We are notationally consistent with [16] and largely also with
[8] and [1]. A brief introductory survey on the
present topic is provided in [17].
Hypotheses We suppose $m$ and $n$ are positive integers, $m < n$,
$1 \leq p \leq \infty$, $U$ is an open subset of $\mathbf R^n$, $V$ is an $m$
dimensional varifold in $U$, the first variation $\delta V$ of $V$ is
representable by integration (equivalently, $\| \delta V \|$ is a Radon
measure), the $m$ dimensional density of the weight $\| V \|$ of $V$ satisfies
the uniform lower bound $\boldsymbol \Theta^m ( \| V \|, x ) \geq 1$ for $\| V
\|$ almost all $x$, and, in case $p>1$, we suppose additionally that $\|
\delta V \|$ is absolutely continuous with respect to $\| V \|$ and that the
generalised mean curvature vector $\mathbf h (V,\cdot)$ of $V$ belongs to
$\mathbf L_p^{\mathrm{loc}} ( \| V \|, \mathbf R^n )$.
Context This group of questions concerns various aspects of the
regularising effect of these hypotheses near $V $ almost all points. Since
the support of $\| V \|$ may fail to be an $m$ dimensional submanifold of
$\mathbf R^n$ near a set of points of positive $\| V \|$ measure (see 8.1 (2) in
[1]), weaker measures of regularity need to be studied.
Most frequently, those are decay rates of tilt-excess or height-excess,
rectifiability of second order, and summability of the approximate second
fundamental form.
Level of difficulty Most questions in this group have already
been solved in the special case of integral varifolds; those that are open
even in the integral case appear to require new methods. On the other hand,
the study of regularity of non-integral varifolds (after Section 8 in
[1], [6], and [7]) has not kept up with the
subsequent developments of the integral case; hence, adaptations may be
feasible.
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