Math Colloquia

Interior regularity for two-dimensional stationary Q-valued maps

by Prof. Jonas Hirsch (Universität Leipzig)

Rectorate/Building-Auditorium (GSSI)



Viale Iacobucci 2

joint work with L. Spolaor


The mathematical study of minimal surfaces, being critical points of the area functional, has a long-lasting history. Starting perhaps with the systematic exper?iments with soap films of Plateau. More or less since then, it has been an active field of research, that inspired and fascinated many mathematicians. It has led to a huge number of mathematical results reaching far beyond the original research topic of minimal surfaces. Despite all the amazing results established so far our under?standing of the regularity of minimal surfaces, being a critical point not necessarily being minimizing, is very limited. In fact in his groundbreaking work, on the first variation of a varifold, Allard proved that the singular set of stationary integral varifolds is meager. Since then little to no progress has been made on the question of the optimal dimension of the singular set for integral stationary varifolds. Which is the first rough description concerning the regularity of a minimal surface. In my talk, I want to address this question under two assumptions: multiplicity 2 and Lipschitz graphicality, i.e. consider a 2-valued Lipschitz graph that is stationary for the area. In particular, I would like to present the recent result obtained in cooperation with Luca Spolaor that the singular set of such a stationary 2-valued Lipschitz graph is of codimension 1. The very rough approach is to apply Almgren’s linearization strategy in the stationary setting. This needs major changes since so far the minimality assumption is used in various places crucially. In this colloquium, I would like to give you an idea about Almgren’s strategy, its difficulties, and our partial resolution for 2-valued Lipschitz graphs. In particular, I want to highlight the differences between the minimizing and stationary case.



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ID riunione: 816 0569 9944
Codice d’accesso: 189217