by gluing solutions of the corresponding partial differential equation on each bounded subsets $K_i$ with ${\bigcup}K_i$ = M.

• 대한수학회지
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• 제41권5호
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• pp.809-820
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• 2004

On $R^{2n}$, n$\geq$2, with the standard symplectic structure we construct compatible almost K hler metrics with negative scalar curvature on a polydisc which are Euclidean away from the polydisc.c.

• 대한수학회보
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• 제48권6호
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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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• 제63권4호
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• pp.639-645
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• 2023

An almost Yamabe soliton is a generalization of the Yamabe soliton. In this article, we deduce some results regarding almost gradient Yamabe solitons. More specifically, we show that a compact almost gradient Yamabe soliton having non-negative Ricci curvature is trivial. Again, we prove that an almost gradient Yamabe soliton with a non-negative potential function and scalar curvature bound admitting an integral condition is trivial. Moreover, we give a characterization of a compact almost gradient Yamabe solitons.

• 대한수학회보
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• 제50권4호
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• pp.1087-1098
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• 2013

We find a $C^{\infty}$-continuous path of Riemannian metrics $g_t$ on $\mathbb{R}^k$, $k{\geq}3$, for $0{\leq}t{\leq}{\varepsilon}$ for some number ${\varepsilon}$ > 0 with the following property: $g_0$ is the Euclidean metric on $\mathbb{R}^k$, the scalar curvatures of $g_t$ are strictly decreasing in $t$ in the open unit ball and $g_t$ is isometric to the Euclidean metric in the complement of the ball. Furthermore we extend the discussion to the Fubini-Study metric in a similar way.

• 대한수학회보
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• 제49권3호
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• pp.581-588
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• 2012

We find a $C^{\infty}$ one-parameter family of Riemannian metrics $g_t$ on $\mathbb{R}^3$ for $0{\leq}t{\leq}{\varepsilon}$ for some number ${\varepsilon}$ with the following property: $g_0$ is the Euclidean metric on $\mathbb{R}^3$, the scalar curvatures of $g_t$ are strictly decreasing in t in the open unit ball and $g_t$ is isometric to the Euclidean metric in the complement of the ball.