Details
| Original language | English |
|---|---|
| Pages (from-to) | 405-442 |
| Number of pages | 38 |
| Journal | Journal of differential geometry |
| Volume | 53 |
| Issue number | 3 |
| Publication status | Published - 1999 |
| Externally published | Yes |
Abstract
We use the Green function of the Yamabe operator (conformai Laplacian) to construct a canonical metric on each locally conformally flat manifold different from the standard sphere that supports a Riemannian metric of positive scalar curvature. In dimension 3, the assumption of local conformai flatness is not needed. The construction depends on the positive mass theorem of Schoen-Yau. The resulting metric is different from those obtained earlier by other methods. In particular, it is smooth and distance nondecreasing under conformai maps. We analyze the behavior of our metric if the scalar curvature tends to 0. We demonstrate that the canonical metrics converge under surgery-type degenerations to the corresponding metric on the limit space. As a consequence, the L2—metric on the moduli space of scalar positive locally conformally flat structures is not complete. The example of S1 × S2 as underlying manifold is studied in detail.
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Algebra and Number Theory
- Mathematics(all)
- Geometry and Topology
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In: Journal of differential geometry, Vol. 53, No. 3, 1999, p. 405-442.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Greenfunctions and conformal geometry
AU - Habermann, Lutz
AU - Jost, Jürgen
PY - 1999
Y1 - 1999
N2 - We use the Green function of the Yamabe operator (conformai Laplacian) to construct a canonical metric on each locally conformally flat manifold different from the standard sphere that supports a Riemannian metric of positive scalar curvature. In dimension 3, the assumption of local conformai flatness is not needed. The construction depends on the positive mass theorem of Schoen-Yau. The resulting metric is different from those obtained earlier by other methods. In particular, it is smooth and distance nondecreasing under conformai maps. We analyze the behavior of our metric if the scalar curvature tends to 0. We demonstrate that the canonical metrics converge under surgery-type degenerations to the corresponding metric on the limit space. As a consequence, the L2—metric on the moduli space of scalar positive locally conformally flat structures is not complete. The example of S1 × S2 as underlying manifold is studied in detail.
AB - We use the Green function of the Yamabe operator (conformai Laplacian) to construct a canonical metric on each locally conformally flat manifold different from the standard sphere that supports a Riemannian metric of positive scalar curvature. In dimension 3, the assumption of local conformai flatness is not needed. The construction depends on the positive mass theorem of Schoen-Yau. The resulting metric is different from those obtained earlier by other methods. In particular, it is smooth and distance nondecreasing under conformai maps. We analyze the behavior of our metric if the scalar curvature tends to 0. We demonstrate that the canonical metrics converge under surgery-type degenerations to the corresponding metric on the limit space. As a consequence, the L2—metric on the moduli space of scalar positive locally conformally flat structures is not complete. The example of S1 × S2 as underlying manifold is studied in detail.
UR - http://www.scopus.com/inward/record.url?scp=0038853057&partnerID=8YFLogxK
U2 - 10.4310/jdg/1214425634
DO - 10.4310/jdg/1214425634
M3 - Article
AN - SCOPUS:0038853057
VL - 53
SP - 405
EP - 442
JO - Journal of differential geometry
JF - Journal of differential geometry
SN - 0022-040X
IS - 3
ER -