401-4530-18L  Topics in Minimal Surface Theory

SemesterFrühjahrssemester 2018
DozierendeT. Rivière
Periodizitäteinmalige Veranstaltung
LehrspracheEnglisch
KommentarMaximale Teilnehmerzahl: 12


KurzbeschreibungThe study of minimal surfaces takes its origins in the works of Euler and Bernouilli from the eigtheen century. Since then, minimal surfaces have become central objects in mathematics much beyond the field of geometry with applications in analysis, applied mathematics, theoretical physics and natural sciences.
Lernziel
InhaltThe study of minimal surfaces takes its origins in the works of Euler and Bernouilli from the eigtheen century. Since then, minimal surfaces have become central objects in mathematics much beyond the field of geometry with applications in analysis, applied mathematics, theoretical physics and natural sciences.

There have been tremendous developments in the field in the last few years, with a particular emphasis on the variational methods, which permitted to solve several old problems.

In the seminar we shall concentrate first on the Almgren-Pitts min-max theory for the construction of codimension one minimal surfaces in arbitrary closed manifolds as well as on the Gromov-Guth min-max widths. We will then carefully study a series of recent works by Marques and Neves about the realization of these widths by minimal hypersurfaces. The paramount of the seminar will be the presentation of their very recent proof of a conjecture by Yau asserting the existence of infinitely many embedded minimal hypersurfaces on any closed manifold of dimension strictly less than 8 for generic metrics.
Literatur1) L. Simon “Lectures on Geometric Measure Theory’’ Australian National University (1983).
2) T. Colding and W.Minicozzi “A course in Minimal Surfaces’’ AMS (2011)

More Advanced Bibliography :
3) J.Pitts,”Existence and regularity of minimal surfaces on riemannian manifolds” Princeton University Press 1981.
4) More bibliography will be given during the course of the seminar.
Voraussetzungen / BesonderesPrerequisite : Fundamental notions of Geometric Measure Theory and Differential Geometry of Minimal Submanifolds from respectively

1) L. Simon “Lectures on Geometric Measure Theory’’ Australian National University (1983).
2) T. Colding and W.Minicozzi “A course in Minimal Surfaces’’ AMS (2011)