Sobolev spaces and harmonic maps for metric space targets

Abstract

vanced Study for its support. This paper was completed while they were visitors, during the academic year 1992-93. We thank Bruce Kleiner for several helpful discussions concerning the geometry of metric spaces.Note added in proof. After this paper was written we received a preprint from J. Jost in which he also obtains some existence results for harmonic maps to (NPC) spaces. CONTENTS N. J. KOREVAAR AND R. M. SCHOEN1.5. The energy-density measure 1.6. Lower semicontinuity, and consistency when X = M 581 1.7. Directional energies 1.8. The calculus of directional energies 1.9. Differentiability theory for directional energies 591 1.10. Absolute continuity of de for p > 1 601 1.11. The calculus of energy-density functions 1.12. Trace theory for Lipschitz domains 605 1.13. Precompactness 612 2. Harmonic Maps into Non-Positively Curved Metric Spaces 2.1. Non-positively curved metric spaces 616 2.2. The solution to the Dirichlet Problem 622 2.3. The pull-back inner product TT 2.4. Geodesic homotopies and interior Lipschitz continuity 628 2.5. Center of mass constructions 639 2.6. Equivariant mapping problems 643 2.7. Homotopy problems 656 References 658

A Counterexample to Metric Differentiability

“…We point out that if G = R n , then any Lipschitz map is metrically differentiable almost everywhere, as proved in [1], [9] and [10]. Furthermore, in [16] it is shown that bi-Lipschitz maps are almost everywhere metric differentiable on stratified groups if one allows the direction v to vary only among the elements of V 1 , namely the horizontal directions.…”

“…When the target is another Euclidean space, by Rademacher's differentiability theorem, many of the classical properties of smooth surfaces can be extended to rectifiable sets (see [4] or [14] for a complete presentation of the subject). Recently, some properties of rectifiable sets as the existence almost everywhere of tangent spaces, the regularity of their Hausdorff measure, area and co-area formulae have been extended to general metric spaces (see [1,9,10]). The key idea is to replace the notion of differentiability with a weaker one, namely metric differentiability, and to prove that any metric-valued Lipschitz map defined on a Euclidean space is metrically differentiable.…”

An intrinsic approach to manifold constrained variational problems

“…Related notions have been introduced in the theory of harmonic maps with metric space targets in the works of Korevaar and Schoen [26] and Jost [22,23], and in the theory of analysis on metric spaces by Heinonen et al [21] and Ohta [34]. Equivalences for all these approaches and the one adopted here have been established partially in Reshetnyak [36] and Jost [24], and fully in Chiron [6].…”

Singularities of harmonic maps

“…Motivated by important applications to algebraic geometry, most work has focused on the existence of regular (e.g. Lipschitz) solutions which usually map to nonpositively curved metric spaces [46,71,73,75,100,120]. A model for nematic liquid crystals leads to harmonic maps to a metric cone over S 2 or RP 2 [79,61] which may exhibit curve singularities.…”