Abstract
We show that harmonic maps from 2-dimensional Euclidean polyhedra to arbitrary NPC spaces are totally geodesic or constant depending on a geometric and combinatorial condition of the links of the 0-dimensional skeleton. Our method is based on a monotonicity formula rather than a codimension estimate of the singular set as developed by Gromov–Schoen or the mollification technique of Korevaar–Schoen.
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References
Alexandrov A.D.: Ruled surfaces in metric spaces. Vestnik LGU Ser. Math. Mech. Astron. vyp. 1, 5–26 (1957)
Bridson M., Haefliger A.: Metric Spaces of Non-positive Curvature. Springer, Berlin (1991)
Chen J.: On energy minimizing mappings between and into singular spaces. Duke Math. J. 79, 77–99 (1995)
Daskalopoulos G., Mese C.: Harmonic maps from a 2-complex. Commun. Anal. Geom. 14, 497–549 (2006)
Daskalopoulos G., Mese C.: Harmonic maps from a simplicial complex and geometric rigidity. J. Differ. Geom. 79, 277–334 (2008)
Eells J., Fuglede B.: Harmonic Maps between Riemannian Polyhedra. Cambridge Tracts in Mathematics 142. Cambridge University Press, Cambridge (2001)
Gromov R., Schoen M.: Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Publ. Math. IHES. 76, 165–246 (1992)
Izeki H., Nayatani S.: Combinatorial harmonic maps and discrete-group actions on Hadamard spaces. Geom. Dedic. 114, 147–188 (2005)
Korevaar N., Schoen R.: Sobolev spaces and harmonic maps into metric space targets. Commun. Anal. Geom. 1, 561–659 (1993)
Korevaar N., Schoen R.: Global existence theorem for harmonic maps to non-locally compact spaces. Commun. Anal. Geom. 5, 333–387 (1997)
Petrunin A.: Metric minimizing surfaces. Electron. Res. Announc. Am. Math. Soc. 5, 47–54 (1999)
Reshetnyak Y.G.: Non-expansive maps in a space of curvature no greater than K. Sib. Math. J. 9, 683–689 (1968)
Wang M.-T.: A fixed point theorem of discrete group actions on Riemannian manifolds. J. Differ. Geom. 50, 249–267 (1998)
Wang M.-T.: Generalized harmonic maps and representations of discrete groups. Commun. Anal. Geom. 8, 545–563 (2000)
Wolpert, S.: Behaviour of Geodesic-length Functions on Teichmüller Space. Preprint (2007)
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Daskalopoulos, G., Mese, C. Fixed point and rigidity theorems for harmonic maps into NPC spaces. Geom Dedicata 141, 33–57 (2009). https://doi.org/10.1007/s10711-008-9342-1
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DOI: https://doi.org/10.1007/s10711-008-9342-1