[Paper Review] Fine structure of the singular set of area minimizing hypersurfaces modulo $p$

Camillo De Lellis, Jonas Hirsch+3 more|arXiv (Cornell University)|Jan 25, 2022

Geometric Analysis and Curvature Flows4 citations

TL;DR

This paper completes the classification of the singular set of area-minimizing hypersurfaces modulo $p$ by proving that, for even $p$, the singular set is a $C^{1,eta}$ hypersurface of dimension $m-1$ outside a set of dimension at most $m-2$, where $N \leq p$ regular sheets with positive integer multiplicities summing to $p$ meet transversally. The result extends prior work on odd $p$ and resolves the remaining case via a refined blow-up analysis and epiperimetric inequality in the linearized problem.

ABSTRACT

Consider an area minimizing current modulo $p$ of dimension $m$ in a smooth Riemannian manifold of dimension $m+1$. We prove that its interior singular set is, up to a relatively closed set of dimension at most $m-2$, a $C^{1,α}$ submanifold of dimension $m-1$ at which, locally, $N\leq p$ regular sheets of the current join transversally, each sheet counted with a positive multiplicity $k_i$ so that $\sum_i k_i = p$. This completes the analysis of the structure of the singular set of area minimizing hypersurfaces modulo $p$, initiated by J. Taylor for $m=2$ and $p=3$ and extended by the authors to arbitrary $m$ and all odd $p$ in arXiv:2105.08135. We tackle the remaining case of even $p$ by showing that the set of singular points admitting a flat blow-up is of codimension at least two in the current. First, we prove a structural result for the singularities of minimizers in the linearized problem, by combining an epiperimetric inequality with an analysis of homogeneous minimizers to conclude that the corresponding degrees of homogeneity are always integers; second, we refine Almgren's blow-up procedure to prove that all flat singularities of the current persist as singularities of the $\mathrm{Dir}$-minimizing limit. An important ingredient of our analysis is the uniqueness of flat tangent cones at singular points, recently established by Minter and Wickramasekera in arXiv:2111.11202.

Motivation & Objective

• To complete the structural classification of the singular set of area-minimizing hypersurfaces modulo $p$ in codimension one, particularly for even $p$, which remained open after prior work on odd $p$.
• To establish that the singular set is a $C^{1,\alpha}$ submanifold of dimension $m-1$ outside a set of dimension at most $m-2$, where $N \leq p$ regular sheets with positive integer multiplicities summing to $p$ meet transversally.
• To resolve the case of even $p$ by proving that all flat singularities persist in the Dir-minimizing blow-up limit and that the set of points admitting a flat blow-up has codimension at least two.
• To extend the regularity theory for area-minimizing currents modulo $p$ by combining linearized analysis with nonlinear blow-up techniques, leveraging recent uniqueness results on flat tangent cones.

Proposed method

• Prove a structural result for homogeneous Dir-minimizers in the linearized problem by combining an epiperimetric inequality with classification of $1$-homogeneous minimizers, showing that degrees of homogeneity are always integers.
• Apply a refined version of Almgren’s blow-up procedure to show that flat singularities of the current persist as singularities in the Dir-minimizing limit, ensuring the singular structure is preserved under blow-up.
• Use the uniqueness of flat tangent cones at singular points—recently established by Minter and Wickramasekera—to rule out non-unique blow-ups and support the $C^{1,\alpha}$ regularity of the singular set.
• Establish that the set of points admitting a flat blow-up has codimension at least two in the current, which allows the singular set to be decomposed into a $C^{1,\alpha}$ part and a lower-dimensional remainder.
• Leverage the fact that $T_n = Q^{-1}T_n$ is a mod 2 current with zero boundary mod 2 and trivial homology to deduce that $T_n$ is a boundary mod 2, enabling the use of strong maximum principle arguments.
• Apply the strong maximum principle to $T_o^i$ and $T_2$ to rule out intersection of regular parts of $T_o$ and $T_n$, thereby proving disjoint supports of orientable and non-orientable components.

Experimental results

Research questions

• RQ1 What is the fine structure of the singular set of area-minimizing hypersurfaces modulo $p$ when $p$ is even, particularly in the absence of the odd-$p$ symmetry?
• RQ2 Can the singular set be decomposed into a $C^{1,\alpha}$ submanifold of dimension $m-1$ and a lower-dimensional remainder, even for even $p$?
• RQ3 Do flat singularities of the current persist as singularities in the Dir-minimizing blow-up limit, and what does this imply for the regularity of the singular set?
• RQ4 How does the epiperimetric inequality in the linearized problem constrain the possible degrees of homogeneity of singularities, and why are they necessarily integers?
• RQ5 What role does the uniqueness of flat tangent cones play in ensuring the $C^{1,\alpha}$ regularity of the singular set for even $p$?

Key findings

• The singular set of an area-minimizing current modulo $p$ in codimension one is, up to a set of dimension at most $m-2$, a $C^{1,\alpha}$ submanifold of dimension $m-1$ for all $p \geq 2$, including even $p$.
• At each point of the singular set, exactly $N \leq p$ regular sheets with positive integer multiplicities $k_i$ such that $\sum k_i = p$ meet transversally, forming a conical structure.
• The set of points admitting a flat blow-up has codimension at least two in the current, implying that such points do not contribute to the main singular stratum.
• The degrees of homogeneity of $1$-homogeneous Dir-minimizers are always integers, a result derived from the epiperimetric inequality and classification of homogeneous minimizers.
• All flat singularities of the current persist as singularities in the Dir-minimizing blow-up limit, ensuring that the singular structure is preserved under the blow-up process.
• The supports of the orientable and non-orientable parts of the current are disjoint, as shown via the strong maximum principle applied to mod 2 currents with zero boundary.