The structure of stable codimension one integral varifolds near classical cones of density 5/2

We prove a multi-valued $C^{1, \alpha}$ regularity theorem for the varifolds in the class $S_2$ (i.e., stable codimension one stationary integral $n$ -varifolds admitting no triple junction classical singularities) which are sufficiently close to a stationary integral cone comprised of 5 half-hyperplanes (counted with multiplicity) meeting along a common axis. Such a result is the first of its kind for non-flat cones of higher (i.e. $\gt 1$ ) multiplicity allowing for branch points to be present in the nearby varifolds. For such varifolds, this completes the analysis of the singular set in the region where the density is $\lt 3$ , up to a set which is countably ( $n-2$ )-rectifiable. Our approach builds upon the blow-up arguments in [Sim93] and [Wic14]. A crucial new element of our work involves inductively using progressively finer blow-up procedures which are used to show that a certain $\epsilon$ -regularity property holds at the blow-up level. Subsequently, we use this property to establish a $C^{1, \alpha}$ boundary regularity theorem for two-valued $C^{1, \alpha}$ harmonic functions arising as blow-ups of sequences of such varifolds. This boundary regularity theory is carried out in the accompanying work [Min21].