Positive scalar curvature and the Dirac operator on complete Riemannian manifolds.

*(English)*Zbl 0538.53047This long and very rich article extends to complete non compact Riemannian manifolds a wealth of results concerning metrics with positive scalar curvature obtained by the authors using the Dirac operator in the compact case. It will certainly be of long lasting interest both because of the depth of the results it contains, and of the efforts made by the authors to present them in a readable manner. Among the notions that the authors have used successfully in the compact situation is that of an enlargeable manifold. They extend it here to the non compact case. This allows them to improve some of their results even in the compact situation.

One part of the article uses Dirac operator techniques. It contains an extension to non compact manifolds of the Atiyah-Singer index theorem for the Dirac operator by means of a relative index theorem on \(L^ 2\)- sections for manifolds whose metrics agree outside a compact set. Among the far-reaching consequences of this theorem to the subject under study, we mention two of them: ”any compact manifold which carries a metric of non positive sectional curvature cannot carry a metric with positive scalar curvatue”; ”any complete hyperbolic manifold with finite volume cannot carry a complete metric with positive scalar curvature”.

A very careful analysis of what happens at infinity allows them also to prove that if M is an enlargeable manifold, then \(M\times {\mathbb{R}}\) cannot have positive scalar curvature, \(M\times {\mathbb{R}}^ 2\) cannot have uniformly positive scalar curvature, but \(M\times {\mathbb{R}}^ 3\) can have uniformly positive curvature. (Recall that M is said to be enlargeable if, for any Riemannian metric on M and any \(\epsilon>0\), one can find an \(\epsilon\)-contracting map from a spin covering of M in a sphere which is constant outside a compact set and with at least one regular level set having non zero \(\hat A\)-genus.)

The other part of the paper involves minimal hypersurfaces. The key to their use in problems dealing with the scalar curvature originates in an expression of the second variation formula of the area for hypersurfaces in a 3-manifold due to R. Schoen and S.-T. Yau. By this technique, the authors show for example that on a complete oriented Riemannian n- manifold (\(n\leq 7)\) with scalar curvature \(s\leq s_ 0<0\), any map of non-zero degree onto the n-sphere is at most \(c_ ns_ 0\)-contracting where \(c_ n\) is a universal constant.

To conclude, we reproduce the table of contents of the article in order to give a better overview of its scope: 0. A glimpse at the main results, 1. Generalized Dirac operators on a complete manifold, 2. Some vanishing theorems, 3. Estimates for the dimension of the kernel and cokernel, 4. The relative index theorem, 5. Hyperspherical and enlargeable manifolds, 6. Manifolds which admit no complete metrics of positive scalar curvature, 7. Manifolds which admit no complete metrics of uniformly positive scalar curvature (introduction of some fundamental techniques), 8. Results for 3-manifolds, 9. Results for 4-manifolds, 10. Some fundamental theorems for incomplete 3-manifolds, 11. Applications to the topology of minimal hypersurfaces, 12. Theorems for incomplete manifolds of dimension \(\leq 7\), 13. Generalizations to manifolds which represent non-trivial homology classes in spaces of non-positive curvature.

One part of the article uses Dirac operator techniques. It contains an extension to non compact manifolds of the Atiyah-Singer index theorem for the Dirac operator by means of a relative index theorem on \(L^ 2\)- sections for manifolds whose metrics agree outside a compact set. Among the far-reaching consequences of this theorem to the subject under study, we mention two of them: ”any compact manifold which carries a metric of non positive sectional curvature cannot carry a metric with positive scalar curvatue”; ”any complete hyperbolic manifold with finite volume cannot carry a complete metric with positive scalar curvature”.

A very careful analysis of what happens at infinity allows them also to prove that if M is an enlargeable manifold, then \(M\times {\mathbb{R}}\) cannot have positive scalar curvature, \(M\times {\mathbb{R}}^ 2\) cannot have uniformly positive scalar curvature, but \(M\times {\mathbb{R}}^ 3\) can have uniformly positive curvature. (Recall that M is said to be enlargeable if, for any Riemannian metric on M and any \(\epsilon>0\), one can find an \(\epsilon\)-contracting map from a spin covering of M in a sphere which is constant outside a compact set and with at least one regular level set having non zero \(\hat A\)-genus.)

The other part of the paper involves minimal hypersurfaces. The key to their use in problems dealing with the scalar curvature originates in an expression of the second variation formula of the area for hypersurfaces in a 3-manifold due to R. Schoen and S.-T. Yau. By this technique, the authors show for example that on a complete oriented Riemannian n- manifold (\(n\leq 7)\) with scalar curvature \(s\leq s_ 0<0\), any map of non-zero degree onto the n-sphere is at most \(c_ ns_ 0\)-contracting where \(c_ n\) is a universal constant.

To conclude, we reproduce the table of contents of the article in order to give a better overview of its scope: 0. A glimpse at the main results, 1. Generalized Dirac operators on a complete manifold, 2. Some vanishing theorems, 3. Estimates for the dimension of the kernel and cokernel, 4. The relative index theorem, 5. Hyperspherical and enlargeable manifolds, 6. Manifolds which admit no complete metrics of positive scalar curvature, 7. Manifolds which admit no complete metrics of uniformly positive scalar curvature (introduction of some fundamental techniques), 8. Results for 3-manifolds, 9. Results for 4-manifolds, 10. Some fundamental theorems for incomplete 3-manifolds, 11. Applications to the topology of minimal hypersurfaces, 12. Theorems for incomplete manifolds of dimension \(\leq 7\), 13. Generalizations to manifolds which represent non-trivial homology classes in spaces of non-positive curvature.

Reviewer: J.P.Bourguignon

##### MSC:

53C20 | Global Riemannian geometry, including pinching |

53C27 | Spin and Spin\({}^c\) geometry |

58J20 | Index theory and related fixed-point theorems on manifolds |

##### Keywords:

positive scalar curvature; Dirac operator; enlargeable manifold; Atiyah- Singer index theorem; sectional curvature; hyperbolic manifold; minimal hypersurfaces; vanishing theorems; homology classes
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\textit{M. Gromov} and \textit{H. B. Lawson jun.}, Publ. Math., Inst. Hautes Étud. Sci. 58, 83--196 (1983; Zbl 0538.53047)

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