Negatively Curved Manifolds with Exotic Smooth Structures

Let M denote a compact real hyperbolic manifold with dimensionm 2: 5 and sectional curvature K = I , and let 1: be an exotic sphere ofdimension m. Given any small number t5 > 0 , we show that there is a finitecovering space M of M satisfying the following properties: the connectedsum M#1: is not diffeomorphic to M, but it is homeomorphic to M; M#1:supports a Riemannian metric having all of its sectional curvature values in theinterval [-I 0, -I + 0]. Thus, there are compact Riemannian manifoldsof strictly negative sectional curvature which are not diffeomorphic but whosefundamental groups are isomorphic. This answers Problem 12 of the list com-piled by Yau [22]; i.e., it gives counterexamples to the Lawson-Yau conjecture.Note that Mostow's Rigidity Theorem [17] implies thatM#1: does not supporta Riemannian metric whose sectional curvature is identically -I. (In fact, itis not diffeomorphic to any locally symmetric space.) Thus, the manifold M#1:supports a Riemannian metric with sectional curvature arbitrarily close to -I ,but it does not support a Riemannian metric whose sectional curvature is iden-tically -I . More complicated examples of manifolds satisfying the propertiesof the previous sentence were first constructed by Gromov and Thurston [II]. DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY, NEW YORK, NEW YORK 10027 DEPARTMENT OF MATHEMATICS, SUNY AT STONY BROOK, STONY BROOK, NEW YORK 11794 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use