Geometry and Geometric Analysis Working Group

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Regularty and compactness for prescribed-mean-curvature hypersurfaces.

Speaker: Neshan Wickramasekera, University of Cambridge

Location: Warren Weaver Hall 101

Date: Tuesday, April 9, 2019, 3:30 p.m.

Synopsis:

In the period 1960-1980, there were two major developments in 
the regularity theory of minimal hypersurfaces: the first is the 
regularity theory for locally area minimizing hypersurfaces (due to De Giorgi, 
Federer and Simons) and the other is the compactness theory for locally 
uniformly area bounded stable hypersurfaces 
with small singular sets (due to Schoen--Simon--Yau in low dimensions and 
Schoen--Simon in general dimensions). Both these theories played 
a key role in the Almgren--Pitts geometric approach to the existence of 
embedded minimal hypersurfaces in compact Riemannian manifolds, 
also developed in the same period.

In a series of works in the past few years (starting with the speaker's 
2014 work and continued in joint work with C. Bellettini), these two 
theories have been unified and extended, providing a sharp embeddedness 
and compactness criterion for locally uniformly area bounded weakly 
stable prescribed-mean-curvature hypersurfaces (integral varifolds). This 
work includes minimal and constant mean curvature (CMC) hypersurfaces as 
important special cases, and is a sharpening of the previous work even in 
the minimal case in that it requires no apriori assumption on the size of 
the sigular set. With less to check on the singular set, the work has 
lead to a considerably simpler PDE alternative to the Almgren--Pitts 
theory (due to the combined work of Guraco, Hutchinson, Tonegawa and the 
speaker). We will discuss these recent developments and the central ideas 
behind them.