A structure theory for stable codimension 1 integral varifolds with applications to area minimising hypersurfaces mod 𝑝

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).