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We study singular surfaces (viz. \(2\)-dimensional integral varifolds)
in \(\mathbf{R}^n\) satisfying additional hypotheses on the
generalised mean curvature. A classical result [Almgren, unpublished
(1965)] states that the class of all \(m\)-dimensional integral
varifolds in \(\mathbf{R}^n\) having locally uniformly bounded mass
and first variation is compact. This makes varifolds a natural
object of study in the calculus of variations. It has also been long
known [Allard, Ann. of Math. (1972)] that if the generalised mean
curvature of a varifold \(V\) is in \(L^p\) for some \(p > m\), then
a relatively open and dense subset of the support of \(V\) is an
embedded \(C^{1,\alpha}\) manifold (\(\alpha = 1 - m/p\)). More recently
[Wickramasekera, Ann. of Math. (2014)] it was proven that if a
codimension \(1\) integral varifold is stationary (mean curvature is
zero) and stable (second variation is nonnegative) and no tangent
cone consists of three or more half-hyperplanes meeting along a
common codimension \(2\) vector spaces, then the support of the
varifold is a smooth hypersurface outside a set of codimension at
least \(8\) (\(\dim V - 7\)).
In all the regularity results concerning varifolds a crucial role is
played by various estimates on the tilt-excess, i.e. mean
deviation of the tangent plane to a given plane measured
in \(L^2\). They turned out to be useful also for proving
perpendicularity of the mean curvature vector for integral varifolds
[Brakke, Math. Notes, (1978)], locality of mean curvature
[Schätzle, J. Differential Geom.(2009)] as well as
\(C^2\)-rectifiability of integral varifolds whose first variation is
a Radon measure [Menne, J. Geom. Anal. (2013)]. Besides being used
as an intermediate step in various proofs, the notion of tilt-excess
decay serves itself as weak measure of regularity.
Optimal decay rates are known for \(m\)-dimensional varifolds having
mean curvature in \(L^p\) for the cases \(m > 2\) and \(p \ge 1\) [Menne,
Arch. Ration. Mech. Anal. (2012)] or \(p > m\) and \(p \ge 2\)
[Schätzle, Ann. Sc. Norm. Super. Pisa Cl. Sci. (2004)]. In a
joint work with Menne, we resolve the only remaining case,
i.e. \(m=2\), \(p=1\) and prove sharpness of our result.
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