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In [13] various estimates for the length of the second fundamental form of stable minimal hypersurfaces immersed in a complete Riemannian manifold, including pointwise estimates in case dimension n of the hypersurface satisfied n S 5, were obtained.There were no known examples to discount the possibility that such estimates might hold for hypersurfaces of dimension n 5 6. Here we show that, at least for properly imbedded stable minimal hypersurfaces, it is indeed possible to obtain such bounds in dimension 6. Perhaps more interestingly, we obtain a regularity theorem (Theorem 1) which is valid in arbitrary dimensions. Roughly speaking, this theorem says that whenever a stable minimal hypersurface is sufficiently close to a hyperplane, it decomposes into a finite union of graphs, which (relative to suitable coordinate axes) are graphs of functions of small Cz norm. Results of this type are familiar in case the hypersurface M is assumed to be area minimizing, or in case the area of M is assumed u priori to be close to the area of a disk (see [4], [l 11, [I], [3]).In contrast to the method of [13] which relies heavily on the identity for Laplacian of the second fundamental form (see [14]), our method uses only "lower order" computations.Using Theorem 1, together with a modification of the "dimension reducing" argument of Federer [6] and the result of J. Simons ([14], Lemma 6.1.1) concerning nonexistence of stable minimal cones in dimension less than or equal to 6, we are able to prove estimates for the singular set, and a compactness theorem. These results (given precisely in Theorems 2, 3, and Corollary 1) apply in the context of stable hypersurfaces with sufficiently small singular set. (We need to know u priori that the singular set has (n -2)-dimensional Hausdorff measure zero.)Finally, in Section 7, we discuss applications of these results to the question of existence of embedded minimal hypersurfaces in a given compact Riemannian manifold, extending recent work of Pitts [9]. In fact we show that if N is an arbitrary compact oriented Riemannian manifold, then N contains a properly embedded boundaryless minimal hypersurface M such that m-M has s-dimensional Hausdorff measure zero for each non-negative s > n

We consider the boundary-value problem for the mean curvature operator in Minkowski space, and give necessary and sufficient conditions for the existence of smooth strictly spacelike solutions. Our main results hold for non-constant mean curvature, and make no assumptions about the smoothness of the boundary or boundary data.

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