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

Abstract:

For any $Q\in \{\frac {3}{2},2,\frac {5}{2},3,\dotsc \}$, we establish a structure theory for the class $\mathcal {S}_Q$ of stable codimension 1 stationary integral varifolds admitting no classical singularities of density $<Q$. This theory comprises three main theorems which describe the nature of a varifold $V\in \mathcal {S}_Q$ when: (i) $V$ is close to a flat disk of multiplicity $Q$ (for integer $Q$); (ii) $V$ is close to a flat disk of integer multiplicity $<Q$; and (iii) $V$ is close to a stationary cone with vertex density $Q$ 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 $V\in \mathcal {S}_Q$ near branch points of density $Q$. 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 $Q=p/2$, are readily applicable to codimension 1 rectifiable area minimising currents mod $p$ for any integer $p\geq 2$, establishing local structure properties of such a current $T$ as consequences of little, readily checked, information. Specifically, applying case (i) it follows 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 of a $\frac {p}{2}$-valued function with $C^{1,\alpha }$ 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 $p=2$ and $p=4$ 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 $P$ has multiplicity $< p/2$ (for $p$ even or odd), it follows from case (ii) that $T$ is smoothly embedded near $y$, 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 $T$ all follow from case (iii).