On the Nature of Stationary Integral Varifolds near Multiplicity 2 Planes

Abstract

We study stationary integral $n$-varifolds $V$ in the unit ball $B_1(0)\subset\mathbb{R}^{n+k}$. Allard's regularity theorem establishes the existence of $\epsilon = \epsilon(n,k)\in (0,1)$ for which if $V$ is $\epsilon$-close (as varifolds) to the plane $P_0 = \{0\}^k\times\mathbb{R}^n$ with multiplicity 1 then, in $B_{1/2}(0)$, $V$ is represented by a single $C^{1,\alpha}$ minimal graph. However, when instead $P_0$ occurs with multiplicity $Q\in \{2,3,\dotsc\}$, simple examples show that this conclusion, now as a multi-valued graph, may fail, even if $V$ corresponds to an area-minimising rectifiable current. In the present work we investigate the structure of such $V$ which are close to planes with multiplicity $Q>1$, focusing primarily on the case $Q=2$. We show that an $\epsilon$-regularity theorem holds when $V$ is close, as a varifold, to $P_0$ with multiplicity $2$, provided $V$ satisfies a certain topological structural condition on the part of its support where the density of $V$ is $<2$. The conclusion then is that, in $B_{1/2}(0)$, $V$ is represented by the graph of a Lipschitz $2$-valued function over $P_0$ with small Lipschitz constant; in fact, the function is $C^{1,\alpha}$ in a precise generalised sense, and satisfies estimates, implying that all tangent cones at singular points in $B_{1/2}(0)$ are unique and comprised of stationary unions of $4$ half-planes (which may form a union of two distinct planes or a single multiplicity $2$ plane). The theorem does not require any additional assumption on the part of $V$ with density $\geq 2$ (which a priori may be a relatively large set in $\mathcal{H}^n$-measure with high topological complexity). As a corollary, we show that our $\epsilon$-regularity theorem applies unconditionally to stationary $2$-valued Lipschitz graphs with arbitrary Lipschitz constant, yielding improved regularity and uniform a priori estimates.