In this paper, we prove the Lipschitz regularity of continuous harmonic maps from an finite dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in [36]. The proof extends the argument of Huang-Wang [26].

Let$C$and$C$′ be two smooth self-transverse immersions of$S$^{1}into ℝ^{2}. Both$C$and$C$′ subdivide the plane into a number of disks and one unbounded component. An isotopy of the plane which takes$C$to$C$′ induces a one-to-one correspondence between the disks of$C$and$C$′. An obvious necessary condition for there to exist an area-preserving isotopy of the plane taking$C$to$C$′ is that there exists an isotopy for which the area of every disk of$C$equals that of the corresponding disk of$C$′. In this paper we show that this is also a sufficient condition.

Introduction. The classical Schwarz-Pick lemma states that any holomorphic map of the unit disk into itself decreases the Poincare metric. Later Ahlfors generalized this lemma to holomorphic mappings between two Riemann surfaces where curvatures of these Riemann surfaces were used in a very explicit way. More recently, Chern initiated the study of holomorphic mappings between higher-dimensional complex manifold by generalizing the Ahlfors lemma to these spaces. Then this lemma was further extended by Kobayashi, Griffiths, Wu and others. It plays a very important role in their theory. In this note, we shall prove the following generalization of the Schwarz lemma.

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In this paper, we shall discuss the existence, uniqueness and regularity of harmonic maps from an Alexandrov space into a geodesic space with curvature 􏰢 1 in the sense of Alexandrov.