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The zero scalar curvature Yamabe problem on noncompact manifolds with boundary. (English) Zbl 1116.53026

Let \((M^n,g)\), \(n\geq 3\) be a complete Riemannian manifold with boundary \(\partial M\). The prescribed mean curvature problem consists in finding a complete scalar flat metric on \(M\) conformal to \(g\) whose mean curvature on \(\partial M\) is a prescribed smooth function \(f\). J. F. Escobar and G. Garcia [J. Funct. Anal. 211, 71–152 (2004; Zbl 1056.53026)] solved this problem for the Euclidean balls if \(f\) is a Morse function that satisfies certain Morse inequalities. In the present paper the author investigates this problem for a certain class of noncompact Riemannian manifolds with boundary. He shows that the prescribed mean curvature problem can be solved for any smooth function on the boundary. This should be seen in contrast to the situation of Euclidean balls where it is known that a solution to the problem is not possible in general, i.e., that some conditions must be imposed on the smooth function on the boundary. The specific conditions on \((M,g)\) in this paper are that \(M\) is noncompact and scalar flat, \(\partial M\) is compact and the mean curvature on \(\partial M\) is positive. Moreover, \((M,g)\) is assumed to have finitely many ends, each of which is large, i.e., \(\int_1^{\infty} t/V_E(t) \,dt\) is finite where \(V(t)\) is the volume of the intersection of a geodesic ball with radius \(t\) and the end. The author shows that such noncompact manifolds \((M,g)\) can be obtained from removing small submanifolds from compact scalar flat manifolds with positive mean curvature on the boundary.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35J60 Nonlinear elliptic equations
53A30 Conformal differential geometry (MSC2010)

Citations:

Zbl 1056.53026
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