The spherical Plateau problem: existence, uniqueness, stability

Abstract: Consider a countable group $G$ acting on the unit sphere $S$ in the space of $L^2$ functions on $G$ by the regular representation. Given a homology class $h$ in the quotient space $S/G$, one defines the spherical Plateau solutions for $h$ as the intrinsic flat limits of volume minimizing sequences of cycles representing $h$. Interestingly in some special cases, for example when $G$ is the fundamental group of a closed hyperbolic manifold of dimension at least $3$, the spherical Plateau solutions are essentially unique and can be identified. However in general not much is known. I will discuss the questions of existence and structure of non-trivial Plateau solutions. I will also explain how uniqueness of spherical Plateau solutions for hyperbolic manifolds of dimension at least $3$ implies stability for the volume entropy inequality of Besson-Courtois-Gallot.

analysis of PDEsdifferential geometry

Audience: researchers in the topic

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NCTS international Geometric Measure Theory seminar

Series comments: We envisage an event built around virtual presentations on progress in geometric measure theory by external speakers. Every researcher is free to register as a participant and thus gain access to a virtual facility which is complete with lobby, lecture hall, and areas with boards for discussion. Thus, it shall recreate the exchange possibilities found at international conferences.

Focus: regularity and singularity theories for submanifolds of Riemannian manifolds and some of its applications.

Frequency: one presentation every other month.

Registration: required for new participants, go to the seminar website (allow at least one working day for processing).

Virtual venue: HyHyve space NCTS iGMT seminar (only for registered participants, opened one hour before the events).

You might want to consult the description of the premises and instructions.

Former organiser: Guido De Philippis (till March 2022).

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