Moduli spaces of singular yamabe metrics

Journal of the American Mathematical Society

Volumn 9, Issue 2, 1996, Pages 303-344

  Mazzeo, Rape ^a   Pollack, Daniel ^b   Uhlenbeck, Karen ^c  

^a STANFORD UNIVERSITY   (United States)

^b UNIVERSITY OF CHICAGO   (United States)

^c UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030526566     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0894-0347-96-00208-1     Document Type: Article

References (40)

1. L. Andersson, P.T. Chruściel, and H. Friedrich, On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal data for Einstein's Field Equations, Comm. Math. Phys. 149 (1992), 587-612. MR 93i:53040
2. P. Aviles and R. McOwen, Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds, Duke Math. Jour. 56 No. 2 (1988), 395-398. MR 89b:58224
3. P. Aviles, N. Korevaar and R. Schoen, The symmetry of constant scalar curvature metrics near point singularities, preprint.
4. K. Grosse-Brauckmann, New surfaces of constant mean curvature (to appear Math. Z.).
5. L. Caffarelli, B. Gidas and J. Spruck, Asymptotic Symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271-297. MR 90c:35075
6. C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constant, J. Math. Pure Appl. 6 (1841), 309-320.
7. H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969. MR 41:1976
8. q = Su, Indiana Univ. Math. J. 42 (1993), 1487-1523. MR 95g:58216
9. A. Fischer and J. Marsden, Deformations of the scalar curvature, Duke Math. J. 42 (1975), 519-547. MR 52:1804
10. _, Linearization stability of nonlinear partial differential equations, Proceedings of Symposia in Pure Math., vol. 27, 1975, pp. 219-263. MR 52:4337
11. R. H. Fowler, The form near infinity of real continuous solutions of a certain differential equation of the second order, Quart. J. Pure Appl. Math. 45 (1914), 289-349.
12. _, Further studies of Emden's and similar differential equations, Quart. J. Math. Oxford Series 2 (1931), 259-287.
13. N. Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. 131 (1990), 239-330. MR 93a:53007a
14. T. Kato, Perturbation theory for linear operators, second ed., Springer-Verlag, Berlin, 1976. MR 53:11389
15. O. Kobayashi, A differential equation arising from the scalar curvature function, J. Math. Soc. Japan 34 (1982), 665-675. MR 84a:53046
16. N. Korevaar and R. Kusner, The global structure of constant mean curvature surfaces, Invent. Math. 114 (1993), 311-332. MR 95f:53015
17. _,_, W. Meeks III, and B. Solomon, Constant mean curvature surfaces in hyperbolic space, Amer. J. of Math. 114 (1992), 1-43. MR 92k:53116
18. _, _ and B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geometry 30 (1989), 465-503. MR 90g:53011
19. R. Kusner, R. Mazzeo and D. Pollack, The moduli space of complete embedded constant mean curvature surfaces, Geom. and Functional Analysis (to appear).
20. J. Lafontaine, Sur la géométrie d'une généralisation de l'équation différentielle d'Obata, J. Math. pures et appl. 62 (1983), 63-72. MR 84i:53047
21. A. Lichnerowicz, Propagateurs et commutateurs en relativité générale, Inst. Hautes Études Sci. Publ. Math. No. 10 (1961). MR 28:967
22. C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contributions to Analysis, Academic Press, New York, 1974, pp. 245-272. MR 50:10543
23. R. Mazzeo, Regularity for the singular Yamabe equation, Indiana Univ. Math. J. 40 (1991), 1277-1299. MR 92k:53071
24. _, Elliptic theory of differential edge operators I, Comm. in P.D.E. 16 (1991), 1615-1664. MR 93d:58152
25. _ and F. Pacard, A new construction of singular solutions for a semilinear elliptic equation, To appear, J. Differential Geometry.
26. _, D. Pollack and K. Uhlenbeck, Connected sum constructions for constant scalar curvature metrics, Preprint.
27. _ and N. Smale, Conformally flat metrics of constant positive scalar curvature on subdomains of the sphere, J. Differential Geometry 34 (1991), 581-621. MR 92i:53034
28. R. McOwen, Singularities and the conformal scalar curvature equation, Geometric Analysis and Nonlinear PDE (I. Bakelman, eds.), Marcel Decker, 1993. MR 94f:53076
29. R. Melrose, The Atiyah-Patodi-Singer index theorem, AK Peters Ltd., Wellesley, MA, 1993. CMP 95:17
30. M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333-340. MR 25:5479
31. F. Pacard, The Yamabe problem on subdomains of even dimensional spheres, preprint.
32. D. Pollack, Nonuniqueness and high energy solutions for a conformally invariant scalar equation, Comm. Anal. and Geom. 1 (1993), 347-414. MR 94m:58051
33. n, Indiana Univ. Math. J. 42 (1993), 1441-1456. MR 95c:53052
34. M. Reed and B. Simon, Methods of modern mathematical physics, vol. IV, Academic Press, 1978. MR 58:12429c
35. R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geometry 20 (1984), 479-495. MR 86i:58137
36. _, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure and Appl. Math. XLI (1988), 317-392. MR 89e:58119
37. _, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations; Lecture Notes in Mathematics # 1365 (M. Giaquinta, eds.), Springer-Verlag, 1989, pp. 120-154. MR 89k:58051
38. _, On the number of constant scalar curvature metrics in a conformal class, Differential Geometry: A symposium in honor of Manfredo Do Carmo (H.B. Lawson and K. Tenenblat, eds.), Wiley, 1991, pp. 311-320. MR 93b:53001
39. _ and S.T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), 47-71. MR 89c:58139
40. C. Taubes, Gauge theory on asymptotically periodic 4-manifolds, J. Differential Geometry 25 (1987), 363-430. MR 88g:58176