• Title/Summary/Keyword: curvature equation

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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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EVOLUTION EQUATIONS ON A RIEMANNIAN MANIFOLD WITH A LOWER RICCI CURVATURE BOUND

  • Chang, Jeongwook
    • East Asian mathematical journal
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    • v.30 no.1
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    • pp.79-91
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    • 2014
  • We consider the parabolic evolution differential equation such as heat equation and porus-medium equation on a Riemannian manifold M whose Ricci curvature is bounded below by $-(n-1)k^2$ and bounded below by 0 on some amount of M. We derive some bounds of differential quantities for a positive solution and some inequalities which resemble Harnack inequalities.

Kelvin Equation and Its Role in Nano Systems (켈빈 식과 이의 나노 계에서의 구실)

  • Lim, Kyung-Hee
    • Journal of the Korean Applied Science and Technology
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    • v.23 no.1
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    • pp.54-62
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    • 2006
  • Kelvin equation is revisited, which accounts for important phenomena observed frequently in nano-dispersion systems. They include vapor pressure increase for curved interfaces, nucleation, capillary condensation, Ostwald ripening and so on. The smaller the radius of curvature is, the more significant Kelvin equation becomes. Therefore, its meaning, curvature effect, and importance are examined and discussed.

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.

GRADIENT ESTIMATE OF HEAT EQUATION FOR HARMONIC MAP ON NONCOMPACT MANIFOLDS

  • Kim, Hyun-Jung
    • Journal of applied mathematics & informatics
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    • v.28 no.5_6
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    • pp.1461-1466
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    • 2010
  • aSuppose that (M, g) is a complete Riemannian manifold with Ricci curvature bounded below by -K < 0 and (N, $\bar{b}$) is a complete Riemannian manifold with sectional curvature bounded above by a constant $\mu$ > 0. Let u : $M{\times}[0,\;{\infty}]{\rightarrow}B_{\tau}(p)$ is a heat equation for harmonic map. We estimate the energy density of u.

STRESS-ENERGY TENSOR OF THE TRACELESS RICCI TENSOR AND EINSTEIN-TYPE MANIFOLDS

  • Gabjin Yun
    • Journal of the Korean Mathematical Society
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    • v.61 no.2
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    • pp.255-277
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    • 2024
  • In this paper, we introduce the notion of stress-energy tensor Q of the traceless Ricci tensor for Riemannian manifolds (Mn, g), and investigate harmonicity of Riemannian curvature tensor and Weyl curvature tensor when (M, g) satisfies some geometric structure such as critical point equation or vacuum static equation for smooth functions.

TUBES OF WEINGARTEN TYPES IN A EUCLIDEAN 3-SPACE

  • Ro, Jin Suk;Yoon, Dae Won
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.3
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    • pp.359-366
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    • 2009
  • In this paper, we study a tube in a Euclidean 3-space satisfying some equation in terms of the Gaussian curvature, the mean curvature and the second Gaussian curvature.

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ON EVOLUTION OF FINSLER RICCI SCALAR

  • Bidabad, Behroz;Sedaghat, Maral Khadem
    • 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.

GRADIENT ESTIMATES OF A NONLINEAR ELLIPTIC EQUATION FOR THE V -LAPLACIAN

  • Zeng, Fanqi
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.4
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    • pp.853-865
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    • 2019
  • In this paper, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold: $${\Delta}_Vu+cu^{\alpha}=0$$, where c, ${\alpha}$ are two real constants and $c{\neq}0$. By applying Bochner formula and the maximum principle, we obtain local gradient estimates for positive solutions of the above equation on complete Riemannian manifolds with Bakry-${\acute{E}}mery$ Ricci curvature bounded from below, which generalize some results of [8].