Harmonic maps into spaces with an upper curvature bound in the sense of Alexandrov
Harmonic maps into spaces with an upper curvature bound in the sense of Alexandrov
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DOI：
10.1007/s002090100372
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发表时间：
2002-05
影响因子：
0.8
通讯作者：
Chikako Mese
中科院分区：
文献类型：
--
作者：
Chikako Mese
In this paper, we study harmonic maps into metric spaces of curvature bounded from above in the sense of Alexandrov. The generalization of the classical harmonic map theory to the case when the target is a metric space of curvature bounded from above was initiated by the work of Gromov and Schoen [GS] who developed the general existence and regularity theory for harmonic maps into nonpositively curved Riemannian simplicial complexes. Korevaar and Schoen [KS1][KS2][KS3] and also Jost [J1][J2][J3][J4][J5] have further generalized the setting in which we consider harmonic map theory. These investigations have explicitly revealed the role that curvature plays in the analysis of solutions to geometric variational problems. Furthermore, the theory has proven to be useful in many applications. For example,[GS] provides a new approach in the study of p-adic representations of lattices in noncompact semisimple Lie groups. A major portion of this paper is devoted to showing that some results which hold for energy minimizing maps into spaces with nonpositive curvature are true with only an assumption of an upper curvature bound. First, we will show the existence of the order function for energy minimizing maps. For a harmonic function u, the order function measures the order with which u attains the value u (x) at x. It has been important in the study of harmonic maps into metric spaces of nonpositive curvature; Gromov and Schoen [GS] utilized the order function to show regularity of energy minimizing maps into a nonpositively curved Riemannian simplicial complex and Hardt and Lin [HL] have used it to study nematic liquid crystals. We will develop the
