Celebratio Mathematica — Uhlenbeck

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Works connected to Rafe Roys Mazzeo

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R. Mazzeo, D. Pollack, and K. Uhlenbeck: “Connected sum constructions for constant scalar curvature metrics,” Topol. Methods Nonlinear Anal. 6 : 2 (1995), pp. 207–233. Dedicated to Louis Nirenberg on the occasion of his 70th birthday. MR 1399537 Zbl 0866.58069 article

We give a general procedure for gluing together possibly noncompact manifolds of constant scalar curvature which satisfy an extra nondegeneracy hypothesis. Our aim is to provide a simple paradigm for making “analytic” connected sums. In particular, we can easily construct complete metrics of constant positive scalar curvature on the complement of certain configurations of an even number of points on the sphere, which is a special case of Schoen’s [1988] well-known, difficult construction. Applications of this construction produces metrics with prescribed asymptotics. In particular, we produce metrics with cylindrical ends, the simplest type of asymptotic behaviour. Solutions on the complement of an infinite number of points are also constructed by an iteration of our construction.

Louis Nirenberg	Related
Daniel Pollack	Related
Rafe Roys Mazzeo	Related

R. Mazzeo, D. Pollack, and K. Uhlenbeck: “Moduli spaces of singular Yamabe metrics,” J. Am. Math. Soc. 9 : 2 (1996), pp. 303–344. MR 1356375 Zbl 0849.58012 article

Complete, conformally flat metrics of constant positive scalar curvature on the complement of \( k \) points in the \( n \)-sphere, \( k \ge 2 \), \( n \ge 3 \), were constructed by R. Schoen in 1988. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally real analytic variety of dimension \( k \). For a generic set of nearby conformal classes the moduli space is shown to be a \( k \)-dimensional real analytic manifold. The structure as a real analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinite dimensional real analytic manifolds.

Daniel Pollack	Related
Rafe Roys Mazzeo	Related