Adapting the blow-up arguments in [Sim93] and [Wic14a], we prove a C 1,α regularity theorem for stable codimension one integral n-varifolds which are sufficiently close to a "classical cone" of density 5 2 , i.e., a stationary integral cone supported on at most 5 half-hyperplanes meeting along a common (n − 1)-dimensional axis with density 5 2 at the origin. Our result is the first of its kind for non-flat cones with multiplicity > 1 when branch points are present in the nearby varifold. The only additional hypothesis necessary for our result is geometric: ruling out singularities comprised of three C 1,α hypersurfaces-withboundary meeting (only) along their common boundary; the corresponding results when the density of the classical cone is 3 2 or 2 -in which case the cone has multiplicity one almost everywhere -have been previously established in [Sim93] and [Wic21] respectively. For such varifolds the present work completes the analysis of the singular set in the region where the density is < 3, up to a set which is countably (n − 2)-rectifiable. A key new ingredient in our work is establishing a C 1,α boundary regularity theory for two-valued C 1,α harmonic functions which arise as blow-ups of sequences of such varifolds; this is discussed more generally in the accompanying work [Min21]. Subject to appropriate conditions on lower density classical singularities and (expected) regularity theorems concerning lower density branch point singularities, our arguments have the potential to be iterated inductively to establish regularity results for stable codimension one integral varifolds near classical cones of density q + 1 2 at the origin.

The regularity theory of the Campanato space L (q,λ) k(Ω) has been successfully used to understand the regularity of solutions to certain geometric variational problems. Here we extend this theory to multi-valued functions, adapting for the most part Campanato's original ideas in [Cam64]. We then give an application within the regularity theory of stationary integral varifolds. More precisely, we prove a regularity theorem for certain blow-up classes of functions, which typically arise when studying blow-ups of sequences of stationary integral varifolds converging to higher multiplicity planes or unions of half-planes. In such a setting, based in part on ideas in [Wic14] and [BKW21], we are able to deduce a boundary regularity theory for multi-valued harmonic functions, which is the first of its kind. In conjunction with [Min21], the results here establish a regularity theorem for stable codimension one integral varifolds near classical cones of density 5 2 .

For any Q ∈ { 3 2 , 2 , 5 2 , 3 , … } Q\in \{\frac {3}{2},2,\frac {5}{2},3,\dotsc \} , we establish a structure theory for the class S Q \mathcal {S}_Q of stable codimension 1 stationary integral varifolds admitting no classical singularities of density > Q >Q . This theory comprises three main theorems which describe the nature of a varifold V ∈ S Q V\in \mathcal {S}_Q when: (i) V V is close to a flat disk of multiplicity Q Q (for integer Q Q ); (ii) V V is close to a flat disk of integer multiplicity > Q >Q ; and (iii) V V is close to a stationary cone with vertex density Q Q and supports the union of 3 or more half-hyperplanes meeting along a common axis. The main new result concerns (i) and gives in particular a description of V ∈ S Q V\in \mathcal {S}_Q near branch points of density Q Q . Results concerning (ii) and (iii) directly follow from parts of previous work of the second author [Ann. of Math. (2) 179 (2014), pp. 843–1007]. These three theorems, taken with Q = p / 2 Q=p/2 , are readily applicable to codimension 1 rectifiable area minimising currents mod p p for any integer p ≥ 2 p\geq 2 , establishing local structure properties of such a current T T as consequences of little, readily checked, information. Specifically, applying case (i) it follows that, for even p p , if T T has one tangent cone at an interior point y y equal to an (oriented) hyperplane P P of multiplicity p / 2 p/2 , then P P is the unique tangent cone at y y , and T T near y y is given by the graph of a p 2 \frac {p}{2} -valued function with C 1 , α C^{1,\alpha } regularity in a certain generalised sense. This settles a basic remaining open question in the study of the local structure of such currents near points with planar tangent cones, extending the cases p = 2 p=2 and p = 4 p=4 of the result which have been known since the 1970’s from the De Giorgi–Allard regularity theory [Ann. of Math. (2) 95 (1972), pp. 417–491] [Frontiere orientate di misura minima, Editrice Tecnico Scientifica, Pisa, 1961] and the structure theory of White [Invent. Math. 53 (1979), pp. 45–58] respectively. If P P has multiplicity > p / 2 > p/2 (for p p even or odd), it follows from case (ii) that T T is smoothly embedded near y y , recovering a second well-known theorem of White [Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1986, pp. 413–427]. Finally, the main structure results obtained recently by De Lellis–Hirsch–Marchese–Spolaor–Stuvard [arXiv:2105.08135, 2021] for such currents T T all follow from case (iii).

For any given Q ∈ { 3 2 , 2, 5 2 , 3, . . .}, we establish a structure theory for the class SQ of stable codimension 1 integral varifolds admitting no classical singularities of density < Q. This theory comprises three main theorems which describe: (i) the nature of a varifold V ∈ SQ, for integer Q, when V is close to a flat disk of multiplicity Q; (ii) the nature of a varifold V ∈ SQ when V is close to a flat disk of integer multiplicity < Q; and (iii) the nature of a varifold V ∈ SQ when V is close, in a ball, to a stationary cone with vertex density Q and support the union of 3 or more half-hyperplanes meeting along a common axis. The main new result in the present work concerns (i) and gives in particular a description of V ∈ SQ near branch points of density Q, while results concerning (ii) and (iii), giving that V is embedded or has the structure of a classical singularity respectively, directly follow from parts of the previous work [Wic14] (and are reproduced in Part 2 below).These three theorems, taken with Q = p/2, are readily applicable to codimension 1 rectifiable area minimising currents mod p for any integer p ≥ 2, establishing local structural properties of such a current T as consequences of little information, namely the (easily checked) stability of the regular part of T and the fact that such a 1-dimensional singular (representative) current in R 2 consists of p rays meeting at a point. Specifically, it follows from (i) that, for even p, if T has one tangent cone at an interior point y equal to an (oriented) hyperplane P of multiplicity p/2, then P is the unique tangent cone at y, and T near y is given by the graph over P of a p 2 -valued function with C 1,α regularity in a certain generalised sense; this settles a basic remaining open question in the study of the local structure of codimension 1 area minimising currents mod p near points with planar tangent cones, extending the cases p = 2 and p = 4 of the result (with classical C 1,α conclusions near y) which have been known since the 1970's from the De Giorgi-Allard regularity theory ([All72]) and the structure theory of White ([Whi79]) respectively. If P has multiplicity < p/2 (for p even or odd), it follows from (ii) that T is smoothly embedded near y, recovering a second wellknown theorem of White ([Whi84]). Finally, the main structure results obtained recently by De Lellis-Hirsch-Marchese-Spolaor-Stuvard ([DLHM + 21]) for such currents T all follow from (iii). The implication to mod p minimising currents of the structure theory for S p/2 is analogous to how the regularity theory for codimension 1 area minimising integral currents is a direct corollary of the regularity theory ([Wic14]) for S∞ = ∩QSQ (the class of stable codimension 1 integral varifolds with no classical singularities).