Optimal decay rates almost everywhere either for subquadratic tilt-excess of integral varifolds or for arbitrary tilt-excess of non-integral varifolds
OpenYear of origin: 2009
Posted online: 2018-11-09 07:10:34Z by Ulrich Menne99
Cite as: P-181109.1
General Description View the group
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.
Problem's Description
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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