# 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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