Geometric Sobolev-like embedding using high-dimensional Menger-like curvature

Abstract:

We study a modified version of Lerman-Whitehouse Menger-like curvature defined for $(m+2)$ points in an $n$-dimensional Euclidean space. For $1 \le l \le m+2$ and an $m$-dimensional set $\Sigma \subset R^n$, we also introduce global versions of this discrete curvature by taking the supremum with respect to $(m+2-l)$ points on $\Sigma$. We then define geometric curvature energies by integrating one of the global Menger-like curvatures, raised to a certain power $p$, over all $l$-tuples of points on $\Sigma$. Next, we prove that if $\Sigma$ is compact and $m$-Ahlfors regular and if $p$ is greater than the dimension of the set of all $l$-tuples of points on $\Sigma$ (i.e. $p > ml$), then the P. Jones’ $\beta$-numbers of $\Sigma$ must decay as $r^{\tau }$ with $r \to 0$ for some $\tau \in (0,1)$. If $\Sigma$ is an immersed $C^1$ manifold or a bilipschitz image of such a set then, it follows that it is Reifenberg flat with vanishing constant; hence (by a theorem of David, Kenig and Toro) an embedded $C^{1,\tau }$ manifold. We also define a wide class of other sets for which this assertion is true. After that, we bootstrap the exponent $\tau$ to $\alpha = 1 - ml/p$, which is optimal due to our theorem with S. Blatt [Adv. Math., 2012]. This gives an analogue of the Morrey-Sobolev embedding theorem $W^{2,p}(\mathbb {R}^{ml}) \subseteq C^{1,\alpha }(\mathbb {R}^{ml})$ but, more importantly, we also obtain a qualitative control over the local graph representations of $\Sigma$ only in terms of the energy.