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@PHDTHESIS{Kfer:820789,
      author       = {Käfer, Bastian},
      othercontributors = {von der Mosel, Heiko and Wagner, Alfred and Strzelecki,
                          Pawel},
      title        = {{S}cale-invariant geometric curvature functionals, and
                      characterization of {L}ipschitz- and ${C}^1$-submanifolds},
      school       = {RWTH Aachen University},
      type         = {Dissertation},
      address      = {Aachen},
      publisher    = {RWTH Aachen University},
      reportid     = {RWTH-2021-05824},
      pages        = {1 Online-Ressource : Illustrationen},
      year         = {2021},
      note         = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
                      University; 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.},
      cin          = {112120 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)112120_20140620$ / $I:(DE-82)110000_20140620$},
      typ          = {PUB:(DE-HGF)11},
      doi          = {10.18154/RWTH-2021-05824},
      url          = {https://publications.rwth-aachen.de/record/820789},
}