Fredholm, Hodge and Liouville theorems on noncompact manifolds

Lockhart, Robert

doi:10.1090/S0002-9947-1987-0879560-0

Fredholm, Hodge and Liouville theorems on noncompact manifolds

Author: Robert Lockhart
Journal: Trans. Amer. Math. Soc. 301 (1987), 1-35
MSC: Primary 58G30; Secondary 47B38, 58A12
DOI: https://doi.org/10.1090/S0002-9947-1987-0879560-0
MathSciNet review: 879560
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Fredholm, Liouville, Hodge, and ${L^2}$ -cohomology theorems are proved for Laplacians associated with a class of metrics defined on manifolds that have finitely many ends. The metrics are conformal to ones that are asymptotically translation invariant. They are not necessarily complete. The Fredholm results are, of necessity, with respect to weighted Sobolev spaces. Embedding and compact embedding theorems are also proved for these spaces.

[1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
[2] M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 0397797, https://doi.org/10.1017/S0305004100049410
[3] Jeff Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 5, 2103–2106. MR 530173
[4] Jeff Cheeger, On the Hodge theory of Riemannian pseudomanifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 91–146. MR 573430
[5] Jeff Cheeger, Mark Goresky, and Robert MacPherson, 𝐿²-cohomology and intersection homology of singular algebraic varieties, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 303–340. MR 645745
[6] Jeff Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), no. 4, 575–657 (1984). MR 730920
[7] Jozef Dodziuk, 𝐿² harmonic forms on rotationally symmetric Riemannian manifolds, Proc. Amer. Math. Soc. 77 (1979), no. 3, 395–400. MR 545603, https://doi.org/10.1090/S0002-9939-1979-0545603-8
[8] Pierre Grisvard, Problème de Dirichlet dans un domaine non régulier, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1615–1617 (French). MR 0352964
[9] Kunihiko Kodaira, Harmonic fields in Riemannian manifolds (generalized potential theory), Ann. of Math. (2) 50 (1949), 587–665. MR 0031148, https://doi.org/10.2307/1969552
[10] Robert B. Lockhart and Robert C. McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 3, 409–447. MR 837256
[11] Werner Müller, Spectral theory for Riemannian manifolds with cusps and a related trace formula, Math. Nachr. 111 (1983), 197–288. MR 725778, https://doi.org/10.1002/mana.19831110109
[12] G. De Rham, Variétés différentiables, Hermann, Paris, 1973.
[13] Steven Zucker, 𝐿₂ cohomology of warped products and arithmetic groups, Invent. Math. 70 (1982/83), no. 2, 169–218. MR 684171, https://doi.org/10.1007/BF01390727