A boundary value problem for Hermitian harmonic maps and applications. (English) Zbl 0864.58009
Author’s abstract: We study the existence and uniqueness problem for Hermitian harmonic maps from Hermitian manifolds with boundary to Riemannian manifolds of nonpositive sectional curvature and with convex boundary. The complex analyticity of such maps and the related rigidity problems are also investigated.
Reviewer: W.Mozgawa (Lublin)
MSC:
| 58E10 | Variational problems in applications to the theory of geodesics (problems in one independent variable) |
| 58E20 | Harmonic maps, etc. |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
References:
| [1] | S. Bochner, Analytic and meromorphic continuation by means of Green’s formula, Ann. of Math. 44 (1943), 652–673. · Zbl 0060.24206 |
| [2] | Jih-Hsin Cheng, Chain-preserving diffeomorphisms and CR equivalence, Proc. Amer. Math. Soc. 103 (1988), 75–80. · Zbl 0661.32024 |
| [3] | Kevin Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. (2) 135 (1992), no. 1, 165 – 182. · Zbl 0768.53025 · doi:10.2307/2946567 |
| [4] | J.Y. Chen and S.Y. Li, On the holomorphic extension of maps from the boundary of Kähler manfolds, preprint, 1994. |
| [5] | Mikhail Gromov and Richard Schoen, Harmonic maps into singular spaces and \?-adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 165 – 246. · Zbl 0896.58024 |
| [6] | Richard S. Hamilton, Harmonic maps of manifolds with boundary, Lecture Notes in Mathematics, Vol. 471, Springer-Verlag, Berlin-New York, 1975. · Zbl 0308.35003 |
| [7] | Howard Jacobowitz, Chains in CR geometry, J. Differential Geom. 21 (1985), no. 2, 163 – 194. · Zbl 0581.32022 |
| [8] | Howard Jacobowitz, An introduction to CR structures, Mathematical Surveys and Monographs, vol. 32, American Mathematical Society, Providence, RI, 1990. · Zbl 0712.32001 |
| [9] | Jürgen Jost and Shing-Tung Yau, A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry, Acta Math. 170 (1993), no. 2, 221 – 254. · Zbl 0806.53064 · doi:10.1007/BF02392786 |
| [10] | ——, The strong rigidity of locally symetric complex manifolds of rank one and finite volume, Math. Ann. 275 (1986), 291–304. · Zbl 0583.32053 |
| [11] | ——, On the rigidity of certain discrete groups and algebriac varieties, Math. Ann. 278 (1987), 481–496. · Zbl 0624.32019 |
| [12] | Ngaiming Mok, The holomorphic or antiholomorphic character of harmonic maps into irreducible compact quotients of polydiscs, Math. Ann. 272 (1985), no. 2, 197 – 216. · Zbl 0604.58021 · doi:10.1007/BF01450565 |
| [13] | ——, Strong rigidity of irreducible quotients of polydisks of finite volume, Math. Ann. 282 (1988), 555–578. |
| [14] | Seiki Nishikawa and Kiyoshi Shiga, On the holomorphic equivalence of bounded domains in complete Kähler manifolds of nonpositive curvature, J. Math. Soc. Japan 35 (1983), no. 2, 273 – 278. · Zbl 0509.53050 · doi:10.2969/jmsj/03520273 |
| [15] | R. Remmert, Holomorphie und meromorphe Abbildungen komplexer Räume, Math. Ann. 133 (1957), 328–370. · Zbl 0079.10201 |
| [16] | Yum Tong Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. (2) 112 (1980), no. 1, 73 – 111. · Zbl 0517.53058 · doi:10.2307/1971321 |
| [17] | Yum Tong Siu, Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems, J. Differential Geom. 17 (1982), no. 1, 55 – 138. · Zbl 0497.32025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
