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Riemannian manifolds of faster-than-quadratic curvature decay. (English) Zbl 0833.53037
Some new results concerning noncompact, complete Riemannian manifolds with sectional curvature going to zero sufficiently rapidly at infinity are presented in this announcement. These results are aspects of the general viewpoint that, in some sense, the geometry at infinity of noncompact Riemannian manifolds is controlled by the curvature behaviour at infinity. Their main theorem says in effect that the results on manifolds which are flat outside some compact set [see R. E. Greene and H. Wu, Invent. Math. 27, 265-298 (1974; Zbl 0342.31003) and Duke Math. J. 49, 731-756 (1982; Zbl 0513.53045)] still hold under the weaker assumption that the sectional curvature of \(M\) goes to zero rapidly enough as the distance from a chosen point increases. Some consequences of this theorem are pointed out. For example, if \(M\) is a complete, noncompact Riemannian manifold of faster than quartic curvature decay and with Ricci curvature everywhere nonnegative, and if one end of \(M\) is simply connected, then \(M\) is isometric on \(\mathbb{R}^n\), with \(n= \dim(M)\). This result supplements previous “gap theorems” [see R. Greene and H. Wu, Duke Math. J. (loc. cit.); G. Drees, Differ. Geom. Appl. 4, No. 1, 77-90 (1994; Zbl 0796.53041)] both under the more general curvature hypotheses which allow some negative sectional curvatures and in the absence of the dimension restrictions \(n\neq 4,8\). Remarks on proof techniques and relationships to various results are also given in a historical perspective.

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