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Pacific Journal of Mathematics Vol. 248 (2010) Pacific Journal of Mathematics Volume 248 No. 1 November 2010 PACIFIC JOURNAL OF MATHEMATICS http://www.pjmath.org Founded in 1951 by E. F. Beckenbach (1906–1982) and F. Wolf (1904–1989) EDITORS V. S. Varadarajan (Managing Editor) Department of Mathematics University of California Los Angeles, CA 90095-1555 pacifi[email protected] Vyjayanthi Chari Darren Long Sorin Popa Department of Mathematics Department of Mathematics Department of Mathematics University of California University of California University of California Riverside, CA 92521-0135 Santa Barbara, CA 93106-3080 Los Angeles, CA 90095-1555 [email protected] [email protected] [email protected] Robert Finn Jiang-Hua Lu Jie Qing Department of Mathematics Department of Mathematics Department of Mathematics Stanford University The University of Hong Kong University of California Stanford, CA 94305-2125 Pokfulam Rd., Hong Kong Santa Cruz, CA 95064 fi[email protected] [email protected] [email protected] Kefeng Liu Alexander Merkurjev Jonathan Rogawski Department of Mathematics Department of Mathematics Department of Mathematics University of California University of California University of California Los Angeles, CA 90095-1555 Los Angeles, CA 90095-1555 Los Angeles, CA 90095-1555 [email protected] [email protected] [email protected] PRODUCTION pacifi[email protected] Silvio Levy, Scientific Editor Matthew Cargo, Senior Production Editor SUPPORTING INSTITUTIONS ACADEMIA SINICA, TAIPEI STANFORD UNIVERSITY UNIV. OF CALIF., SANTA CRUZ CALIFORNIA INST. OF TECHNOLOGY UNIV. OF BRITISH COLUMBIA UNIV. OF MONTANA INST. DE MATEMÁTICA PURA E APLICADA UNIV. OF CALIFORNIA, BERKELEY UNIV. OF OREGON KEIO UNIVERSITY UNIV. OF CALIFORNIA, DAVIS UNIV. OF SOUTHERN CALIFORNIA MATH. SCIENCES RESEARCH INSTITUTE UNIV. OF CALIFORNIA, LOS ANGELES UNIV. OF UTAH NEW MEXICO STATE UNIV. UNIV. OF CALIFORNIA, RIVERSIDE UNIV. OF WASHINGTON OREGON STATE UNIV. UNIV. OF CALIFORNIA, SAN DIEGO WASHINGTON STATE UNIVERSITY UNIV. OF CALIF., SANTA BARBARA These supporting institutions contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its contents or policies. See inside back cover or www.pjmath.org for submission instructions. The subscription price for 2010 is US $420/year for the electronic version, and $485/year for print and electronic. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163, U.S.A. Prior back issues are obtainable from Periodicals Service Company, 11 Main Street, Germantown, NY 12526-5635. The Pacific Journal of Mathematics is indexed by Mathematical Reviews, Zentralblatt MATH, PASCAL CNRS Index, Referativnyi Zhurnal, Current Mathematical Publications and the Science Citation Index. The Pacific Journal of Mathematics (ISSN 0030-8730) at the University of California, c/o Department of Mathematics, 969 Evans Hall, Berkeley, CA 94720-3840, is published monthly except July and August. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. POSTMASTER: send address changes to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163. PJM peer review and production are managed by EditFLOW™ from Mathematical Sciences Publishers. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS at the University of California, Berkeley 94720-3840 A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2010 by Pacific Journal of Mathematics PACIFIC JOURNAL OF MATHEMATICS Vol. 248, No. 1, 2010 AN EXISTENCE THEOREM OF CONFORMAL SCALAR-FLAT METRICS ON MANIFOLDS WITH BOUNDARY SÉRGIO DE MOURA ALMARAZ Let .M; g/ be a compact Riemannian manifold with boundary. We address the Yamabe-type problem of finding a conformal scalar-flat metric on M whose boundary is a constant mean curvature hypersurface. When the boundary is umbilic, we prove an existence theorem that finishes some of the remaining cases of this problem. 