Stationary Integral Varifolds near Multiplicity 2 Planes

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Allard’s regularity theorem provides regularity for stationary integral varifolds near a plane of multiplicity one: they must be smooth minimal graphs. What can be said when the plane instead has (integer) multiplicity larger than 1 has remained an open question and is crucial to the global regularity of stationary integral varifolds, as it currently prevents us concluding almost everywhere regularity. Examples such as the catenoid and complex varieties illustrate the complication. Even uniqueness of a multiplicity 2 plane as a tangent cone is not known in general, and those cases where there are results have additional variational assumptions, such as being area minimising.

In this talk, I will discuss a new regularity result for stationary integral varifolds near planes of multiplicity 2. This holds in arbitrary dimension and codimension, providing full regularity as a Lipschitz 2-valued graph (with certain improved estimates). The only additional assumption one needs is of a topological nature on the support of the varifold: no other variational assumptions are needed other than stationarity. This assumption can be directly checked in several cases of interest. I will also discuss hopes for the future. This is all based on joint work with Spencer Becker-Kahn and Neshan Wickramasekera.