Abstract

If two smooth, connected, embedded minimal hypersurfaces with no singularities satisfy the property that locally near every common point $p$, one hypersurface lies on one side of the other, then it is a direct consequence of the Hopf maximum principle that either the hypersurfaces are disjoint or they coincide. Given that singularities in minimal hypersurfaces are generally unavoidable, it is a natural question to ask if the same result must extend to pairs of singular minimal hypersurfaces (stationary codimesion 1 integral varifolds) with connected supports; in this case the above ``one hypersurface lies locally on one side of the other’’ hypothesis can naturally be imposed for each common point $p$ which is a regular point of at least one hypersurface.

The answer to this question in general is no in view of simple examples such as two pairs of transversely interecting hyperplanes with a common axis. The answer however is yes if the singular set of one of the hypersurfaces has $(n-1)$-dimesional Hausdorff measure zero, where $n$ is the dimension of the hypersurfaces. I will discuss this result, which generalizes and unifies previous maximum principles of Ilmanen and Solomon-White (and thereby unifies all previously known strong maximum principles for singular minimal hypersurfaces).