Minimal cones, plateau's problem, and the bernstein conjecture.

The work of Fleming,5 along with that of De Giorgi,2 has shown that the Bernstein conjecture for graphs in Rnf+l would follow from an interior regularity theorem for (n 1)-dimensional minimal integral currents in Rn. The work of FedererFleming,4 along with that of De Giorgi' and Triscari,7 has shown that such an interior regularity would follow from a theorem showing that the cone in Rn over an (n 2)-dimensional, nontotally geodesic, closed minimal variety in Sl"is unstable with respect to its boundary. Almgren' showed that the cone over such a 2-dimensional variety in S3 is unstable, and this yielded the interior regularity theorem in R4 and the Bernstein conjecture in R5. In this note we announce an instability theorem which is valid for the cone over any such subvariety of Sn for n < 6, and give an example of a cone over such a subvariety in S' which is locally stable in the sense that every deformation initially increases area. These results yield the Bernstein conjecture through R8, interior regularity through R7, and a good candidate for a counterexample to interior regularity in R8. We are indebted to F. J. Almgren for a number of very useful conversations on these topics. Let M denote a closed, codimension 1, oriented minimal variety immersed in S". Let CM denote the cone over M and CM, the truncated cone over M, i.e., CM = {tm t e [0,1]and m e Ml, and CM, = {tm t e [E,1]and m e M}. CM-0 and CM. are immersed minimal varieties Rt+1. THEOREM 1. IfM is not totally geodesic and if n < 6, then CM does not minimize area with respect to its boundary. Proof: Let N(m,t) denote the unit normal field on CMe. Let F(m,t) be a C function on CM. such that F(m,c) = F(m,1) = 0. Then the normal field V(m,t) = F(m,t)N(m,t) is a variation of CM, which vanishes on OCM., and every variation is of this type. Let I(VV) denote the index form of this variation, i.e., I(V,V) is the second derivative of the area of any 1-parameter family of immersed submanifolds which starts at CMX and moves away in the V direction. Let A (m) denote the second fundamental form of M at m when that variety is considered as a submanifold of Sn. For a < 0, set