Quantitative estimates for singular strata of minimizing maps between manifolds

27 November 2018
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Venue: Seminar Room "-1" – Via Sommarive, 14 Povo - Trento
Hour: 3.00 p.m.

  • Mattia Vedovato - PhD in Mathematics

Abstract:
Regularity properties for minimizing harmonic maps between Riemannian manifolds have been known since the classical work of Schoen and Uhlenbeck (1982); here an estimate on the Hausdorff dimension of the singular set S(u) of a map u is given: dim(S(u))≤n-3, where n is the dimension of the domain manifold.  In this seminar, we are looking deeper into some more recent quantitative results, which describe precisely the structure of S(u). As a starting point, we are recalling the main tools introduced in the work of Cheeger and Naber (2013), which allow to extend the previous estimate to the Minkowski dimension of S(u): in particular, we are stratifying the singular set according to how close the map u is, at any point of S(u), to be invariant with respect to an affine plane. New techniques are then used to improve the aforesaid estimate: following the work of Naber and Valtorta (2017), we obtain an upper bound on the (n-3)-dimensional Minkowski content of S(u). Furthermore, by the use of a suitable version of the Reifenberg Theorem, we manage to show that the above defined singular strata are rectifiable, thus gaining very powerful information on the structure of the singular set.

Supervisor: 
Francesco Serra Cassano