On the Nature of Stationary Integral Varifolds near Multiplicity 2 Planes

We study stationary integral n-varifolds V in the unit ball B₁(0)ⁿ⁺ᵏ. Allard's regularity theorem establishes the existence of ε= ε(n,k)∈ (0,1) for which if V is $ε$-close (as varifolds) to the plane P₀ = \0\ᵏⁿ with multiplicity 1 then, in B1/2(0), V is represented by a single C1,α minimal graph. However, when instead P₀ occurs with multiplicity Q∈ \2,3,\, 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 $ε$-regularity theorem holds when V is close, as a varifold, to P₀ 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 B1/2(0), V is represented by the graph of a Lipschitz $2$-valued function over P₀ with small Lipschitz constant; in fact, the function is C1,α in a precise generalised sense, and satisfies estimates, implying that all tangent cones at singular points in B1/2(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 ≥ 2 (which a priori may be a relatively large set in Hⁿ-measure with high topological complexity). As a corollary, we show that our $ε$-regularity theorem applies unconditionally to stationary $2$-valued Lipschitz graphs with arbitrary Lipschitz constant, yielding improved regularity and uniform a priori estimates.