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Extra info for Journal of Differential Geometry-Volume 12

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1. The basic construction and the results In n-dimensional Riemannian space M, n > 2 (of regularity class C') we consider a connected region which has a compact closure T and is bounded by a nonempty, possibly disconnected regular hypersurface F (of class C'). The surface P divides a sufficiently small ball neighborhood of any of its points into two components ; we suppose that only one of them belongs to T. ) Let the boundary F of the region T be strictly convex in the following sense : all the normal curvatures of F on the side of the interior normal are not less than some positive x.

3. If x < 0 or ks < -/C2, then in such a region infinitely long geodesic can exist. 4. The requirement of regularity of class C' is explained by the references in [4]. It could be reduced to regularity of class C2 by means of a suitable approximation. Later on we will remark only on the fact that certain objects are regular without mentioning the class. 5. L. is a strictly increasing function of. the arguments x and X and a strictly decreasing function of the arguments x and k, in the region >0, ks> -K 2, 0

Then by our previous reasoning the same is true for all \v / with x in a neighborhood I x - xb ll < e and v unre- stricted. Exactly the same argument applies to each TkP°o. We would be done, except that the neighborhood may shrink to a point as k --f oo, and we need to show that all the TkP°O are continuous and tame on some fixed neighborhood of xb. This follows by a slightly more complicated reasoning. z) is continuous and tame for all x e V° and all Lemma. TkPo(x, v, w, v, w, , z without restriction.

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