Stability, complex-analyticity and constancy of pluriharmonic maps from compact Kaehler manifolds

Ohnita, Yoshihiro; Udagawa, Seiichi

doi:10.1007/bf02571268

Cited by 17 publications

(12 citation statements)

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“…Though in general the pluriharmonicity is much stronger than the usual harmonicity, the class of pluriharmonic maps contains many interesting examples of harmonic maps (cf. [15]). There are many beautiful results on harmonic maps from Riemann surfaces (cf.…”

Section: Introductionmentioning

confidence: 99%

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Pluriharmonic Maps into Compact Lie Groups and Factorization into Unitons

Ohnita

Valli

1990

Proceedings of the London Mathematical Society

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Grassmann manifold. The interesting problems are the removability or resolution of the singularity in the factorization for a pluriharmonic map and the explicit construction of pluriharmonic maps from a specific complex manifold into U(N).We shall introduce the notion of meromorphically pluriharmonic maps, which is the smallest class of maps invariant under the addition of meromorphic unitons, and prove a factorization theorem for meromorphically pluriharmonic maps by the energy-reduction process with Harder-Narasimhan filtration.Moreover, by the methods of [33,34,25] and [18,22,10], we also can make the action of the loop algebra and loop group on the space of pluriharmonic maps into a compact Lie group.The authors wish to thank Professor J. Eells for his kind suggestions. The firstnamed author wishes to thank Dr S. Udagawa and Dr M. Bergvelt for some useful communications. He was supported by the Max-Planck-Institut fur Mathematik in Bonn and partially by the London Mathematical Society during the preparation of this paper. He wishes to thank the Max-Planck-Institut and the Universities of Leeds, Warwick and Durham for their hospitality. The second-named author wishes to thank the University of Catania and the International Centre for Theoretical Physics in Trieste for their hospitality and financial support. Pluriharmonic mapsLet M be a connected complex manifold and N be a connected Riemannian manifold with a Riemannian metric g N . Let cp: Af -»N be a smooth map from M to N. The differential dcp: TM-*cp~1TN extends by complex linearity to dcp: TM c ->cp~1TN c . Relative to the complex structure / of M we have a decomposition TM C = rM ( 1 0 ) © rM (01) . By restricting d

cp^TN* 2 and dcp: TM (0A)^> cp~xTN c . Using the induced connection V v and the 3-operator of TM (X ' Q) , we define the (0, l)-exterior derivative of dcp by (z>& d0) ). Then cp is called pluriharmonic if cp satisfies D" dcp = 0. We immediately see the following. PROPOSITION 1.1. A smooth map cp from a complex manifold M to a Riemannian manifold N is pluriharmonic if and only if for any holomorphic curve i: C-+M, the composite cp°i is always harmonic.Note that a pluriharmonic map cp: M-*N is harmonic with respect to any Kahler metric on M (we can always give a Kahler metric in a small neighborhood of M).Assume that M is a Kahler manifold. Denote by g M and V M the Kahler metric of M and its Riemannian connection. The second fundamental form V dcp of the map cp is defined byfor each X, Y e C°°(TM C ). Since V M is a Riemannian connection of a Kahler metric g M , the (0, l)-part of V M is the 3-operator of TM (1 -0) . Hence the (1, l)-part of the second fundamental form V dcp coincides with D" dcp. 548 YOSHIHIRO OHNITA AND GIORGIO VALLI LEMMA 1.2. Let N be a pluriharmonic map from a complex manifold M to a Riemannian manifold N. Then we have R N (d(W) = 0 for each Z,V,We T X M^X' O) and each xeM, where R N denotes the curvatu...

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Section: Introductionmentioning

confidence: 99%

“…LEMMA 1.5. (1) If q) is a pluriharmonic map, then the (1, \)-form e(q>) is closed [14,15]. Moreover, if M is compact and M has a cosymplectic Hermitian metric of fundamental 2-form a), then the energy E w (q)) depends only on the cohomology classes [e(cp)] e H 2 (M, U) and [<o m " 1 ] e H 2 " 1 " 2^, R).…”

Section: Introductionmentioning

confidence: 99%

Pluriharmonic Maps into Compact Lie Groups and Factorization into Unitons

Ohnita

Valli

1990

Proceedings of the London Mathematical Society

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“…Or, on sait que R α est une 2-formeà valeurs dans les endomorphismes antisymétriques de Ω 1 (φ * T N), ceci entraine la nullité du premier terme dans le membre de droite de l'expression (13). Maintenant, pour toute 1-forme σ ∈ Ω 1 (φ * T N),…”

Section: Opérateurs De Diracunclassified

“…Dans le cas particulier où M est une variété kählérienne munie de sa forme de Kähler α, on retrouve dans l'assertion (i) de ce théorème certains résultats de Sampson [15], Hernandez [6] et Ohnita-Udagawa [13].…”

Section: Estimations Sur Le Rang Et Applicationsunclassified

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Applications harmoniques, applications pluriharmoniques et existence de 2-formes parall�les non nulles

Soufi

Petit

1998

Commentarii Mathematici Helvetici

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Abstract. In this paper we study harmonic maps from a compact riemannian manifold equiped with a non trivial parallel 2-form, to a Kähler manifold of strongly negative curvature tensor or a riemannian manifold of strictly negative complex sectional curvature. In a first part we set up some rigidity results of Siu type. Then we obtain an upper bound for the rank of such maps in terms of the rank of the 2-form and deduce some vanishing theorems.Mathematics Subject Classification (1991). 58E20, 53C55, 53C20.

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Some Kind of Stabilities and Instabilities of Energies of Maps Between Kähler Manifolds

Taniguchi

Udagawa

2014

Springer Proceedings in Mathematics &Amp; Statistics

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Stability, complex-analyticity and constancy of pluriharmonic maps from compact Kaehler manifolds

Cited by 17 publications

References 25 publications

Pluriharmonic Maps into Compact Lie Groups and Factorization into Unitons

Pluriharmonic Maps into Compact Lie Groups and Factorization into Unitons

Applications harmoniques, applications pluriharmoniques et existence de 2-formes parall�les non nulles

Some Kind of Stabilities and Instabilities of Energies of Maps Between Kähler Manifolds

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