Second order rectifiability of non-integral varifolds

OpenYear of origin: 2009

Posted online: 2018-11-09 07:10:34Z by Ulrich Menne69

Cite as: P-181109.2

  • Differential Geometry
  • Analysis of PDEs
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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

Definitions Suppose $A = U \cap \{ x \mathop : \boldsymbol \Theta^m ( \| V \|, x ) \geq 1 \}$, $X$ is the set of $x \in U$ such that $\mathrm{Tan}^m ( \| V \|, x )$ is an $m$ dimensional plane, $Y$ is the vectorspace of symmetric endomorphisms of $\mathbf R^n$, the tangent plane map $\tau : X \to Y$ is defined by requiring that $\tau (x) \in Y$ is the canonical orthogonal projection of $\mathbf R^n$ onto $\mathrm{Tan}^m ( \| V \|, x )$ for $x \in X$, $\Theta = \boldsymbol \Theta^m ( \| V \|, \cdot )$, and the trace $T : \mathrm{Hom} ( \mathbf R^n, Y) \to \mathbf R^n$ satisfies $T ( g ) = \sum_{i=1}^n g(u_i)(u_i)$ whenever $g \in \mathrm{Hom} ( \mathbf R^n, Y )$ and $u_1, \ldots, u_n$ form an orthonormal basis of $\mathbf R^n$.

Questions In (1) below, we employ the notion of approximate pointwise differentiability of higher order defined in 3.8 in [20].

(1) Is the set $A$ approximately differentiable of order $(2,0)$ at $\mathcal H^m$ almost all $x \in A$?

(2) Are the tangent plane map $\tau$ and the density $\Theta$ approximately differentiable with respect to $(\| V \|, m)$ at $\| V \|$ almost all $x$ and, if so, does there necessarily hold, for $\| V \|$ almost all $x$, \begin{equation*} \mathbf h (V,x) \bullet u = T ( \mathrm{ap} \, \mathrm D \tau (x) \circ \tau (x)) \bullet u + ( \mathrm{ap} \, \mathrm D ( \mathrm{log} \circ \Theta ) \circ \tau (x)) (u) \quad \text{for $u \in \mathbf R^n$}? \end{equation*}

(3) Suppose $0 < q < \infty$. If $\tau$ is $(\| V \|,m)$ approximately differentiable at $\| V \|$ almost all $x$, does there hold, for $\| V \|$ almost all $x$, \begin{equation*} {\textstyle\int_{\mathbf B(x,r)}} \| \mathrm{ap} \, \mathrm{D} \, \tau \|^q \, \mathrm d \| V \| < \infty \quad \text{for some $0 < r < \infty$}? \end{equation*}

(1) was originally formulated (in an equivalent fashion) for $p \geq m$ on p. 5 in [19]. (2) is taken from Question 3 on p. 5 in [18]. (3) was inspired by a result in the theory of viscosity solutions (see Proposition 7.4 in [4]) and was posed for integral varifolds with $m \geq 2$ and $p = \infty$, in Question 2 on p. 5 in [10].

Remarks By 3.23 and 5.6 in [20], the condition in (1) is equivalent to the existence of a countable collection of $m$ dimensional submanifolds of class $2$ of $\mathbf R^n$ whose union covers $\| V \|$ almost all of $U$. Clearly, if (1) has an answer in the affirmative, then $\tau$ is $(\| V \|, m)$ approximately differentiable at $\| V\|$ almost all $x$. The converse is not obvious by the fact from [9] described in the previous problem. If $V$ corresponds to a pair consisting of an $m$ dimensional submanifold of class $2$ and a positive density function of class $1$ thereon, then the equation in (2) holds. Even for integral varifolds and $p = \infty$, the possible approximate differentiability of $\Theta$ considered in (2) may, for $m \geq 2$, not be strengthened to differentiability in an integral sense (such a notion for functions on varifolds is considered in 3.9 in [12]) by 10.8 in [10]. In relation to (3), we notice that simple examples show that a stationary integral varifold need not to be a curvature varifold (see 3.11 in [11]).

Known cases In case $V$ is integral, $\Theta$ is evidently $(\| V \|, m )$ approximately differentiable at $\| V \|$ almost all $x$ with $\mathrm{ap} \, \mathrm{D} \, \Theta (x) =0$. Therefore, it follows from the second order rectifiability result in 4.8 in [15] that the answers to (1) and (2) are in the affirmative in the special case of integral varifolds. For non-integral varifolds, the answer to (1) is in the affirmative in case $p = \infty$ and $U = \mathbf R^n$; in fact, this follows combining 6.10 in [19] and 2.3 in [21] see the footnote below. Moreover, it is noted on p. 5 in [19] that 6.10 in [19] is tailored for a possible extension to the case $p = m$. For $p < m$, a strategy more in line with the case of integral varifolds seems more promising for both (1) and (2). Finally, (3) was answered in the negative for $q > 1$ in 10.5 in [10]. No positive results are known for (3) even for integral varifolds.

Relation to tilt-excess decay If the decay condition of the preceding problem is satisfied by some $V$ with $q=\alpha = 1$, then $\tau$ is $\| V \|$ almost everywhere differentiable both in an approximate and an integral sense by 3.7 (i) and 3.9 in [12].

Footnote As, by 3.5 (1) and 8.3 in [1] and 11.3 in [16], the condition $p \geq m$ implies $\mathcal H^m \mathop{\llcorner} A \leq \| V \|$ and $\mathrm{Tan} (A,a) \in \mathbf G(n,m)$ for $\mathcal H^m$ almost all $a \in A$, in order to apply 6.10 in [19], one records, in addition to the assertion of 2.3 in [21], that, locally, $A$ is an $(m,h)$ set, for some $0 \leq h < \infty$, and that $A$ has the $m$ dimensional Lusin ($N$) property from the proof of 2.3 in [21].

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