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PDE Real Analysis Seminar
Seminar information archive ～04/24｜Next seminar｜Future seminars 04/25～
| Date, time & place | Tuesday 10:30 - 11:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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2019/01/29
10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Salvatore Stuvard (The University of Texas at Austin)
The regularity of area minimizing currents modulo $p$ (English)
Salvatore Stuvard (The University of Texas at Austin)
The regularity of area minimizing currents modulo $p$ (English)
[ Abstract ]
The theory of integer rectifiable currents was introduced by Federer and Fleming in the early 1960s in order to provide a class of generalized surfaces where the classical Plateau problem could be solved by direct methods. Since then, a number of alternative spaces of surfaces have been developed in geometric measure theory, as required for theory and applications. In particular, Fleming introduced currents modulo $2$ to treat non-orientable surfaces, and currents modulo $p$ (where $p \geq 2$ is an integer) to study more general surfaces occurring as soap films.
It is easy to see that, in general, area minimizing currents modulo $p$ need not be smooth surfaces. In this talk, I will sketch the proof of the following result, which achieves the best possible estimate for the Hausdorff dimension of the singular set of an area minimizing current modulo $p$ in the most general hypotheses, thus answering a question of White from the 1980s: if $T$ is an area minimizing current modulo $p$ of dimension $m$ in $R^{m+n}$, then $T$ is smooth at all its interior points, except those belonging to a singular set of Hausdorff dimension at most $m-1$.
The theory of integer rectifiable currents was introduced by Federer and Fleming in the early 1960s in order to provide a class of generalized surfaces where the classical Plateau problem could be solved by direct methods. Since then, a number of alternative spaces of surfaces have been developed in geometric measure theory, as required for theory and applications. In particular, Fleming introduced currents modulo $2$ to treat non-orientable surfaces, and currents modulo $p$ (where $p \geq 2$ is an integer) to study more general surfaces occurring as soap films.
It is easy to see that, in general, area minimizing currents modulo $p$ need not be smooth surfaces. In this talk, I will sketch the proof of the following result, which achieves the best possible estimate for the Hausdorff dimension of the singular set of an area minimizing current modulo $p$ in the most general hypotheses, thus answering a question of White from the 1980s: if $T$ is an area minimizing current modulo $p$ of dimension $m$ in $R^{m+n}$, then $T$ is smooth at all its interior points, except those belonging to a singular set of Hausdorff dimension at most $m-1$.
