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Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary. (English) Zbl 0766.53033

Let \((M^ n,g)\) be a compact Riemannian manifold with boundary. Does there exist a conformally related metric \(g'=u^{2\alpha} g\) on \(M\) (where \(\alpha=2/(n-2)\)) such that (1) \((M,g')\) has zero scalar curvature, (2) \(\partial M\) has constant mean curvature with respect to \(g'\)? The special case where \(M\) is a bounded domain with smooth boundary in \(\mathbb{R}^ n\) with its induced metric \(g\) can be considered as an \(n\)- dimensional analogue of the Riemann mapping theorem. In the present paper, an affirmative answer is given under one of the following assumptions on \((M,g)\): a) \(n=3\), b) \(n=4\) or 5, and \(\partial M\) is totally umbilic, c) \(n\geq 6\), with \(M\) conformally flat and \(\partial M\) umbilic, d) \(n\geq 7\), and \(\partial M\) is not totally umbilic.
In particular, the answer in the above special case is yes for \(n=3\) and \(n\geq 7\). The differential equations for the function \(u\) arising from (1) and (2) are the Euler-Lagrange equations of the functionals \(Q(u)=(\int_ M u\cdot Lu+\int_{\partial M} hu^ 2/\alpha)/(\int_{\partial M} u^{2\beta})^{1/\beta}\) with \(\beta=(n- 1)/(n-2)\), where \(L=\Delta-R/(4\beta)\) denotes the conformal Laplacian (\(R\) being the scalar curvature) and \(h\) the mean curvature of \(\partial M\). We look for a minimizer of \(Q\). However, this functional does not satisfy the Palais-Smale condition, since the exponent \(2\beta\) is critical for the Sobolev trace embedding \(H^ 1(M)\to L^{2\beta}(\partial M)\), i.e. the restriction map \(H^ 1(M)\to L^{2\beta'}(\partial M)\) is compact for any \(\beta'<\beta\), but no longer for \(\beta\) itself [cf. R. A. Adams, Sobolev spaces (1978; Zbl 0314.46030), p. 114]. Therefore, instead of \(Q\) the functional \(Q'\) is minimized where \(\beta\) is replaced with some \(\beta'<\beta\), and then the limit \(\beta'\to\beta\) is considered.
It is shown in Chapter 2 that the limit exists and is a smooth minimizer of \(Q\) provided that the Sobolev quotient \(Q(M,\partial M):=\inf Q\) (which is a conformal invariant, cf. the author [Indiana Univ. Math. J. 37, No. 3, 687-698 (1988; Zbl 0666.35014) and Differential geometry. A symposium in honor of Manfredo do Carmo, Proc. Inf. Conf., Rio de Janeiro/Bras. 1988, Pitman Monogr. Surv. Pure Appl. Math. 52, 171-177 (1991; Zbl 0733.53022)]) is strictly smaller than its counterpart on the Euclidean unit ball \(B\subset \mathbb{R}^ n\). By transplanting the Euclidean minimizer onto the manifold \(M\), it is easy to show that \(Q(M,\partial M)\leq Q(B,\partial B)\), but the strict inequality is needed to compensate for the error \(| Q'-Q|\). The remainder of the paper is devoted to prove this inequality in the various cases a)–d) which are very different. E.g. for the cases a) and c), a positive mass theorem [the author, J. Differ. Geom. 35, 21-84 (1992)] is used.

MSC:

53C20 Global Riemannian geometry, including pinching
58E30 Variational principles in infinite-dimensional spaces
58J32 Boundary value problems on manifolds
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