A sharp extension of Allard’s boundary regularity theorem for area minimizing currents with arbitrary boundary multiplicity

Abstract: In the context of area-minimizing currents, Allard boundary regularity theorem asserts that an oriented current with boundary that minimizes area cannot have boundary singularities of minimum density. Indeed, in a neighborhood of a point of minimum density, the surface must coincide with a classical smooth minimal surface that attaches smoothly to the boundary.

In this talk, I will discuss a series of papers, one of them in collaboration with Reinaldo Resende, that extend Allard’s boundary regularity theory to a higher boundary multiplicity setting. Specifically, for an area-minimizing current with a multiplicity $Q$ boundary, we study density $Q/2$ boundary points. In this context, a regular point is one where smooth submanifolds with multiplicity attach transversally to the boundary. We establish that the set of singular boundary points of minimum density is of boundary codimension at most 2 and rectifiable, extending the corresponding result in 2d by De Lellis - Steinbrüchel - Nardulli to higher dimensional currents. The sharpness of this regularity theory is confirmed by my construction of a 3-dimensional area mininimizing current in $\mathbb R^5$ with a singular boundary point of minimum density.

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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