Classification of Stable Currents in the Product of Spheres

Shihshu Walter Wei

Tamkang Journal of Mathematics

42卷4期（2011 / 12 / 01）

427 - 440

英文

In [11] we constructed the first set of examples of area-minimizing hypersurfaces with isolated singularities in various high dimensional locally symmetric spaces of both compact type and non-compact type. These are not the minimizing cones in Euclidean spaces R(superscript n) constructed by Bombieri, de Giorgi and Giusti [2] and later by Lawson [6].Among our first known examples in various locally symmetric spaces, the ones in the product of hyperbolic spaces H(superscript n+1)×H(superscript n+1) are complete, while the examples in the product of Euclidean spheres S(superscript n+1)×S (superscript n+1) have boundaries, where n ＞ 2. From the analytic viewpoint, this is due to the fact that the non-linear elliptic partial differential inequalities involved in S(superscript n+1)×S (superscript n+1) unlike the ones in the non-compact type case, have only local solutions.This striking contrast leads immediately to the conjecture that no closed stable rectifiable (2n +1)-current (or roughly speaking, no closed stable minimal hypersurface with singularities) exists in S(superscript n+1)×S(superscript n+1) , and hence motivates a general study in S(superscript p)×S(superscript q). A stable rectifiable current (resp. stable minimal submanifold) is a stationary or minimal rectifiable current (resp. minimal submanifold) that has no mass (resp. area) decreasing variations. An extension of Synge lemma due to Simons [9] which states that there are no closed stable hypersurfaces in a Riemannian manifold with positive Ricci curvature supports a piece of evidence of the conjecture.In this paper, we give a positive answer to the conjecture and classify the stable currents in the product of Euclidean spheres S(superscript p)×S(superscript q).