Impossible Metric Conditions on Exotic R’s

There are many theorems in the differential geometry literature of the following sort. Let M be a complete Riemannian manifold with some conditions on various curvatures, diameters, volumes, etc. Then M is homotopy equivalent to a finite CW complex, or M is the interior of a compact, topological manifold with boundary. At first glance it seems unlikely that such theorems have anything to say about smooth manifolds homeomorphic to R. However, there is a common theme to all the proofs which forbids the existence of such metrics on most (and possibly all) exotic R’s. 1. Definitions and the main result We say a smooth manifold E homeomorphic to R satisfies the DFT condition (for De Michelis, Freedman and Taubes) provided that, for every compact subset K ⊂ E, there exists an open neighborhood U of K such that (1) K ⊂ U (2) the closure of U , U , is homeomorphic to D (3) U can be engulfed by itself rel K Precisely, condition (3) means that there exists a smooth, ambient, isotopy of E from the identity to ι such that ι : U → E satisfies U ⊂ ι(U) and the isotopy is the identity on K. As discussed in section 2, all exotic R’s (smoothings of R not diffeomorphic to the standard smoothing) known to the author do not have the DFT -property. Differential geometry enters the picture via critical points of functions related to the distance function. Such functions are not necessarily smooth so the notion of critical point needs to be interpreted and there is a standard way to do this going back at least to Grove and Shiohama [8]. The idea involves constructing vector fields and using differential geometry to get enough similarity to the gradient-like vector fields of Morse theory to prove results. Here is the main observation of this note. Theorem 1.1. Let E be a smooth, manifold homeomorphic to R with a proper Lipschitz function which has bounded critical values. Then E satisfies the DFT -condition. Partially supported by the N.S.F.