Joint Geometry and Analysis seminar: On the behaviour of stationary integral varifolds near multiplicity 2 planes

Professor Neshan Wickramasekera (Cambridge) will present his research in Analysis and Geometry.

Consider a singular $n$-dimensional minimal submanifold (i.e. a stationary integral $n$-varifold) $V$ in an open ball in $R^{n+k}$ lying close to a plane of some integer multiplicity $q$ passing through the centre of the ball. In 1972 Allard proved a fundamental regularity theorem, generalising earlier pioneering work of De Giorgi, that implies that if $q=1$ then near the centre of the ball the varifold is smoothly embedded. This celebrated De Giorgi--Allard theory in fact says that near the centre of the ball the varifold is the graph of a smooth function over the plane with small gradient and satisfying estimates on all derivatives. This result implies that for any stationary integral varifold, the (relatively open) set $\Omega$ of points of mass density $<2$ is fairly regular; if non-empty, $\Omega$ is an embedded submanifold away from a closed set whose Hausdorff dimension is at most $(n-1)$, and in the absence of triple-junction singularities (e.g. when the varifold is the limit of embedded minimal submanifolds) $\Omega$ is embedded everywhere if $n=2$ and is embedded away from a closed set of Hausdorff dimension at most $(n-3)$ if $n \geq 3.$

It is a long standing open question what one can say about $V$ when $q \geq 2$. We will discuss some 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, near the centre of the ball: (a) $V$ is a Lipschitz 2-valued graph with small Lipschitz constant and (b) each tangent cone to $V$ is unique, and is equal to either a single plane of multiplicity 1 or 2, or a pair of distinct multiplicity 1 planes or a union of four multiplicity 1 half-planes meeting along an $(n-1)$-dimensional axis. 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 in the ball and the regular part of $V$ is stable. The analysis involves, among other things, a new energy non-concentration estimate for a class of $q$-valued harmonic functions that approximate stsationary integral varifolds close to multiplicity $q$ planes, and a novel non-variational argument based on this estimate to establish mononotonicity of the Almgren frequency function for this class when $q=2$.

 

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