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Lectures by Andrejs Treibergs, May 23 & May 26, 1:30 - 2:30 pmSome Variational Problems in GeometrySome examples of geometric variational problems will be discussed, beginning with two related problems for closed embedded plane curves. The ordinary isoperimetric problem, solved by the circle, asks for the curve that encloses the largest area among curves of fixed length; the reverse problem asks for the curve of least area for such curves. Of course this problem has only degenerate solutions unless additional constraints are imposed. First we shall solve the problem under the added constraint that the curve has bounded curvature. Second we restrict to curves with bounded bending energy. The second problem is equivalent to Euler's elastic deformation of a plane ring under hydroststic pressure. As an example in higher dimensions, we shall discuss the existence of minimal immersions of two spheres into compact manifolds after the work of Sachs/Uhlenbeck and Schoen/Yau. The talks will be mostly elementary, and geometric background will be developed as needed. |
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| Department of Mathematics VIGRE University of Utah | ||