• Title, Summary, Keyword: scalar curvature

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A NOTE ON DECREASING SCALAR CURVATURE FROM FLAT METRICS

  • Kim, Jongsu
    • Honam Mathematical Journal
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    • v.35 no.4
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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.

ZERO SCALAR CURVATURE ON OPEN MANIFOLDS

  • Kim, Seong-Tag
    • Communications of the Korean Mathematical Society
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    • v.13 no.3
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    • pp.539-544
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    • 1998
  • Let (M, g) be a noncompact complete Riemannian manifold of dimension n $\geq$ 3 with scalar curvature S, which is close to O. With conditions on a conformal invariant and scalar curvature of (M, g), we show that there exists a conformal metric (equation omitted), near g, whose scalar curvature (equation omitted) = 0 by gluing solutions of the corresponding partial differential equation on each bounded subsets $K_{i}$ with ∪$K_{i}$ = M.

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Constant scalar curvature on open manifolds with finite volume

  • Kim, Seong-Tag
    • 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 $\cup K_i = M$ .

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RICCI AND SCALAR CURVATURES ON SU(3)

  • Kim, Hyun-Woong;Pyo, Yong-Soo;Shin, Hyun-Ju
    • Honam Mathematical Journal
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    • v.34 no.2
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    • pp.231-239
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    • 2012
  • In this paper, we obtain the Ricci curvature and the scalar curvature on SU(3) with some left invariant Riemannian metric. And then we get a necessary and sufficient condition for the scalar curvature (resp. the Ricci curvature) on the Riemannian manifold SU(3) to be positive.

DEFORMATION OF CARTAN CURVATURE ON FINSLER MANIFOLDS

  • Bidabad, Behroz;Shahi, Alireza;Ahmadi, Mohamad Yar
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.6
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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.

SPACE-LIKE SUBMANIFOLDS WITH CONSTANT SCALAR CURVATURE IN THE DE SITTER SPACES

  • Liu, Ximin
    • 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.

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TOTAL SCALAR CURVATURE AND EXISTENCE OF STABLE MINIMAL SURFACES

  • Hwang, Seung-Su
    • Honam Mathematical Journal
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    • v.30 no.4
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    • pp.677-683
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    • 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.

DIFFERENTIAL EQUATIONS ON WARPED PRODUCTS (II)

  • JUNG, YOON-TAE
    • Honam Mathematical Journal
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    • v.28 no.3
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    • pp.399-407
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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 zero scalar curvature.

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DIFFERENTIAL EQUATIONS ON WARPED PRODUCTS

  • JUNG, YOON-TAE;LEE, SANG CHEOL
    • 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.

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