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Talks by Edoardo Cavalloto

Minimal boundary cones

Université Paris-Sacaly

The Plateau problem arises from physics, and in particular from soap bubbles and soap films. Solving the Plateau problem means to find a surface with minimal area among all surfaces with a given boundary. Part of the problem actually consists in giving a suitable sense to the notions of "surface", "area" and "boundary". Given \(0 < d < n\) we will consider a setting, due to Almgren, in which the considered objects are sets with locally finite d-dimensional Hausdorff measure, the functional we will try to minimize is the Hausdorff area \(H^d\), and the boundary condition is given in terms of a one-parameter family of deformations. Almgren minimizers turn out to have nice regularity properties, in particular an Almgren minimizer is a \(C^{1,\alpha}\) embedded submanifold of \(\mathbb{R}^n\) up to a negligible set, and the tangent cone to any point of such a minimizer is a minimal cone. Therefore in order to give a complete characterisation of these object we need to know how minimal cones look like. The complete list of minimal cones of \(\mathbb{R}^2\) and \(\mathbb{R}^3\) has been well known long time ago while in higher dimensions the list is far from being complete and we only know few examples. My talk will focus to a small variation of this setting which we call "sliding boundary" and to minimal cones that arise in this frame.

Seminario 1ª Planta, IEMATH

Edoardo Cavalloto

Université Paris-Sacaly

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