## 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

### Citations:

Zbl 0314.46030; Zbl 0666.35014; Zbl 0733.53022
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