Image structures of harmonic maps into singular spaces

In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space (X, dX) with curvature bounded above by a constant κ (κ ⩾ 0) in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng (1980) and Choi (1982) to harmonic maps into singular spaces.

Harmonic maps into singular spaces and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -adic superrigidity for lattices in groups of rank one

The last five decades have witnessed many developments in the theory of harmonic maps. To become acquainted to some of these, the reader is referred to two reports and a survey paper by Eells and Lemaire [119, 122, 124] about the developments of harmonic maps up to 1988 for details. Several books on harmonic maps [203, 205, 206, 389, 425] are also available. In this chapter, we follow the notions and notations of harmonic maps between Riemannian manifolds by Eells- Sampson [129] in the introduction.We discuss the crucial topics in harmonic maps including fundamentals, regularity, maps of surfaces, maps of KRahler manifolds, maps into groups and Grassmannians, harmonic maps, loop groups, and integrable systems, harmonicmorphisms, maps of singular spaces, and transversally harmonic maps. Since the theory of harmonic maps has been developed over half a century, it is impossible to provide full details. However, we try to present the most important components of the topics.

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