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University of Cambridge > Talks.cam > Partial Differential Equations seminar > A sharp strong maximum principle for singular minimal hypersurfaces
A sharp strong maximum principle for singular minimal hypersurfacesAdd to your list(s) Download to your calendar using vCal
If you have a question about this talk, please contact Filip Rindler. 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). This talk is part of the Partial Differential Equations seminar series. This talk is included in these lists:
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