Regularity of weakly stable codimension 1 CMC varifolds

The lecture will discuss recent joint work with C. Bellettini and O. Chodosh. The work taken together establishes sharp regularity conclusions, compactness and curvature estimates for any family of codimension 1 integral $n$-varifolds satisfying: (i) locally uniform mass and $L^{p}$ mean curvature bounds for some $p > n;$ (ii) two structural conditions and (iii) two variational hypotheses on the orientable regular parts, namely, stationarity and (weak) stability with respect to the area functional for volume preserving deformations (supported on the regular parts). The work builds on the earlier work for zero mean curvature, strongly stable varifolds and on the fundamental theories of Schoen--Simon--Yau and of Schoen--Simon for strongly stable hypersurfaces with small singular sets. The lecture will focus on how to handle the new difficulties that arise. These stem from the combinatinon of higher multiplicity, lack of a two sided strong maximum principle and the absence of any a priori hypothesis on the size of the singular set.