Nonclassical minimizing surfaces with smooth boundary
Dr. Jonas Hirsch, University of Leipzig
Aula C29 (Lavandino), Dipartimento di Matematica – Giovedì 19 Dicembre 2019 h.14:30
Abstract. We construct a Riemannian metric g on R4 (arbitrarily close to the euclidean one) and a smooth simple closed curve Γ ⊂ R4 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 ⊂ Rn 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 (Ck,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.