1. Introduction J. Escobar [1992a] has studied the following Yamabe-type problem for manifolds with boundary: Yamabe problem. Let .Mn; g/ be a compact Riemannian manifold of dimension n ≥ 3 with boundary @ M. Is there a scalar-flat metric on M that is conformal to g and has @ M as a constant mean curvature hypersurface? In dimension two, the classical Riemann mapping theorem says that any simply connected, proper domain of the plane is conformally diffeomorphic to a disk. This theorem is false in higher dimensions since the only bounded open subsets of Rn for n ≥ 3 that are conformally diffeomorphic to Euclidean balls are the Euclidean balls themselves. The Yamabe-type problem proposed by Escobar can be viewed as an extension of the Riemann mapping theorem for higher dimensions. In analytical terms, this problem corresponds to finding a positive solution to Lgu D 0 in M; (1-1) n=.n−2/ Bgu C K u D 0 on @ M D − 1 − − for some constant K , where Lg 1g 4 .n 2/=.n 1/Rg is the conformal D − 1 − Laplacian and Bg @=@η 2 .n 2/hg. Here 1g is the Laplace–Beltrami operator, Rg is the scalar curvature, hg is the mean curvature of @ M and η is the inward unit normal vector to @ M. MSC2000: primary 53C25; secondary 35J65. Keywords: Yamabe problem, manifold with boundary, scalar curvature, mean curvature, Weyl tensor. The author was supported at IMPA by CNPq-Brazil. 1 2 SÉRGIODEMOURAALMARAZ The solutions of the equations (1-1) are the critical points of the functional Z n−2 n−2 Z jr uj2 C R u2dv C h u2dσ g 4.n−1/ g g 2 g g Q u D M @ M . / Z .n−2/=.n−1/ ; 2 u .n − 1/=.n − 2/dσg @ M where dvg and dσg denote the volume forms of M and @ M, respectively. Escobar introduced the conformally invariant Sobolev quotient Q.M;@ M/ D inffQ.u/ V u 2 C1.M/; u 6≡ 0 on @ Mg and proved that it satisfies Q.M;@ M/ ≤ Q.Bn;@ B/. Here, Bn denotes the unit ball in Rn endowed with the Euclidean metric. Under the hypothesis that Q.M;@ M/ is finite (which is the case when Rg ≥ 0), he also showed that the strict inequality (1-2) Q.M;@ M/ < Q.Bn;@ B/ implies the existence of a minimizing solution of the equations (1-1). Notation. We denote by .Mn; g/ a compact Riemannian manifold of dimension n ≥ 3 with boundary @ M and finite Sobolev quotient Q.M;@ M/. Theorem 1.1 [Escobar 1992a]. Assume that one of the following conditions holds: (1) n ≥ 6 and M has a nonumbilic point on @ M; (2) n ≥ 6, M is locally conformally flat and @ M is umbilic; (3) n D 4 or 5 and @ M is umbilic; (4) n D 3. Then Q.M;@ M/ < Q.Bn;@ B/ and there is a minimizing solution to (1-1). The proof for n D 6 under condition (1) appeared later, in [Escobar 1996b]. Further existence results were obtained by F. Marques in [Marques 2005] and [Marques 2007]. Together, these results can be stated as follows: Theorem 1.2 [Marques 2005; Marques 2007]. Assume that one of the following conditions holds: (1) n ≥ 8, W.x/ 6D 0 for some x 2 @ M and @ M is umbilic; (2) n ≥ 9, W.x/ 6D 0 for some x 2 @ M and @ M is umbilic; (3) n D 4 or 5 and @ M is not umbilic. Then Q.M;@ M/ < Q.Bn;@ B/ and there is a minimizing solution to (1-1). AN EXISTENCE THEOREM OF CONFORMAL SCALAR-FLAT METRICS 3 Here, W denotes the Weyl tensor of M and W the Weyl tensor of @ M. Our main result deals with the remaining dimensions n D 6; 7 and 8 when the boundary is umbilic and W 6D 0 at some boundary point: Theorem 1.3. Suppose that n D6; 7 or 8, that @ M is umbilic and that W.x/6D0 for some x 2 @ M. Then Q.M;@ M/ < Q.Bn;@ B/ and there is a minimizing solution to the equations (1-1). These cases are similar to the case of dimensions 4 and 5 when the boundary is not umbilic, studied in [Marques 2007]. Other works concerning conformal deformation on manifolds with boundary include [Ahmedou 2003; Ambrosetti et al. 2002; Ben Ayed et al. 2005; Brendle 2002; Djadli et al. 2003; 2004; Escobar 1992b; 1996a; Escobar and Garcia 2004; Felli and Ould Ahmedou 2003; 2005; Han and Li 1999; 2000] We will now discuss the strategy in the proof of Theorem 1.3. We assume that @ M is umbilic and choose x0 2 @ M so that W.x0/ 6D 0. Our proof is explicitly based on constructing a test function such that (1-3) Q. / < Q.Bn;@ B/: The function has support in a small half-ball around the point x0. The usual strategy in this kind of problem (which goes back to [Aubin 1976]) is to define the function in the small half-ball as one of the standard entire solutions to the corresponding Euclidean equations. In our context those are − .n 2/=2 (1-4) U .x/ D ; 2 C· · ·C 2 C C 2 x1 xn−1 ( xn/ where x D .x1;:::; xn/ and xn ≥ 0. The next step would be to expand the quotient of in powers of and, by exploiting the local geometry around x0, show that the inequality (1-3) holds if is small. To simplify the asymptotic analysis, we use conformal Fermi coordinates centered at x0. This concept, introduced in [Marques 2005], plays the same role that conformal normal coordinates (see [Lee and Parker 1987]) did in the case of manifolds without boundary. When n ≥ 9, the strict inequality (1-3) was proved in [Marques 2005]. The difficulty arises because, when 3 ≤ n ≤ 8, the first correction term in the expansion does not have the right sign.
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