OpenYear of origin: 2009

Posted online: 2018-11-09 07:10:34Z by Ulrich Menne65

Cite as: P-181109.1

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.

**Definition** Whenever $a \in \mathbf R^n$, $0 < r < \infty$,
$\mathbf B(a,r) \subset U$, and $T \in \mathbf G(n,m)$, we define
\begin{align*}
\phi_q (a,r,T) & = r^{-m/q} \big ( {\textstyle\int_{\mathbf B(a,r)
\times \mathbf G(n,m)}} | S_\natural - T_\natural |^q \, \mathrm d V
\, (x,S) \big)^{1/q}\ \text{for $1 \leq q < \infty$}, \\
\phi_\infty (a,r,T) & = \inf \big \{ t \mathop : V ( ( \mathbf B(a,r)
\times \mathbf G (n,m) ) \cap \{ (x,S) \mathop : | S_\natural -
T_\natural | > t \} ) = 0 \big \}.
\end{align*}
This quantity is usually referred to as the tilt-excess with exponent $q$ of
the varifold $V$ in $\mathbf B(a,r)$ with respect to $T$.

**Question** Suppose $1 \leq q \leq \infty$ and $0 < \alpha \leq 1$.
Do the hypotheses of this group of problems on $V$ imply that, for $V$ almost
all $(a,T)$, there holds
\begin{equation*}
\limsup_{r \to 0+} r^{-\alpha} \phi_q (a,r,T) < \infty?
\end{equation*}
This question was formulated for the special case of integral varifolds in Problem (ii) on p. 248 in [12] and re-iterated, for the subcase $q
\geq 2$, in Question 3 on p. 5 in [10]. Near points at which the
limit superior is finite, the tangent planes of $V$ may be said to exhibit a
*pointwise Hölder type behaviour with exponent $\alpha$, measured in
Lebesgue spaces with exponent $q$*.

**Known cases** For *integral varifolds*, the state-of-the-art
may be summarised as follows (see the introduction of [10] for
detailed references also to earlier results):

(1) the cases $q=2$ and $q = \infty$ are completely solved (see [12], [15], and [10]); for $m=2$, $p=1$, and $q=2$, even the exact optimal decay rate is known (see [10]);

(2) the case $2 < q < \infty$ is largely solved (see [10]); only the subcase $m=2$, $p = 1$, and $\alpha = 2/q$ remains open;

(3) *the case $q < 2$ is completely open* apart of the
information obtainable by combining the preceding cases with Hölder's
inequality.

*For non-integral varifolds, no positive results are known as yet.* The
negative results for integral varifolds (see [12] and [10])
still apply.

**General context** Decay estimates of the classical tilt-excess, that
is for $q=2$, (see, e.g., 5.5–5.7 in [3] and 10.2 in
[14]) lie at the foundation of many subsequent results for
integral varifolds. This includes perpendicularity of mean curvature (see 5.8 in
[3]), second order rectifiability (see 4.8 in
[15]), and a characterisation of curvature varifolds (see
15.6 in [16]). Estimates of the tilt-excess with $q>2$ play a
central role in the regularity theory for absolutely area-minimising currents
if $n-m>1$, see 3.29 and 3.30 in [2] and its modern reformulation in
Theorem 7.11 in [5]. In both cases, control on the tilt-excess
entails a precise local approximation by multiple-valued functions – a scheme
introduced in 3.1–3.12 in [2].

**Relation to second order rectifiability** In dealing with second
order rectifiability it is important to be aware of the fact that for general
submanifolds of class $1$, first order rectifiability of its tangent plane map
(as, under our hypotheses of $V$, would be entailed tilt-excess decay with
$\alpha=1$ by 3.7 (i) in [12]), does *not* imply second order
rectifiability by [9]. However, due to our conditions on the
first variation, decay of tilt-excess often entails decay of height-excess via
suitable Sobolev-Poincaré inequalities (see 4.11 in [13] and 10.7 in
[16]) whence the corresponding order of rectifiability
follows (see the appendix in [22] or p. 372 in [13]).
Nonetheless, the above-mentioned optimal decay rates for integral varifolds
were instead proven through a "boot-strap" procedure: firstly, non-optimal
decay rates where proven in 5.7 in [3], then second order
rectifiability in 4.8 in [15], and finally optimal rates in
5.2 in [15] and 9.2, 11.1, and 12. 4 in [10].

**Remark** To bring the theory of curvature varifolds up to date, it
would be useful to answer the analogous question for that subclass; a possible
strategy is available on p. 990 in [16].

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Created at: 2018-11-09 07:10:34Z

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