On the Radial Behavior of Minimal Surfaces and the Uniqueness of their Tangent Cones

Abstract

Suppose V is an m + 1-dimensional minimal surface in R containing 0 as an isolated singular point. What can one say about the structure of V near O? This question initiated the present investigation and, in certain special cases, has a satisfactory answer. ( 1 ) If one of the tangent cones to V at 0 is of the form 0 * M where M is an m-dimensional minimal submanifold of S'-' and if also for each Jacobi normal vectorfield Z of M in Sn-1 there is a one parameter family of minimal surfaces in S'-1 having velocity Z at M then 0* M is the unique tangent cone to V at 0 with V converging to 0 * M for small radii r with rate rX', It > 0. ( 2 ) In case M is the cartesian product of two standard spheres of appropriate radii then each Jacobi vectorfield on M arises from isometric motions of Sn-1 and the Jacobi vectorfield hypothesis in (1) is satisfied. This is shown in Chapter 6. The uniqueness of tangent cones to one dimensional stationary varifolds with positive densities was shown in [AA], and the uniqueness of tangent cones to certain two dimensional area minimizing singular surfaces in R3 was shown by J. Taylor in [TJ1I, [TJ2], [TJ3]. Also B. White has shown such uniqueness for area minimizing hypersurfaces mod 4 in space dimen-