[Turkmath:6310] Plateau problem for minimal surfaces in differential geometry - Geometry Seminar - 8/12, 16:00

[Mustafa Kalafat]

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Title: Plateau problem for minimal surfaces in differential geometry

Speaker: Mustafa Kalafat

Abstract:

Plateau problem can be stated as follows: Given an
(n−1)-manifold(surface) as a boundary in an (n+k)-manifold, find an
n-surface that is bounded by that boundary and has minimal area. The
problem was first posed by Lagrange in 1760, and named after the
Belgian Physicist Joseph Plateau, who studied soap films and observed
several laws of their geometric properties.


Depending on the conditions we impose on the boundary and enclosing
surface, the ambient manifold M, the codimension k and the
interpretation of ”bounded by Γ”, we have variants of the Plateau
problem. In this talk we are mainly focused on the oriented
codimension one Plateau problem: Given a closed oriented immersed
(n−1)-surface in the Euclidean (n+1)-space, find a oriented bounding
surface which has minimal area among other candidates. To better
understand the bounding condition, consider the following example.
Take two parallel circles in R3 that are closed to each other. The
oriented solution will be a catenoid if the two components are
equipped with different orientations, and two disks if the two
components are given the same orientation. Also, we know from the
example that the oriented solution may not be minimizing among all
surfaces that span.

To solve the Plateau problem, one wants to take a minimizing sequence
of surfaces Σi, and hope that Σi converges to some minimal surface Σ.
However, in general we do not have convergence as the area bound is
not strong enough to control the surface. In the same spirit as the
weak solution of a PDE, we want to find a space of ”weak manifolds” in
which a notion of ”mass” is defined, and has nice functional analysis
properties:

1. The space has good compactness property, so for a mass-minimizing
sequence Σi, we can find a convergent subsequence.

2. The mass functional is lower semicontinuous, so the limit is minimizing.

3. The ”weak solution” generated above is actually regularity, thus a
”classical solution”.

In 1960, Federer and Fleming came up with a very powerful setting,
called integral currents, which is suitable for the discussion of the
oriented Plateau problem.

We will be using the following resources.

References:


1. Leon Simon - Geometric Measure Theory.
Stanford Univ. lecture notes.

2. James Simons. - Minimal varieties in riemannian manifolds.
Ann. of Math. (2), 88:62–105, 1968.


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Time:  Friday, December 8, 4:00 pm

Seminar website: http://kalafat.droppages.com/seminars/gt23.html

Zoom link:

https://boun-edu-tr.zoom.us/j/99710229552?pwd=Ti9HRFRTdVN6QXNUd3BESUhvNlVDQT09

  Meeting ID: 997 1022 9552
  Passcode: geometri




 |Mustafa Kalafat -------------------------------------------------
 |    Rheinische Friedrich-Wilhelms-Universität Bonn   |
 |    Office: Endenicher Allee 60, Zimmer 1.036            |
 |    Email: kalafat at math.uni-bonn.de                          |
 |    Web  : http://kalafat.droppages.com/                      |
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