PIMS Differential Geometry and Analysis Summer School
August 13 - August 17, 2007
University of Washington
Department of Mathematics
Seattle, WA 98195
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PIMS Differential Geometry and Analysis Summer School August 13 - August 17, 2007 |
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University of Washington |
Robert Hardt
Rice University
"Some Basic Results of Geometric Measure Theory in a
Metric Space"
Here we will develop and discuss notions of
differentiability, area, co-area, rectifiable sets,
measures, and currents, as well as flat chains with
coefficient in a normed abelian group, in the context of a
metric space. We will outline the proof of the basic
rectifiability theorems and applications to the Plateau
problem of mass-minimization.(first considered by
H.Federer and W.Fleming in 1960). We will combine some of
the techniques of L.Ambrosio-B.Kirchheim (2000) on
currents in a metric space with those of B.White (1999) on
chains with group coefficients and also introduce some new
notions occuring in current work with T. De Pauw.
Useful books to read are:
F. Morgan, "Geometric Measure Theory, A Beginner's Guide",
Associated Press, 1995.
H. Federer, "Geometric Measure Theory", Springer, New
York, 1969.
(Especially sections 3.1, 3.2, 4.1-4.3)
L. Simon, "Lectures on Geometric Measure Theory",
Australian National Unversity, 1984
(Especially sections 2, 3.11-3.12, 6)
F. Lin and Y. Xiaoping, "Geometric Measure Theory",
International Press, Boston, 2002.
(Especially chapters 3, 4, 5, 7)
Useful papers to read are:
L. Ambrosio and B. Kirchheim, "Currents in metric
spaces". Acta Math. 185 (2000), no. 1, 1--80.
White, Brian, "Rectifiability of flat chains". Ann.Math.
(2) 150(1999), no.1, 165-184.
R.Hardt and T.De Pauw, "Size minimization and
approximating problems." Calc.Var. PDE 17 (2003), no.4,
405-442.