Giovedì 19 dicembre 2019, alle ore 14.30 precise, presso l'aula C29
(Lavandino) del
Dipartimento di Matematica "F. Casorati", il

Dr. Jonas Hirsch, University of Leipzig

terrà un seminario dal titolo:

NONCLASSICAL MINIMIZING SURFACES WITH SMOOTH BOUNDARY

nell'ambito del Seminario di Matematica Applicata (IMATI-CNR e
Dipartimento di Matematica, Pavia),
http://matematica.unipv.it/it/seminari-matematica-applicata

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Abstract. We construct a Riemannian metric g on R^4 (arbitrarily close
to the euclidean one) and a smooth simple closed curve Γ ⊂ R^4 such
that the unique area minimizing surface spanned by Γ has infinite
topology. Furthermore the metric is almost Kaehler and the area
minimizing surface is calibrated. This example suggests that a
conjecture by B. White is sharp. It states that the Federer-Fleming
solution has finite topology if the boundary curve Γ ⊂ R^n is real
analytic. If White's conjecture were true, then for real analytic
boundary curves the Federer-Fleming solution T would coincide with the
Douglas-Rado solution for some genus g. In co-dimension one this holds
true already if the boundary curve Γ is sufficiently regular (C^{k,α}
for k + α > 2) as a consequence of De Giorgi's interior regularity
theorem and Hardt-Simon’s boundary regularity result. In contrast by
our example the situation seems to change dramatically if we go to
higher co-dimension. In my talk I would like to present the
construction of our example and its link to the known boundary
regularity result in higher co-dimension.
Joint work with C. De Lellis and G. De Philippis.