Analysis Seminar

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Stationary varifolds near multiplicity 2 planes

Speaker: Neshan Wickramasekera, Cambridge

Location: Warren Weaver Hall 1314

Date: Friday, October 25, 2019, 1:30 p.m.

Synopsis:

Consider a stationary integral $n$-varifold in an open ball in $R^{n+k}$ lying close to a plane of integer multiplicity $q$. The celebrated Allard regularity theorem says that if $q=1$ then $V$ is an embedded submanifold in the interior. This implies that the (relatively open) set $\Omega$ of points (of any stationary integral varifold) with mass density $<2$ is fairly regular, i.e. an embedded submanifold away from a closed set of Hausdorff dimension at most $(n-1)$. It is a long standing open question what one can say about $V$ when $q \geq 2$. We will discuss some on going work (joint with Spencer Becker-Kahn) that considers this question when $q=2$. The work gives a necessary and sufficient toplogical condition on the region $\Omega$ under which, in the interior: (a) $V$ is a Lipschitz 2-valued graph with small Lipschitz constant and (b) all tangent cones to $V$ are unique. This condition on $\Omega$ is automatically satisfied if $V$ is a Lipschitz 2-valued graph (of arbitrary Lipschitz constant) or if the codimension is 1, $V$ corresponds to a current without boundary and the regular part of $V$ is stable. The analysis involves, among other things, a new energy non-concentration estimate for a class of multi-valued harmonic functions and a non-variational argument to establish mononotonicity of the Almgren frequency function for this class.