Controlling area blow-up in minimal or bounded mean curvature varieties (Q273221)

Let \(M_i\) be a sequence of \(m\)-dimensional minimal varieties in a Riemannian manifold \(\Omega \) or, more generally, varieties with mean curvature bounded by some \(h<\infty\). Let \(Z\) be the set of points at which the areas of \(M_i\) blow up; equivalently, \(Z\) is the smallest closed subset of \(\Omega \) such that the areas of the \(M_i\) are uniformly bounded as \(i\rightarrow \infty \) on compact subsets of \(\Omega \setminus Z\).NEWLINENEWLINENEWLINEFirstly, it is proved that every such set \(Z\) satisfies the following maximum principle: NEWLINENEWLINENEWLINE Theorem 1.1. If \(f:\Omega \rightarrow \mathbb R\) is a \(C^2\) function and if \(f| z\) has a local maximum at \(p\), then \(\mathrm{Trace} _m(D^2f(p))\leq h| Df(p)| \) where \(\mathrm{Trace}_m(D^2f(p))\) is the sum of the \(m\) lowest eigenvalues of the Hessian of \(f\) at \(p\).NEWLINENEWLINENEWLINE A closed set that satisfies the conclusion of Theorem 1.1 is called in the following \textit{an \((m, h)\) set}. It is also proved that any \((m, h)\) set satisfies the same barrier principle that is satisfied by \(m\)-dimensional submanifolds of mean curvature bounded by \(h\):NEWLINENEWLINENEWLINE Theorem 1.2. Let \(\Omega \) be a \(C^1\) Riemannian manifold without boundary and let \(Z\) be an \((m, h)\) subset of \(\Omega \). Let \(N\) be a closed region in \(\Omega \) with smooth boundary such that \(Z\subset N\) and let \(p\in Z\cap \partial N\). Then \(k_1+\dots +k_m\leq h\) where \(k_1\leq\dots \leq k_{n-1}\) are the principal curvatures of \(\partial N\) at \(p\) with respect to the unit normal that points into \(N\).NEWLINENEWLINENEWLINEIn Section 5, these results (specifically, Corollary 1.4) are used to prove Theorem 5.4 that extends Allard's Regularity Theorem in the case of integer-multiplicity varifolds.