| The main theorem in this thesis is a rigidity theorem for hypercones in I&d1; , the varifold closure of n-dimensional, smooth, stable, minimal hypersurfaces immersed in Rn+1 . This theorem asserts that if the vertex density of a cone in I&d1; is greater than or equal to 2 and sufficiently close to 2, and if the cone is sufficiently close, in a weak sense, to a pair of hyperplanes (two transeverse hyperplanes or a single hyperplane with multiplicity 2), then it must be a pair of hyperplanes. The proof of the theorem involves a two-valued blow up procedure together with a dimension reducing argument using a frequency function, followed by a second blow-up.; We apply this rigidity theorem to study the set S of singularities of a varifold in I&d1; where the density is not much greater than 2. Two is the minimum density at points of self intersection of a smooth hypersurface. Thus, understanding the nature of singularities with density not much greater than 2 is a first step in the study of compactness properties of immersed, stable hypersurfaces. We prove that if a tangent cone at a point in S is a pair of hyperplanes, then so is every other tangent cone there, and that except for points in a low dimensional subset, every point in S has tangent cones equal to pairs of hyperplanes. Furthermore, we show that the set of singularities where the tangent cones are pairs of hyperplanes is relatively open in the closed set of points of density greater than or equal to 2. Whenever a tangent cone at a point is equal to a transverse pair of hyperplanes, more is true; namely, locally near that point, the varifold is the union of two smooth manifolds intersecting transversely along a smooth (n - 1)-dimensional submanifold. In particular, every singularity near that point has a unique tangent cone equal to a transverse pair of hyperplanes. |
