Scale-invariant geometric curvature functionals, and characterization of Lipschitz- and $C^1$-submanifolds

Käfer, Bastian; von der Mosel, Heiko (Thesis advisor); Wagner, Alfred (Thesis advisor); Strzelecki, Pawel (Thesis advisor)

Aachen : RWTH Aachen University (2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

In this thesis, we investigate the connection of local flatness and the existence of graph representations of certain regularity for subsets of $\mathbb R^n$ with arbitrary dimension $m\leq n$. In this process, we formulate sufficient conditions providing local graph representations of class $C^{0,1}$ and $C^1$. We identify sets satisfying those local representations at each point as Lipschitz- and $C^1$-submanifolds, respectively. Based on the concept of $\delta$-Reifenberg-flat sets, we introduce a characterization for the class of $m$-dimensional $C^1$-submanifolds of $\mathbb R^n$. We apply the gained information in the study of two families of geometric curvature functionals for different classes of $m$-dimensional admissible sets. Reifenberg-flatness remains to be a crucial tool to achieve additional topological and analytical properties assuming finite energy. The first class of functionals is given by the tangent-point energies $TP^{(k,l)}$ with focus on the scale-invariant case $k=l+2m$. We prove that admissible sets with locally finite energy are embedded submanifolds of $\mathbb R^n$ with local graph representations satisfying Lipschitz continuity. In a second step, using a technique of S. Blatt, we characterize the energy space of $TP^{(k,l)}$ for all $l>m$ and $k\in[l+2m,2l+m)$ as submanifolds of class $C^{0,1}\cap W^{\frac {k-m}l,l}$. In contrast to the first step, the proof of this characterization requires a priori given graph representations by Lipschitz functions in order to guarantee the existence of tangent planes for $\mathscr{H}^m$-almost all points in the computation of the tangent-point energy. Following the work of R. B. Kusner and J. M. Sullivan, we then define a family $\mathcal{E}^\tau$ of Möbius-invariant energies for $m$-dimensional subsets of $\mathbb R^n$. As for $TP^{(k,l)}$, locally finite $\mathcal{E}^\tau$-energy for admissible sets provides local graph representations satisfying Lipschitz continuity. We also prove that for $\tau>0$, each locally compact $C^{0,1}\cap W^{1+\frac 1{1+\tau},(1+\tau)m}$-submanifold of $\mathbb R^n$ has locally finite $\mathcal{E}^\tau$-energy.