# 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 Menne60

Cite as: P-181109.1

• Differential Geometry
• Analysis of PDEs

### 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].

1. ## Article On the first variation of a varifold

Annals of Mathematics. Second Series 95, 417-491, 1972fulltext

2. ## Book Almgren's big regularity paper

pp. xvi+955, year of publication: 2000fulltext

3. ## Book The motion of a surface by its mean curvature

pp. i+252, year of publication: 1978fulltext

4. ## Book Fully nonlinear elliptic equations

pp. vi+104, year of publication: 1995fulltext

5. ## Article Regularity of area minimizing currents I: gradient $L^p$ estimates

Geometric And Functional Analysis 24, 1831-1884, 2014fulltext

6. ## Article Regularity theorems for varifolds with mean curvature

Indiana University Mathematics Journal 35, 117-144, 1986fulltext

7. ## Article $W^{2,p}$ regularity for varifolds with mean curvature

Communications in Partial Differential Equations 11, 903-926, 1986fulltext

8. ## Book Geometric measure theory

pp. xiv+676, year of publication: 1969fulltext

9. ## Article An example concerning approximate differentiation

Indiana University Mathematics Journal 26, 393-397, 1977fulltext

10. ## ArticleIs an originDecay rates for the quadratic and super-quadratic tilt-excess of integral varifolds

NoDEA : Nonlinear Differential Equations and Applications 24, Art. 17, 56, 2017fulltext

11. ## Article Curvature varifolds with boundary

Journal of Differential Geometry 43, 807-843, 1996fulltext

12. ## ArticleIs an originSome applications of the isoperimetric inequality for integral varifolds

Advances in Calculus of Variations 2, 247-269, 2009fulltext

13. ## Article A Sobolev Poincaré type inequality for integral varifolds

Calculus of Variations and Partial Differential Equations 38, 369-408, 2010fulltext

14. ## Article Decay estimates for the quadratic tilt-excess of integral varifolds

Archive for Rational Mechanics and Analysis 204, 1-83, 2012fulltext

15. ## Article Second order rectifiability of integral varifolds of locally bounded first variation

Journal of Geometric Analysis 23, 709-763, 2013fulltext

16. ## Article Weakly differentiable functions on varifolds

Indiana University Mathematics Journal 65, 977-1088, 2016fulltext

17. ## Article The concept of varifold

Notices of the American Mathematical Society 64, 1148-1152, 2017fulltext

arXiv

19. ## ArticleIs an originCurvature of closed subsets of Euclidean space and minimal submanifolds of arbitrary codimension

pp. 1-35, year of publication: 2017arXiv

20. ## Article Rectifiability and approximate differentiability of higher order for sets

Indiana University Mathematics Journal, 2017

21. ## Article ABP inequalities for singular submanifolds of bounded mean curvature

pp. 1-14, year of publication: 2018arXiv

22. ## Article Lower semicontinuity of the Willmore functional for currents

Journal of Differential Geometry 81, 437-456, 2009fulltext