Gluing and moduli for noncompact geometric problems.

*(English)*Zbl 0976.53065
Bourguignon, Jean-Pierre (ed.) et al., Geometric theory of singular phenomena in partial differential equations. Proceedings of the workshop, Cortona, Italy, May 15-19, 1995. Cambridge: Cambridge University Press. Symp. Math. 38, 17-51 (1998).

From the introduction: We survey a number of recent results concerning the existence and moduli spaces of solutions of various geometric problems on noncompact manifolds. The three problems which we discuss in detail are:

I. Complete properly immersed minimal surfaces in \(\mathbb{R}^3\) with finite total curvature.

II. Complete embedded surfaces of constant mean curvature in \(\mathbb{R}^3\) with finite topology.

III. Complete conformal metrics of constant positive scalar curvature on \(M^n\setminus\Lambda\) where \(M^n\) is a compact Riemannian manifold, \(n\geq 3\) and \(\Lambda\subset M\) is closed.

The existence results we discuss for each of these problems are ones whereby known solutions (sometimes satisfying certain nondegeneracy hypotheses) are glued together to produce new solutions. Although this sort of procedure is quite well-known, there have been some recent advances on which we wish to report here. We also discuss what has been established about the moduli spaces of all solutions to these problems, and report on some work in progress concerning global aspects of these moduli spaces. In the final section, we present a new compactness result for the ‘unmarked moduli spaces’ for problem III.

For the entire collection see [Zbl 0893.00031].

I. Complete properly immersed minimal surfaces in \(\mathbb{R}^3\) with finite total curvature.

II. Complete embedded surfaces of constant mean curvature in \(\mathbb{R}^3\) with finite topology.

III. Complete conformal metrics of constant positive scalar curvature on \(M^n\setminus\Lambda\) where \(M^n\) is a compact Riemannian manifold, \(n\geq 3\) and \(\Lambda\subset M\) is closed.

The existence results we discuss for each of these problems are ones whereby known solutions (sometimes satisfying certain nondegeneracy hypotheses) are glued together to produce new solutions. Although this sort of procedure is quite well-known, there have been some recent advances on which we wish to report here. We also discuss what has been established about the moduli spaces of all solutions to these problems, and report on some work in progress concerning global aspects of these moduli spaces. In the final section, we present a new compactness result for the ‘unmarked moduli spaces’ for problem III.

For the entire collection see [Zbl 0893.00031].

##### MSC:

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

58D27 | Moduli problems for differential geometric structures |

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |