• Title, Summary, Keyword: harmonic map

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ON A RIGIDITY OF HARMONIC DIFFEOMORPHISM BETWEEN TWO RIEMANN SURFACES

  • KIM, TAESOON
    • Honam Mathematical Journal
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    • v.27 no.4
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    • pp.655-663
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    • 2005
  • One of the basic questions concerning harmonic map is on the existence of harmonic maps satisfying a certain condition. Rigidity of a certain harmonic map can be considered as an answer for this kinds of questions. In this article, we study a rigidity property of harmonic diffeomorphisms under the condition that the inverse map is also harmonic. We show that every such a harmonic diffeomorphism is totally geodesic or conformal in two dimensional case.

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HARMONIC GAUSS MAP AND HOPF FIBRATIONS

  • Han, Dong-Soong;Lee, Eun-Hwi
    • The Pure and Applied Mathematics
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    • v.5 no.1
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    • pp.55-63
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    • 1998
  • A Gauss map of m-dimensional distribution on a Riemannian manifold M is called a harmonic Gauss map if it is a harmonic map from the manifold into its Grassmann bundle $G_m$(TM) of m-dimensional tangent subspace. We calculate the tension field of the Gauss map of m-dimensional distribution and especially show that the Hopf fibrations on $S^{4n+3}$ are the harmonic Gauss map of 3-dimensional distribution.

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Harmonic maps into open manifolds with nonnegative curvature

  • Kim, Young-Heon;Yim, Jin-Whan
    • Communications of the Korean Mathematical Society
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    • v.11 no.3
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    • pp.789-796
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    • 1996
  • A complete open manifold with nonnegative curvature is diffeomorphic to the normal bundle of the soul, and the projection map is a Riemannian submersion. Under certain circumstances, we prove that a harmonic map from a compact manifold followed by the projection is again harmonic. Therefore we obtain a harmonic map onto the soul when there is a harmonic map into an open manifold.

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HARMONIC TRANSFORMATIONS OF THE HYPERBOLIC PLANE

  • Park, Joon-Sik
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.4
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    • pp.771-776
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    • 2009
  • Let (H, g) denote the upper half plane in $R^2$ with the Riemannian metric g := ($(dx)^2$ + $(dy)^2$)$/y^2$. First of all we get a necessary and sufficient condition for a diffeomorphism $\phi$ of (H, g) to be a harmonic map. And, we obtain the fact that if a diffeomorphism $\phi$ of (H, g) is a harmonic function, then the following facts are equivalent: (1) $\phi$ is a harmonic map; (2) $\phi$ is an affine transformation; (3) $\phi$ is an isometry (motion).

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ON THE EXISTENCE OF SOLUTIONS OF THE HEAT EQUATION FOR HARMONIC MAP

  • Chi, Dong-Pyo;Kim, Hyun-Jung;Kim, Won-Kuk
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.533-545
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    • 1998
  • In this paper, we prove the existence of solutions of the heat equation for harmonic map on a compact manifold with a boundary when the target manifold is allowed to have positively curved parts.

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ASYMPTOTIC BEHAVIOR OF HARMONIC MAPS AND EXPONENTIALLY HARMONIC FUNCTIONS

  • Chi, Dong-Pyo;Choi, Gun-Don;Chang, Jeong-Wook
    • Journal of the Korean Mathematical Society
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    • v.39 no.5
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    • pp.731-743
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    • 2002
  • Let M be a Riemannian manifold with asymptotically non-negative curvature. We study the asymptotic behavior of the energy densities of a harmonic map and an exponentially harmonic function on M. We prove that the energy density of a bounded harmonic map vanishes at infinity when the target is a Cartan-Hadamard manifold. Also we prove that the energy density of a bounded exponentially harmonic function vanishes at infinity.

A MONOTONICITY FORMULA AND A LIOUVILLE TYPE THEOREM OF V-HARMONIC MAPS

  • Zhao, Guangwen
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1327-1340
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    • 2019
  • We establish a monotonicity formula of V-harmonic maps by using the stress-energy tensor. Use the monotonicity formula, we can derive a Liouville type theorem for V-harmonic maps. As applications, we also obtain monotonicity and constancy of Weyl harmonic maps from conformal manifolds to Riemannian manifolds and ${\pm}holomorphic$ maps between almost Hermitian manifolds. Finally, a constant boundary-value problem of V-harmonic maps is considered.

ON STABILITY OF EINSTEIN WARPED PRODUCT MANIFOLDS

  • Pyo, Yong-Soo;Kim, Hyun-Woong;Park, Joon-Sik
    • Honam Mathematical Journal
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    • v.32 no.1
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    • pp.167-176
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    • 2010
  • Let (B, $\check{g}$) and (N, $\hat{g}$) be Einstein manifolds. Then, we get a complete (necessary and sufficient) condition for the warped product manifold $B\;{\times}_f\;N\;:=\;(B\;{\times}\;N,\;\check{g}\;+\;f{\hat{g}}$) to be Einstein, and obtain a complete condition for the Einstein warped product manifold $B\;{\times}_f\;N$ to be weakly stable. Moreover, we get a complete condition for the map i : ($B,\;\check{g})\;{\times}\;(N,\;\hat{g})\;{\rightarrow}\;B\;{\times}_f\;N$, which is the identity map as a map, to be harmonic. Under the assumption that i is harmonic, we obtain a complete condition for $B\;{\times}_f\;N$ to be Einstein.

A NOTE ON ENERGY MINIMIZING MAP ON MANIFOLD WITH ISOLATED PEAKS

  • SHIN, HEAYONG
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.6 no.1
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    • pp.59-65
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    • 2002
  • In this paper, we consider some homogeneous maps from a cone over 2-spheres and determines whether they become energy minimizing maps or not. In fact, any homogeneous map from a standard cone over 2-sphere of radius smaller than 1 can not be a minimizing harmonic map.

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LIOUVILLE TYPE THEOREM FOR p-HARMONIC MAPS II

  • Jung, Seoung Dal
    • Communications of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.155-161
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    • 2014
  • Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that $Ric^M{\geq}-\frac{4(p-1)}{p^2}{\mu}_0$ at all $x{\in}M$ and Vol(M) is infinite, where ${\mu}_0$ > 0 is the infimum of the spectrum of the Laplacian acting on $L^2$-functions on M. Then any p-harmonic map ${\phi}:M{\rightarrow}N$ of finite p-energy is constant Also, we study Liouville type theorem for p-harmonic morphism.