×

zbMATH — the first resource for mathematics

Harmonic maps of V-manifolds. (English) Zbl 0679.58014
The theory of harmonic maps of Riemannian manifolds is generalized to the case of V-manifolds. By using Sampson’s method, we first obtain Rellich’s theorem and Sobolev’s theorem on a compact V-manifold. With these two fundamental theorems, we then construct the Green’s function and the heat kernel. By all the preceding results established, we finally prove the existence of a harmonic map from a V-manifold into a Riemannian manifold with non-positive curvature. Harmonic maps of V-manifolds are also discussed.
Reviewer: Y.-J.Chiang

MSC:
58E20 Harmonic maps, etc.
58J99 Partial differential equations on manifolds; differential operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Baily, W. L.: The decomposition theorem for V-manifolds. Amer. J. Math. 76 (1965), 862-888. · Zbl 0173.22705
[2] Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété Riemannienne. Springer Lecture Notes Math. 194 (1971). · Zbl 0223.53034
[3] Brown, A. B., Koopman, B. C.: On the covering of analytical loci by complexes. Trans. Amer. Math. Soc. 34 (1932), 231-251. · Zbl 0004.13203 · doi:10.1090/S0002-9947-1932-1501636-9
[4] Yuan-Jen Chiang, Thesis, The Johns Hopkins University, 1989.
[5] Eells, J., Sampson, J. H.: Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964), 109-160. · Zbl 0122.40102 · doi:10.2307/2373037
[6] Friedman, A.: Partial differential equations of parabolic type. Prentice-Hall Inc. 1964. · Zbl 0144.34903
[7] Hartman, P.: On homotopic harmonic harmonic maps. Canadian J. Math. 19 (1967), 673-687. · Zbl 0148.42404 · doi:10.4153/CJM-1967-062-6
[8] Hildebrandt, S.: Harmonic Mappings and Minimal Immersions. C.I.M.E. Corso Estivo on Harmonic Mappings and Minimal Immersions, Montecatini 1984. Springer Lecture Notes Math. 1161 (1985), p. 1-117. · doi:10.1007/BFb0075136
[9] Hildebrandt, S., Kaul, H., Widman, K. O.: An existence theorem for harmonic maps of riemannian manifolds. Acta Math. 138 (1977), 1-16. · Zbl 0356.53015 · doi:10.1007/BF02392311
[10] Laugwitz, K.: Differential and Riemannian Geometry. Academic Press, New York 1965. · Zbl 0139.38903
[11] Lojasiewicz, S.: Triangulation of semi-analytic sets. Scuola Norm. Sup. Pisa (3) 18 (1964), 449-474.
[12] Lefschetz, S., Whitehead, J. H. C.: On analytical complexes. Trans. Amer. Math. Soc. 35 (1933), 510-517. · Zbl 0006.37006 · doi:10.1090/S0002-9947-1933-1501698-X
[13] Minakshisundaram, S.: A generalization of Epstein Zeta functions. Canadian Math. 1 (1949), 320-329. · Zbl 0034.05103 · doi:10.4153/CJM-1949-029-3
[14] Pogorzelski, W.: Proprietes des integrales de l’equation parabolique normale. Ann. polon. Math. 4 (1957), 61-92. · Zbl 0080.30602
[15] Riesz, F., Sz.-Nagy, B.: Vorlesungen über Funktionalanalysis. VEB Deutscher Verlag der Wissenschaften, Berlin 1956. · Zbl 0072.11902
[16] Sampson, J. H.: Cours de topologie algebraique. Department de Mathématiques, rue René Descartes, Strasbourg, 1969.
[17] Sampson, J. H.: Applications of harmonic maps tähler geometry. Contemp. Math. 49 (1986), 125-134. · Zbl 0605.58019
[18] Satake, I.: On a generalization of the notion of manifold. Proc. Nat. Acad. Sci. USA 42 (1956), 359-363. · Zbl 0074.18103 · doi:10.1073/pnas.42.6.359
[19] Satake, I.: The Gauss-Bonnet theorem for V-manifolds. J. Math. Soc. Japan 9, 4 (1957), 464-492. · Zbl 0080.37403 · doi:10.2969/jmsj/00940464
[20] Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Diff. Geom. 17 (1982), 307-435. · Zbl 0521.58021
[21] Schoen, R., Uhlenbeck, K.: Boundary regularity and the Dirichlet problem for harmonic maps. J. Diff. Geom. 18 (1983), 253-268. · Zbl 0547.58020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.