Nonlinear gluing methods in geometry
with applications to the singular Yamabe problem
and constant mean curvature surfaces

These researchers have collaborated for the past three and a half years, and have previously written three papers together.

Their work to date concerns the conformal scalar curvature equation, which is sometimes known as the Yamabe problem, and specifically the analysis of singular solutions of this equation. More precisely, the Yamabe problem involves the search for canonical metrics in a conformal class on a (compact) Riemannian manifold. Here canonical is interpreted as having constant scalar curvature. This problem can be analyzed by studying a certain semilinear elliptic partial differential equation


In this equation, tex2html_wrap_inline13 is some number depending only on the dimension n of the manifold M, and tex2html_wrap_inline19 and R(g) are, respectively, the scalar curvature functions of the initial metric tex2html_wrap_inline23 and the new conformally related metric g. The point is to find a positive function u solving this equation, where R(g) is a constant. It can therefore be regarded as some sort of nonlinear eigenvalue problem. When M is compact, this equation can be studied using variational methods, and the final, successful resolution of this problem, by R. Schoen, in 1984, was a landmark in geometric analysis.

In what is called the singular Yamabe problem, one seeks a conformally related metric g only on the complement of some prescribed closed set tex2html_wrap_inline35 and requires both that g have constant scalar curvature and also that it be complete on tex2html_wrap_inline39 . This may be translated to seeking a solution of the equation above which blows up at some rate at tex2html_wrap_inline41 . The hardest case of this problem is when the specified constant R(g) is positive. The problem becomes one of knowing for which sets tex2html_wrap_inline41 is it possible to solve the problem. Then solutions were known only in some special cases. The best work had been done by R. Schoen in 1985, where (in particular) it was shown that tex2html_wrap_inline41 could be any finite set of points. R. Mazzeo and F. Pacard individually had studied this positive case and obtained solutions in some other cases. In their first collaborative effort, they proved the existence of solutions when tex2html_wrap_inline41 is any finite disjoint union of closed submanifolds of positive dimension (subject to the previously known constraint that there is an upper bound on the allowable dimension). They developed a number of new scaling techniques for this. In their next paper they extended their result so that tex2html_wrap_inline41 was also allowed to have components of dimension zero. Although this was covered by R. Schoen's earlier work, their method showed that certain degeneracies of the problem that R. Schoen had had to work around are in fact illusory; the result was a shorter and simpler existence proof in that case. After this, and also in collaboration with N. Korevaar and R. Schoen, they studied the fine asymptotics of (locally-defined) solutions in the neighbourhood of isolated singularities. (R. Mazzeo had previously analyzed the asymptotics near higher dimensional singularities). The rougher asymptotics had been proven in 1988 by L. A. Caffarelli, B. Gidas and J. Spruck, and also by N. Korevaar and R. Schoen (although the latter proof was never published). This new paper contained that proof of N. Korevaar and R. Schoen, which is a much simpler and more geometric proof of these cruder asymptotics, then extended this and gave a number of geometric applications, in particular to the moduli space of all solutions of the problem.

In their continuing work, R. Mazzeo and F. Pacard are now proving the existence of complete surfaces in three-dimensional Euclidean space which have constant mean curvature. Many examples of these surfaces had been constructed by N. Kapouleas in 1986, but the proof there, which is related to the analogous work of R. Schoen's on isolated singularities, also involves working around some apparent degeneracies in the problem. In this new approach, again these degeneracies are shown to be essentially illusory. In fact, a quite different new method for this sort of problem is introduced. The earlier method, here and in other so-called nonlinear gluing problems, is to build a one-parameter family of approximate solutions from known `components' which already solve the problem exactly on simpler domains. These approximate solutions usually have some sort of degenerating geometry. One then shows that at least for some members of this family, there is a perturbation to an exact solution of the problem. This final step involves a careful study of the linear analysis on the elements of this degenerating family. In the new approach one again has simpler component pieces which must be patched together. Now one constructs all possible constant mean curvature surfaces near to each of these component pieces, and then tries to match these up along their common boundaries. Since it is necessary to match these boundaries up not only continuously, but also with their first derivatives, this involves matching the Cauchy data of solutions on the components. The Cauchy data for all constant mean curvature surfaces on each component is essentially an infinite dimensional nonlinear Lagrangian submanifold of some fixed symplectic Hilbert space. The matching problem becomes one of finding intersection points of these various Lagrangian submanifolds. Such Lagrangian intersection problems have been the subject of intensive study at least in finite dimensions. In any case, it may be shown that such an intersection does exist, and hence that the appropriate complete constant mean curvature surfaces do exist. This paper is now nearing completion.

This Lagrangian intersection technique appears to have many further applications. Amongst these are applications to various connected sum constructions. Here one seeks again to construct various complicated geometric objects, which typically may be described (at least locally) as solutions of some nonlinear elliptic equations, by building them out of simpler constituent components. Now however these components are `bridged' together by a small `neck'. Such bridge constructions for minimal and constant mean curvature surfaces with boundary have been known for at least the past decade when the bridge is constructed at boundary points of the component surfaces. Recently an interior bridge construction was completed by S. D. Yang in the minimal surface case, but the proof is quite technical and seems to introduce a number of degeneracies which are not necessarily present. R. Mazzeo and F. Pacard, in collaboration with D. Pollack, intend to use this Lagrangian intersection technique to prove a connected sum theorem in a number of cases, including higher dimensional minimal hypersurfaces and constant mean curvature surfaces. This paper is meant to be partly expository, exhibiting the flexibility and utility of this new method.

In other work, there remains a large set of open problems in the theory of area-minimizing hypersurfaces concerning the existence of such objects with what are called `line singularities.' These hypersurfaces are smooth submanifolds except along a very small set, the properties of which in general are still quite poorly understood. Only very limited and special examples of this phenomenon are known too. In any case, the goal of this research is to construct area-minimizing hypersurfaces, the singular sets of which are not just of the trivial product type. This is a severely over-determined problem, and numerous efforts to understand it, by R. Mazzeo, F. Pacard and others, have failed thus far. What these researchers hope to do is study some simpler, related problems, where the nonlinearity is less severe, but which still exhibit the same over-determined character. The hope is that they will be able to make some progress in the construction of solutions in these other problems, and that this will shed some new light on the problem for area-minimizing hypersurfaces. One possible approach, which might at least give the existence of solutions in a number of cases, is to find some new version of the Baouendi-Goulaouic extension, from the mid 1970's, of the classical Cauchy-Kovalevsky theorem for the existence of solutions of quite general analytic partial differential equations. This project is expected to be quite difficult, and any progress at all would represent a real breakthrough.