Planar frequency, asymptotic normal forms, and local topology of area-minimizing currents.

Abstract: A fundamental problem in geometric measure theory is to understand the local structure of $n$-dimensional area-minimizing rectifiable currents $T$ of codimension at least 2. Almgren's 1983 theory provides a powerful general framework establishing the sharp Hausdorff dimension upper bound $n-2$ for the singular set (subsequently made more accessible by De Lellis–Spadaro). The work of White, Chang, and Micallef–White gives a remarkably complete structure theory when $T$ is 2-dimensional, in which case the singularities are isolated. In higher dimensions, however, the local structure of $T$ and the nature of its singularities remain much more subtle, particularly at $branch$ $points$, where one tangent cone is a plane.

In a series of papers with Brian Krummel, we develop a new framework for this problem in arbitrary dimension $n$. Its geometric philosophy differs from the classical theory: it unifies decay estimates at branch points with the dimension and structure of the singular set, as well as with the structure of $T$. A central conceptual novelty is the introduction of a new intrinsic frequency function, called the $planar$ $frequency$. Unlike the frequency used in the classical theory, planar frequency is defined directly in terms of geometric quantities integrated over the current, without first constructing auxiliary center manifolds at branch points. The approximate monotonicity of planar frequency provides quantitative control of the rate at which $T$ approaches planes and leads to a natural decomposition of the singular set according to planar decay.

I will describe this framework and some of its main consequences. Among these are a more direct proof of Almgren's $n-2$ bound, ${\mathcal H}^{n-2}$-almost everywhere uniqueness of tangent cones, and a detailed asymptotic description of $T$ at typical branch points. In particular, at ${\mathcal H}^{n-2}$-almost every branch point $Z$ there is a unique tangent plane, an intrinsic rational invariant—the $branching$ $order$ ${\mathcal O}_{T}(Z) >1$—and a unique, nonzero, ${\mathcal O}_{T}(Z)$-homogeneous cylindrical multi-valued tangent function. Together, these provide an $asymptotic$ $normal$ $form$ for $T$ at $Z$ with quantitative decay for the remainder. Corollaries of this normal form include a locally finite decomposition of the singular set into disjoint, locally compact, locally $(n-2)$-rectifiable sets with locally finite ${\mathcal H}^{n-2}$ measure, and a sharp branching order criterion under which a branch point $Z$ is $classical$; that is, near $Z$, the support of $T$ is homeomorphic to an $n$-disk and admits a $C^{1, \mu}$ parameterization, while the entire singular set is an $(n-2)$-dimensional $C^{1, \mu}$ submanifold consisting only of branch points with the same density and branching order as $Z$. This is a natural higher-dimensional analogue of the Chang–Micallef–White structural description in dimension $2$.

A central theme of the talk will be how planar frequency avoids the need to construct center manifolds uniformly across all branch points as in the classical framework. Instead, it identifies precisely the regime in which a center manifold becomes necessary and reduces its use to a canonical case. This reduction is crucial for our asymptotic normal form and also leads to substantial technical simplifications over the classical approach.

I will also briefly comment on related contemporaneous work of De Lellis, Minter, and Skorobogatova.

analysis of PDEsdifferential geometry

Audience: researchers in the topic

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NCTS international Geometric Measure Theory seminar

Series comments: We envisage an event built around virtual presentations on progress in geometric measure theory by external speakers. Every researcher is free to register as a participant and thus gain access to a virtual facility which is complete with lobby, lecture hall, and areas with boards for discussion. Thus, it shall recreate the exchange possibilities found at international conferences.

Focus: regularity and singularity theories for submanifolds of Riemannian manifolds and some of its applications.

Frequency: one presentation every other month.

Registration: required for new participants, go to the seminar website (allow at least one working day for processing).

Virtual venue: HyHyve space NCTS iGMT seminar (only for registered participants, opened one hour before the events).

You might want to consult the description of the premises and instructions.

Former organiser: Guido De Philippis (till March 2022).

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