Julian Scheuer:
Two-point functions in differential geometry and the Lawson conjecture
Zeit und Ort
Donnerstag, 12.7.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Zusammenfassung
The Lawson conjecture from 1970 states the following:\n\nConjecture (Lawson, 1970). Any embedded minimal torus in the 3-dimensional unit-sphere is the Cliford torus.\n\n\n\nIn 2013, Brendle confirmed the validity of this conjecture thereby complementing the case of genus zero due to Almgren. Although Lawson himself provided many crucial ingredients used in the final proof by Brendle, the missing\npiece in the completion of the proof was finally given in form of a sophisticated use\nof the maximum principle on the surface T: Instead of trying to come up with an\nauxiliary function on T, an auxiliary function on the product T × T is constructed,\nwhich contains much more geometric information at points where a maximum is\nachieved. This method followed up similar techniques used earlier in the context of\nnon-collapsing for curvature flows by Andrews and Huisken. This auxiliary\nfunction led to the conclusion that the second fundamental form on T must have\nconstant length and due to an earlier result of Lawson, Brendle was able to conclude\nthat T must be the Clifford torus up to isometries.\nIn this talk we present this powerful method of two-point functions and sketch\nits various mentioned applications.