On the holomorphicity of proper harmonic maps between unit balls with the Bergman metrics
نویسندگان
چکیده
Let M and N be two Kähler manifolds with Kähler metrics h = hijdzidzj and g = gαβdwdwβ , respectively. Let u : M → N be a map from M to N . When both M and N are compact, in his proof of the celebrated strong rigidity theorem for compact Kähler manifolds, Siu [S1] proved that any harmonic map u must be holomorphic or antiholomorphic, under the assumption that N has strongly negative curvature in the sense of Siu and the rank of du at one point is greater than or equal to four (the last condition excludes the case of complex dimension one when the theorem is obviously false). The key of the proof is Siu’s ∂∂-Bochner formula:
منابع مشابه
Holomorphic maps from the complex unit ball to Type IV classical domains
The first part of this paper is devoted to establish new rigidity results for proper holomorphic maps from the complex unit ball to higher rank bounded symmetric domains. The rigidity properties have been extensively studied in the past decades for proper holomorphic maps F : Ω1 → Ω2, between bounded symmetric domains Ω1,Ω2. The pioneer works are due to Poincaré [P] and later to Alexander [Al] ...
متن کاملBergman Approximations of Harmonic Maps into the Space of Kähler Metrics on Toric Varieties
We generalize the results of Song-Zelditch on geodesics in spaces of Kähler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kähler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the C topology by harmonic maps into the spaces of Bergman metrics. In pa...
متن کاملCaculus of Variation and the L-Bergman Metric on Teichmüller Space
The canonical metric on a surface is of nonpositive curvature, so it is natural to study harmonic maps between canonical metrics on a surface in a fixed homotopy class. Through this approach, we establish the LBergman metric on Teichmüller space as the second variation of energy functionals of chosen families of harmonic maps.
متن کاملOn Conformal Biharmonic Immersions
This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a 2-parameter family of non-minimal conformal biharmonic immersions of cylinder into R and ...
متن کاملHarmonic Bergman Kernel for Some Balls
We treat the complex harmonic function on the Np–ball which is defined by the Np–norm related to the Lie norm. As a subspace, we treat Hardy spaces and consider the Bergman kernel on those spaces. Then, we try to construct the Bergman kernel in a concrete form in 2–dimensional Euclidean space. Introduction. In [2], [4], [6] and [7], we studied holomorphic functions and analytic functionals on t...
متن کامل