Generalized stereographic projections with prescribed scalar curvature.

*(English)*Zbl 0770.53027
Geometry and nonlinear partial differential equations, Proc. AMS Spec. Sess., Fayetteville/AR (USA) 1990, Contemp. Math. 127, 17-25 (1992).

[For the entire collection see Zbl 0744.00039.]

In this paper the author considers the following question: Given a compact Riemannian \(N\)-manifold \((M,g)\), a closed \(d\)-dimensional submanifold \(\Sigma\) and a real function \(S\) on \(M-\Sigma\) is there a complete metric on \(M-\Sigma\) that is pointwise conformal to \(g\) and has scalar curvature \(S\)? This is called the Yamabe problem on \(M-\Sigma\) in the case that \(S\) is a constant function. The case \(N=2\), \(d=0\) is treated by D. Hulin and M. Troyanov in [Prescribing curvature on open surfaces, to appear in Math. Ann.]. Only the case \(N\geq 3\) is considered here. Each metric \(g\) on \(M\) determines a Yamabe invariant \(\mu(g)\), which is a real number constructed from the conformal Laplacian of \(g\). For a metric \(g\) and a smooth positive function \(u\) on \(M\) let \(g_ u\) denote the conformal metric \(u^{N^*-2}g\), where \(N^*=2N/N- 2\). Note that \(\mu(g)=\mu(g_ u)\). The author considers smooth positive functions \(u\) on \(M-\Sigma\) which blow up sufficiently fast near \(\Sigma\) so that the metric \(g_ u\) is complete on \(M-\Sigma\). P. Aviles and R. McOwen have proved Theorem 1. There exists on \(M-\Sigma\) a complete pointwise conformal metric \(g_ u\) with scalar curvature \(S\equiv -1\) if and only if \(d>(N-2)/2\). In this article the author proves Theorem 2. If \(d\leq(N-2)/2\) and \(\mu(g)>0\), then there exists on \(M-\Sigma\) a complete pointwise conformal metric with scalar curvature \(S\equiv 0\). Let \(R_ u\) denote the scalar curvature of an arbitrary metric \(g_ u\) pointwise conformal to \(g\). The author also proves the following two results: Theorem 3 If \(\mu(g)\leq 0\), then there is no complete metric \(g_ u\) on \(M-\Sigma\) with \(R_ u\geq 0\). Theorem 4 If \(d>(N-2)/2\), then there is no complete metric \(g_ u\) on \(M-\Sigma\) with \(R_ u\geq 0\) and Ricci curvature bounded below. A conformal metric \(g_ u\) of the type that appears as a solution in Theorem 2 is called a stereographic projection of \((M,g)\) from \(\Sigma\). Theorems 3 and 4 combine to yield the following partial converse to Theorem 2: Corollary. If \(g_ u\) is a stereographic projection of \((M,g)\) from \(\Sigma\) with bounded Ricci curvature, then \(\mu(g)>0\) and \(d\leq(N-2)/2\). If \((M,g)\) and \(\Sigma\) are as in Theorem 2 and if \(g'\) is a stereographic projection of \((M,g)\) from \(\Sigma\), then any metric \(g^*\) on \(M-\Sigma\) that is quasi-isometric to \(g'\) – with scalar curvature vanishing on \(\Sigma\) is called a generalized stereographic projection. The author also obtains some results on generalized stereographic projections with prescribed scalar curvature.

In this paper the author considers the following question: Given a compact Riemannian \(N\)-manifold \((M,g)\), a closed \(d\)-dimensional submanifold \(\Sigma\) and a real function \(S\) on \(M-\Sigma\) is there a complete metric on \(M-\Sigma\) that is pointwise conformal to \(g\) and has scalar curvature \(S\)? This is called the Yamabe problem on \(M-\Sigma\) in the case that \(S\) is a constant function. The case \(N=2\), \(d=0\) is treated by D. Hulin and M. Troyanov in [Prescribing curvature on open surfaces, to appear in Math. Ann.]. Only the case \(N\geq 3\) is considered here. Each metric \(g\) on \(M\) determines a Yamabe invariant \(\mu(g)\), which is a real number constructed from the conformal Laplacian of \(g\). For a metric \(g\) and a smooth positive function \(u\) on \(M\) let \(g_ u\) denote the conformal metric \(u^{N^*-2}g\), where \(N^*=2N/N- 2\). Note that \(\mu(g)=\mu(g_ u)\). The author considers smooth positive functions \(u\) on \(M-\Sigma\) which blow up sufficiently fast near \(\Sigma\) so that the metric \(g_ u\) is complete on \(M-\Sigma\). P. Aviles and R. McOwen have proved Theorem 1. There exists on \(M-\Sigma\) a complete pointwise conformal metric \(g_ u\) with scalar curvature \(S\equiv -1\) if and only if \(d>(N-2)/2\). In this article the author proves Theorem 2. If \(d\leq(N-2)/2\) and \(\mu(g)>0\), then there exists on \(M-\Sigma\) a complete pointwise conformal metric with scalar curvature \(S\equiv 0\). Let \(R_ u\) denote the scalar curvature of an arbitrary metric \(g_ u\) pointwise conformal to \(g\). The author also proves the following two results: Theorem 3 If \(\mu(g)\leq 0\), then there is no complete metric \(g_ u\) on \(M-\Sigma\) with \(R_ u\geq 0\). Theorem 4 If \(d>(N-2)/2\), then there is no complete metric \(g_ u\) on \(M-\Sigma\) with \(R_ u\geq 0\) and Ricci curvature bounded below. A conformal metric \(g_ u\) of the type that appears as a solution in Theorem 2 is called a stereographic projection of \((M,g)\) from \(\Sigma\). Theorems 3 and 4 combine to yield the following partial converse to Theorem 2: Corollary. If \(g_ u\) is a stereographic projection of \((M,g)\) from \(\Sigma\) with bounded Ricci curvature, then \(\mu(g)>0\) and \(d\leq(N-2)/2\). If \((M,g)\) and \(\Sigma\) are as in Theorem 2 and if \(g'\) is a stereographic projection of \((M,g)\) from \(\Sigma\), then any metric \(g^*\) on \(M-\Sigma\) that is quasi-isometric to \(g'\) – with scalar curvature vanishing on \(\Sigma\) is called a generalized stereographic projection. The author also obtains some results on generalized stereographic projections with prescribed scalar curvature.

Reviewer: P.Eberlein (Chapel Hill)