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Connected sum constructions for constant scalar curvature metrics. (English) Zbl 0866.58069
The paper is a very illuminating survey of some of the authors results concerning the Yamabe problem on noncompact manifolds.
The noncompactness of the background manifold, on top of the critical Sobolev exponents, makes this problem virtually intractable by variational methods. Instead, the authors use the grafting technique pioneered by C. H. Taubes in gauge theory.
This technique relies on a refined version of the inverse function theorem. It states that in order to find a solution of the (functional) equation \(F(x)=0\) it suffices to find a “nondegenerate approximate solution”, that is, an \(x_0\) such that \(F(x_0)\) is very close to zero and the differential of \(F\) at \(x_0\) is invertible (with some control on the norm of the inverse).
The grafting technique has as input two solutions \(x_1\), \(x_2\) of a functional equation \(F\) on two manifolds \(M_1\) and \(M_2\). Then, in many concrete situations, one can “connect sum” these two solutions to obtain an approximate solution \(x_1 \# x_2\) on the connected sum \(M_1 \# M_2\). In this survey the equation is \(F(x)=0\) is the Yamabe problem and \(M_1\) and \(M_2\) are two complete Riemannian manifolds with constant scalar curvature. A careful connect sum produces a metric on \(M_1 \# M_2\) which is constant away from the gluing region and measurable close to being constant on this region.
The most delicate part is the nondegeneracy issue, and the reviewer believes this is the most important conceptual contribution of the authors. It is an unusual variation of the notion of nondegeneracy suggested and imposed by the geometry of the situation. There are two reasons behind this subtle notion of nondegeneracy. The most important one is the noncompactness of the background manifold, which renders the usual \(L^2\)-spectral theory essentially useless. This couples in an intricate way with the nonlinear features of this equation. The authors do a splendid job introducing and motivating their concept of nondegeneracy and almost half the survey is devoted to this issue.
The heart of the proof of their gluing theorem consists in establishing the nondegeneracy of the approximate solution on the connected sum. This requires additional technical ingenuity. To conclude, the reviewer believes this survey serves as an invaluable guide for anyone interested in these new advances in global geometry and should find a place on the desk of any geometer analyst.

MSC:
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58D27 Moduli problems for differential geometric structures
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35J60 Nonlinear elliptic equations
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