19 mar 2025 -- 12:00   [open in google calendar]

Agenda: Get-together (30 min), presentation Ian Fleschler (60 min), questions and discussions (30 min).

Registration required for new participants. Please go to our seminar website (allow one work day for processing).

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.

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