• Title, Summary, Keyword: total curvature

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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.

TOTAL CURVATURE FOR SOME MINIMAL SURFACES

  • Jun, Sook Heui
    • Korean Journal of Mathematics
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    • v.7 no.2
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    • pp.285-289
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    • 1999
  • In this paper, we estimate the total curvature of non-parametric minimal surfaces by using the properties of univalent harmonic mappings defined on ${\Delta}=\{z:{\mid}z:{\mid}>1\}$.

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CRITICAL POINT METRICS OF THE TOTAL SCALAR CURVATURE

  • Chang, Jeong-Wook;Hwang, Seung-Su;Yun, Gab-Jin
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.3
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    • pp.655-667
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    • 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.

THREE DIMENSIONAL CRITICAL POINT OF THE TOTAL SCALAR CURVATURE

  • Hwang, Seungsu
    • 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.

SOME REMARKS ON STABLE MINIMAL SURFACES IN THE CRITICAL POINT OF THE TOTAL SCALAR CURVATURE

  • Hwang, Seung-Su
    • Communications of the Korean Mathematical Society
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    • v.23 no.4
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    • pp.587-595
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    • 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.

WEAKLY EINSTEIN CRITICAL POINT EQUATION

  • Hwang, Seungsu;Yun, Gabjin
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.4
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    • pp.1087-1094
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    • 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.

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.

CRITICAL POINTS AND WARPED PRODUCT METRICS

  • Hwang, Seung-Su;Chang, Jeong-Wook
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.1
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    • pp.117-123
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    • 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.

REGULARITY OF SOAP FILM-LIKE SURFACES SPANNING GRAPHS IN A RIEMANNIAN MANIFOLD

  • Gulliver, Robert;Park, Sung-Ho;Pyo, Jun-Cheol;Seo, Keom-Kyo
    • Journal of the Korean Mathematical Society
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    • v.47 no.5
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    • pp.967-983
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    • 2010
  • Let M be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $-{\kappa}^2$. Using the cone total curvature TC($\Gamma$) of a graph $\Gamma$ which was introduced by Gulliver and Yamada [8], we prove that the density at any point of a soap film-like surface $\Sigma$ spanning a graph $\Gamma\;\subset\;M$ is less than or equal to $\frac{1}{2\pi}\{TC(\Gamma)-{\kappa}^2Area(p{\times}\Gamma)\}$. From this density estimate we obtain the regularity theorems for soap film-like surfaces spanning graphs with small total curvature. In particular, when n = 3, this density estimate implies that if $TC(\Gamma)$ < $3.649{\pi}\;+\;{\kappa}^2\inf\limits_{p{\in}F}Area(p{\times}{\Gamma})$, then the only possible singularities of a piecewise smooth (M, 0, $\delta$)-minimizing set $\Sigma$ are the Y-singularity cone. In a manifold with sectional curvature bounded above by $b^2$ and diameter bounded by $\pi$/b, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.