[Turkmath:5102] Minimal Surfaces - Geometry Seminar: 20/8, 15:00

[Mustafa Kalafat]

Dear Geometry Fans,


Below is the advertisement for today's talk in
our Geometry seminar.

stay healthy, best wishes !

m. kalafat



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Title: Minimal Immersions into Einstein-Hermitian Manifolds

Speaker: Mustafa Kalafat


Abstract: A minimal surface is a surface that locally minimizes its area.
This is equivalent to having zero mean curvature. They are
2-dimensional analog to geodesics,
which are analogously defined as critical points of the length functional.

Minimal surface theory originates with Lagrange who in 1762 considered
the variational problem
of finding the surface z = z(x, y) of least area stretched across a
given closed contour.
He derived the Euler–Lagrange equation for the solution and did not
succeed in finding any solution beyond the plane.
In 1776 Jean Baptiste Marie Meusnier discovered that the helicoid and
catenoid satisfy the equation
and that the differential expression corresponds to twice the mean
curvature of the surface,
concluding that surfaces with zero mean curvature are area-minimizing.

By expanding Lagrange's equation, Gaspard Monge and Legendre in 1795
derived representation formulas for the solution surfaces.
While these were successfully used by Heinrich Scherk in 1830 to
derive his surfaces,
they were generally regarded as practically unusable.
Catalan proved in 1842/43 that the helicoid is the only ruled minimal surface.

Progress had been fairly slow until the middle of the century when the
Björling problem was solved using complex methods.
The "first golden age" of minimal surfaces began.
Schwarz found the solution of the Plateau problem for a regular
quadrilateral in 1865
and for a general quadrilateral in 1867 using complex methods.
Weierstrass and Enneper developed more useful representation formulas,
firmly linking minimal surfaces to complex analysis and harmonic functions.
Other important contributions came from Beltrami, Bonnet, Darboux,
Lie, Riemann, Serret and Weingarten.

Between 1925 and 1950 minimal surface theory revived, now mainly aimed
at nonparametric minimal surfaces.
The complete solution of the Plateau problem by Jesse Douglas and
Tibor Radó was a major milestone.
Bernstein's problem and Robert Osserman's work on complete minimal
surfaces of finite total curvature were also important.

Another revival began in the 1980s. One cause was the discovery in
1982 by Celso Costa
of a surface that disproved the conjecture that the plane, the
catenoid, and the helicoid
are the only complete embedded minimal surfaces in R^3 of finite
topological type.
This not only stimulated new work on using the old parametric methods,
but also demonstrated the importance of computer graphics
to visualize the studied surfaces and numerical methods to solve the
"period problem"
(when using the conjugate surface method to determine surface patches
that can be assembled into a larger symmetric surface,
certain parameters need to be numerically matched to produce an
embedded surface).
Another cause was the verification by H. Karcher that the triply
periodic minimal surfaces
originally described empirically by Alan Schoen in 1970 actually exist.
This has led to a rich menagerie of surface families and methods of
deriving new surfaces from old,
for example by adding handles or distorting them.

Currently the theory of minimal surfaces has diversified to minimal
submanifolds in other ambient geometries,
becoming relevant to mathematical physics (e.g. the positive mass
conjecture, the Penrose conjecture)
and three-manifold geometry (e.g. the Smith conjecture, the Poincaré
conjecture, the Thurston Geometrization Conjecture).

This is a continuation of the basic minimal submanifold theory.
Topics to be covered in this weeks seminar is as follows:

   "Jacobi Operator and Higher dimensional fundamental forms."

We will be using the following resources.

References:

N. Ejiri - The Index of Minimal Immersions of S^2 into S^2n.
Mathematische Zeitschrift. 184,127-132 (1983).

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Time:  August 20, 15:00

Seminar website: http://kalafat.droppages.com/seminars/gt21.html

Zoom link:

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

  Meeting ID: 997 1022 9552
  Passcode: geometri



  |Mustafa Kalafat ---------------------------------
  |   Associate Professor of Mathematics    |
  |    Web : http://kalafat.droppages.com/   |
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