• 대한수학회보
• /
• 제61권1호
• /
• pp.281-290
• /
• 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.

• 호남수학학술지
• /
• 제30권4호
• /
• pp.677-683
• /
• 2008

On a compact n-dimensional manifold M, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of volume 1, should be Einstein. The purpose of the present paper is to prove that a 3-dimensional manifold (M,g) is isometric to a standard sphere if ker $s^*_g{{\neq}}0$ and there is a lower Ricci curvature bound. We also study the structure of a compact oriented stable minimal surface in M.

• 호남수학학술지
• /
• 제35권4호
• /
• pp.647-655
• /
• 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
• /
• 제50권3호
• /
• pp.867-871
• /
• 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.

• 대한수학회보
• /
• 제53권4호
• /
• pp.1087-1094
• /
• 2016

On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, after derivng an interesting curvature identity, we show that the conjecture is true in dimension three and four when g is weakly Einstein. In higher dimensional case $n{\geq}5$, we also show that the conjecture is true under an additional Ricci curvature bound. Moreover, we prove that the manifold is isometric to a standard n-sphere when it is n-dimensional weakly Einstein and the kernel of the linearized scalar curvature operator is nontrivial.

• 대한수학회보
• /
• 제49권3호
• /
• pp.655-667
• /
• 2012

In this paper, we deal with a critical point metric of the total scalar curvature on a compact manifold $M$. We prove that if the critical point metric has parallel Ricci tensor, then the manifold is isometric to a standard sphere. Moreover, we show that if an $n$-dimensional Riemannian manifold is a warped product, or has harmonic curvature with non-parallel Ricci tensor, then it cannot be a critical point metric.

• 대한수학회논문집
• /
• 제23권4호
• /
• pp.587-595
• /
• 2008

It is well known that critical points of the total scalar curvature functional S on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of S is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is isometric to a standard sphere. In this paper we investigate the relationship between the first Betti number and stable minimal surfaces, and study the analytic properties of stable minimal surfaces in M for n = 3.

• 대한수학회보
• /
• 제41권1호
• /
• pp.117-123
• /
• 2004

It has been conjectured that, on a compact orient able manifold M, a critical point of the total scalar curvature functional restricted the space of unit volume metrics of constant scalar curvature is Einstein. In this paper we show that if a manifold is a 3-dimensional warped product, then (M, g) cannot be a critical point unless it is isometric to the standard sphere.

• 대한수학회보
• /
• 제37권3호
• /
• pp.641-648
• /
• 2000

It has been conjectured that, on a compact 3-dimensional manifold, a critical point of the total scalar curvature functional restricted to the space of constant scalar curvature metrics of volume 1 is Einstein. In this paper we find a sufficient condition that a critical point is Einstein. This condition is equivalent for a critical point ot be conformally flat. Its relationship with the Fisher-Marsden conjecture is also discussed.

• 대한수학회보
• /
• 제26권2호
• /
• pp.175-177
• /
• 1989

The following is the basic problem about the stability in Riemannian Geometry; given a Riemannian manifold N, find all stable complete minimal submanifolds of N. As answers of this problem, do Carmo-Peng [1] and Fischer-Colbrie and Schoen [3] showed that the stable minimal surfaces in R$^{3}$ are planes and Schoen-Yau [5] and Fischer-Colbrie and Schoen [3] gave a solution for the case where the ambient space is a three dimensional manifold with nonnegative scalar curvature. In this paper we will remove the assumption of finite absolute total curvature in [3, Theorem 3].