Titles and Abstracts

Title: Existence of Willmore type surfaces with small area

Abstract: In this talk, we shall study the Willmore functional in 3 dimensional Riemannian manifolds, in particular, we are interested in the existence of Willmore type surfaces. Here we call a surface Willmore type surface if it is a critical point of the Willmore functional under area constraint. We treat both genus zero and genus one surfaces. Under some conditions on the curvature of 3 dimensional manifold, we will prove the existence of Willmore type surfaces. This talk is based on joint work with Andrea Malchiodi (SNS,Pisa) and Andrea Mondino (University of Warwick).

Title: Finite gap solutions for horizontal minimal surfaces of finite type in 5-sphere

Abstract: pdf

Title: Almgren-Pitts Min-max Theory I-III

Abstract: In these three talks, I will give an expository account of the Almgren-Pitts min-max theory including its recent breakthrough and applications. In the first talk, I will begin with Birkhoff’s idea of finding closed geodesics by min-max method. We will use this as a key example to understand the fundamental ideas of min-max constructions. Combining with the Lysternik-Schnirelmann theory, this gives the existence of at least three simple closed geodesics on any closed Riemannian surfaces. In the second talk, I will describe Almgren-Pitts min-max theory in the higher dimensional settings. We then give the main existence and regularity results, and also a survey of recent applications and advances in the theory led by the work of Marques and Neves. In the third talk, I will explain some technical ideas in the existence and regularity issues in Almgren-Pitts min-max theory. The first talk is elementary with focus on the basic ideas and should be accessible to advanced undergraduates. The second talk is mainly a survey of results. It is understandable for graduate students if one is willing to accept some results from geometric measure theory. The last talk is more technical and is targeted towards researchers working on similar field. These works are partially supported by an RGC grant from the Hong Kong Government.

Title: Waist inequalities I-IV

Abstract: I will try to give an expository talk about some min-max theory related with waists (though I don't have any results on this topic). Waists are a version of widths. In the definition we consider volumes of fibers of  continuous maps instead of volumes of cycles. I will try to explain some recent work by Bo'az Klartag and some of  the papers in the following references of waists for beginners:

[Gu] L. Guth, The waist inequality in Gromov's work,  Link

[AlGu] H. Alpert and L. Guth, A family of maps with many small fibers, Link

[DTU]D. Dotterrer, T. Kaufman, and U. Wagner,  On expansion of topological overlap, Link

[Gr] M.Gromov, Isoperimetry of waists and concentration of maps, Link

[Kla16] B. Klartag, Convex geometry and waist inequalities, Link

[Kla17] B. Klartag, Eldan's stochastic localization and tubular neighborhoods of complex-analytic sets, Link

[AHK] A. Akopyan, A. Hubard, and R. Karasev, Lower and upper bounds for the waists of different spaces, Link

[AK1]A. Akopyan and R. Karasev, A tight estimate for the waist of the ball, Link

[AK2]A. Akyopyan and R. Karasev, Waist of balls in hyperbolic and spherical spaces, Link

[Mem] Y. Memarian, On Gromov's waist of the sphere theorem, Link

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