Uniqueness of certain cylindrical tangent cones

Abstract

Abstract We show that the cylindrical tangent cone $C\times \mathbf{R}$ C × R for an area-minimizing hypersurface is unique, where $C$ C is the Simons cone $C_{S}= C(S^{3}\times S^{3})$ C S = C ( S 3 × S 3 ) . Previously Simon proved a uniqueness result for cylindrical tangent cones that applies to a large class of cones $C$ C , however not to the Simons cone. The main new difficulty is that the cylindrical cone $C_{S}\times \mathbf{R}$ C S × R is not integrable, and we need to develop a suitable replacement for Simon’s infinite dimensional Łojasiewicz inequality in the setting of tangent cones with non-isolated singularities.