Area minimizing hypersurfaces modulo $p$: a geometric free-boundary problem

TL;DR

This work develops a comprehensive geometric measure theory framework for area minimizing currents modulo $p$ in complete $C^2$ ambient manifolds, addressing both odd and even moduli and uncovering a robust free-boundary-type structure for the singular set.The core method combines a quantitative tangent-cone decay theory (in the spirit of Leon Simon) with graphical multigraph parametrizations, a linear-selection/binding-function mechanism to handle multiplicities up to $\lfloor p/2\rfloor$, and Scott-style No-hole and spine-estimates to control the singular set along the spine of key cones.For odd $p$, the singular set away from the boundary is locally a union of smooth minimal hypersurfaces meeting along a $C^{1,\alpha}$ free boundary, with the remaining singular set countably $(m-2)$-rectifiable and locally finite in $\mathcal{H}^{m-2}$; for even $p$ the results are local near points whose tangent cones have an $(m-1)$-dimensional spine, with a precise description of the tangent-cone structure and decay toward a reference cone.A key novelty is extending Simon's multiplicity-one cylindrical-tangent-cone framework to higher multiplicities in the mod($p$) setting, addressing the new challenges posed by the presence of multiple sheets and spine structures, and thereby providing optimal, sharp regularity conclusions for the interior singular set.

Abstract

We consider area minimizing $m$-dimensional currents $\mathrm{mod}(p)$ in complete $C^2$ Riemannian manifolds $Σ$ of dimension $m+1$. For odd moduli we prove that, away from a closed rectifiable set of codimension $2$, the current in question is, locally, the union of finitely many smooth minimal hypersurfaces coming together at a common $C^{1,α}$ boundary of dimension $m-1$, and the result is optimal. For even $p$ such structure holds in a neighborhood of any point where at least one tangent cone has $(m-1)$-dimensional spine. These structural results are indeed the byproduct of a theorem that proves (for any modulus) uniqueness and decay towards such tangent cones. The underlying strategy of the proof is inspired by the techniques developed by Leon Simon in "Cylindrical tangent cones and the singular set of minimal submanifolds" (J. Diff. Geom. 1993) in a class of multiplicity one stationary varifolds. The major difficulty in our setting is produced by the fact that the cones and surfaces under investigation have arbitrary multiplicities ranging from $1$ to $\lfloor \frac{p}{2}\rfloor$.

Figures (7)

• Figure 1: All the multiplicities $\kappa(i)$ have the same sign
• Figure 2: The two hemispheres $S^+$ and $S^-$ and the rays $\ell^+ (j)$ and $\ell^- (l)$. The points $v^+ (j)$ and $v^- (l)$ are the intersections of the rays with the appropriate hemisphere.
• Figure 3: The restriction of currents $W$ and $Z$ in $B_1^{n+1}$: in this example, the modulus is $p=3$ and the multiplicities $m^+ (j)$ are all equal to $1$. The current $W$ is represented by the thicker lines connecting the points $v^+ (j)$ to the center of the circle, while the current $Z$ is represented by the lighter lines connecting the points $v^+ (j)$ to the point $z$.
• Figure 4: The Whitney decomposition of $[0,2] \times [-2,2]^{m-1}$. In the above example the parameter $M$ equals $2$.
• Figure 5: An example of $\mathscr{W}(Q_0)$.

Theorems & Definitions (97)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Proposition 1.8
• Theorem 1.10: Uniqueness of cylindrical blow-ups
• Corollary 3.1