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Moduli spaces of singular Yamabe metrics. (English) Zbl 0849.58012
This paper considers the problem of determining the moduli space of complete, conformally flat metrics of constant positive scalar curvature on the complement of $$k$$ points in $$S^n$$. A Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds is developed. It is shown that the moduli space is locally a real analytic variety of formal dimension $$k$$: this is obtained by writing the space as the intersection of a Fredholm pair of real analytic manifolds.

##### MSC:
 58D27 Moduli problems for differential geometric structures
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##### References:
 [1] Lars Andersson, Piotr T. Chruściel, and Helmut Friedrich, On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations, Comm. Math. Phys. 149 (1992), no. 3, 587 – 612. · Zbl 0764.53027 [2] Patricio Aviles and Robert C. McOwen, Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds, Duke Math. J. 56 (1988), no. 2, 395 – 398. · Zbl 0645.53023 [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 Math. Z. (to appear). · Zbl 0878.49026 [5] Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271 – 297. · Zbl 0702.35085 [6] C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constant, J. Math. Pure Appl. 6 (1841), 309-320. [7] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801 [8] David L. Finn and Robert C. McOwen, Singularities and asymptotics for the equation \Delta _{\?}\?-\?^{\?}=\?\?, Indiana Univ. Math. J. 42 (1993), no. 4, 1487 – 1523. · Zbl 0791.35010 [9] Arthur E. Fischer and Jerrold E. Marsden, Deformations of the scalar curvature, Duke Math. J. 42 (1975), no. 3, 519 – 547. · Zbl 0336.53032 [10] Arthur E. Fischer and Jerrold E. Marsden, Linearization stability of nonlinear partial differential equations, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 219 – 263. · Zbl 0274.58003 [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. · JFM 45.0479.01 [12] —, Further studies of Emden’s and similar differential equations, Quart. J. Math. Oxford Series 2 (1931), 259–287. · Zbl 0003.23502 [13] Nicolaos Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131 (1990), no. 2, 239 – 330. , https://doi.org/10.2307/1971494 Nicolaos Kapouleas, Compact constant mean curvature surfaces in Euclidean three-space, J. Differential Geom. 33 (1991), no. 3, 683 – 715. · Zbl 0699.53007 [14] Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. · Zbl 0342.47009 [15] Osamu Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan 34 (1982), no. 4, 665 – 675. · Zbl 0486.53034 [16] Nick Korevaar and Rob Kusner, The global structure of constant mean curvature surfaces, Invent. Math. 114 (1993), no. 2, 311 – 332. · Zbl 0803.53040 [17] Nicholas J. Korevaar, Rob Kusner, William H. Meeks III, and Bruce Solomon, Constant mean curvature surfaces in hyperbolic space, Amer. J. Math. 114 (1992), no. 1, 1 – 43. · Zbl 0757.53032 [18] Nicholas J. Korevaar, Rob Kusner, and Bruce Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465 – 503. · Zbl 0726.53007 [19] R. Kusner, R. Mazzeo and D. Pollack, The moduli space of complete embedded constant mean curvature surfaces, Geom. and Functional Analysis (to appear). · Zbl 0966.58005 [20] Jacques Lafontaine, Sur la géométrie d’une généralisation de l’équation différentielle d’Obata, J. Math. Pures Appl. (9) 62 (1983), no. 1, 63 – 72 (French). · Zbl 0513.53046 [21] André Lichnerowicz, Propagateurs et commutateurs en relativité générale, Inst. Hautes Études Sci. Publ. Math. 10 (1961), 56 (French). · Zbl 0098.42607 [22] Charles Loewner and Louis Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 245 – 272. · Zbl 0298.35018 [23] Rafe Mazzeo, Regularity for the singular Yamabe problem, Indiana Univ. Math. J. 40 (1991), no. 4, 1277 – 1299. · Zbl 0770.53032 [24] Rafe Mazzeo, Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations 16 (1991), no. 10, 1615 – 1664. · Zbl 0745.58045 [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. · Zbl 0866.58069 [27] Rafe Mazzeo and Nathan Smale, Conformally flat metrics of constant positive scalar curvature on subdomains of the sphere, J. Differential Geom. 34 (1991), no. 3, 581 – 621. · Zbl 0759.53029 [28] Robert C. McOwen, Prescribed curvature and singularities of conformal metrics on Riemann surfaces, J. Math. Anal. Appl. 177 (1993), no. 1, 287 – 298. · Zbl 0806.53040 [29] R. Melrose, The Atiyah-Patodi-Singer index theorem, AK Peters Ltd., Wellesley, MA, 1993. CMP 95:17 · Zbl 0796.58050 [30] Morio Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333 – 340. · Zbl 0115.39302 [31] F. Pacard, The Yamabe problem on subdomains of even dimensional spheres, preprint. · Zbl 0854.53037 [32] Daniel Pollack, Nonuniqueness and high energy solutions for a conformally invariant scalar equation, Comm. Anal. Geom. 1 (1993), no. 3-4, 347 – 414. · Zbl 0848.58011 [33] Daniel Pollack, Compactness results for complete metrics of constant positive scalar curvature on subdomains of \?$$^{n}$$, Indiana Univ. Math. J. 42 (1993), no. 4, 1441 – 1456. · Zbl 0794.53025 [34] Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0242.46001 [35] Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479 – 495. · Zbl 0576.53028 [36] Richard M. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure Appl. Math. 41 (1988), no. 3, 317 – 392. · Zbl 0674.35027 [37] M. Giaquinta , Topics in calculus of variations, Lecture Notes in Mathematics, vol. 1365, Springer-Verlag, Berlin, 1989. Lectures given at the Second 1987 C.I.M.E. Session held in Montecatini Terme, July 20 – 28, 1987. · Zbl 0613.49015 [38] Blaine Lawson and Keti Tenenblat , Differential geometry, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 52, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. A Symposium in Honor of Manfredo do Carmo. · Zbl 0718.00010 [39] R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 47 – 71. · Zbl 0658.53038 [40] Clifford Henry Taubes, Gauge theory on asymptotically periodic 4-manifolds, J. Differential Geom. 25 (1987), no. 3, 363 – 430. · Zbl 0615.57009
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