Structure almost everywhere of varifolds with locally bounded first variation and uniform lower density bound

Year of origin: 2009

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

Cite as: G-181109.1

  • Differential Geometry
  • Analysis of PDEs

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Problem's Description

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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  1. Open Optimal decay rates almost everywhere either for subquadratic tilt-excess of integral varifolds or for arbitrary tilt-excess of non-integral varifolds

  2. Open Second order rectifiability of non-integral varifolds

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