Riemannian metrics of constant mass and moduli spaces of conformal structures. (English) Zbl 0964.58008
Lecture Notes in Mathematics. 1743. Berlin: Springer. xii, 116 p. (2000).
In order to study the moduli space of conformal structures on a closed manifold it is desirable to have a canonical Riemannian metric in each conformal class. These metrics then induce a metric on the moduli space itself so that its geometry can be examined. One approach to pick a representative in each conformal class of metrics consists of taking a metric of constant scalar curvature. That this is possible is the famous Yamabe problem solved by Trudinger, Aubin, and Schoen. In the case of positive scalar curvature the problem arises that even after normalizing this metric is in general not unique.
The author proposes a different choice of canonical metric in a conformally flat but scalar positive class on a closed manifold of dimension \(n \geq 3\). Pick any metric \(g\) in the given conformal class and take the Green function \(G\) of the conformal Laplacian \(L = 4 {n-1 \over n-2}\Delta + \text{ scal}\) and define the function \[ \alpha(p) := \lim_{q\to p} (G(p,q)-\text{dist}(p,q)^{2-n})^{1 \over (n-2)}. \] It turns out that \(\alpha\) is a smooth positive function unless \(M\) is the sphere with the standard structure in which case \(\alpha=0\). Moreover, \(\alpha^2\cdot g\) depends only on the conformal class of \(g\) and hence defines a canonical metric in this class. The theory of Kleinian groups is used to study this metric. It turns out that the canonical metric of the standard class on certain lens spaces does not have constant scalar curvature, hence is not given by the standard metric. This construction is used to study the moduli space of flat and scalar positive conformal structure on spherical space forms and on \(S^1\times S^2\). The moduli space is in general not complete, and to understand its completion one has to examine certain degeneration phenomena. In low dimensions \((3,4,5)\) the construction of canonical metric can be generalized to non-flat conformal structures. This is based on the notion of mass originating from relativity theory.
The book is well-written and self-contained since it provides the necessary background on analysis and conformal geometry.
The author proposes a different choice of canonical metric in a conformally flat but scalar positive class on a closed manifold of dimension \(n \geq 3\). Pick any metric \(g\) in the given conformal class and take the Green function \(G\) of the conformal Laplacian \(L = 4 {n-1 \over n-2}\Delta + \text{ scal}\) and define the function \[ \alpha(p) := \lim_{q\to p} (G(p,q)-\text{dist}(p,q)^{2-n})^{1 \over (n-2)}. \] It turns out that \(\alpha\) is a smooth positive function unless \(M\) is the sphere with the standard structure in which case \(\alpha=0\). Moreover, \(\alpha^2\cdot g\) depends only on the conformal class of \(g\) and hence defines a canonical metric in this class. The theory of Kleinian groups is used to study this metric. It turns out that the canonical metric of the standard class on certain lens spaces does not have constant scalar curvature, hence is not given by the standard metric. This construction is used to study the moduli space of flat and scalar positive conformal structure on spherical space forms and on \(S^1\times S^2\). The moduli space is in general not complete, and to understand its completion one has to examine certain degeneration phenomena. In low dimensions \((3,4,5)\) the construction of canonical metric can be generalized to non-flat conformal structures. This is based on the notion of mass originating from relativity theory.
The book is well-written and self-contained since it provides the necessary background on analysis and conformal geometry.
Reviewer: Christian Bär (Hamburg)
MSC:
| 58D27 | Moduli problems for differential geometric structures |
| 53A30 | Conformal differential geometry (MSC2010) |
| 53C20 | Global Riemannian geometry, including pinching |
