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@PHDTHESIS{Wings:1013814,
      author       = {Wings, Axel},
      othercontributors = {von der Mosel, Heiko and Wagner, Alfred},
      title        = {{C}ritical embeddings for tangent-point energies in
                      arbitrary dimensions},
      school       = {RWTH Aachen University},
      type         = {Dissertation},
      address      = {Aachen},
      publisher    = {RWTH Aachen University},
      reportid     = {RWTH-2025-05718},
      pages        = {1 Online-Ressource : Illustrationen},
      year         = {2025},
      note         = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
                      University; Dissertation, RWTH Aachen University, 2025},
      abstract     = {In this thesis, we introduce the generalized tangent-point
                      energy $\mathcal{E}_p^q$ for $\mathcal{C}^1$-embeddings of
                      an abstract smooth manifold $M$ into $\mathbb{R}^n.$ So far,
                      the generalized tangent-point energy $\mathrm{TP}^{(p,q)}$
                      has been studied for a general class of subsets $\Sigma$ of
                      $\mathbb{R}^n.$ We prove that, for
                      $\mathcal{C}^1$-embeddings $e:M\to \mathbb{R}^n,$
                      $\mathcal{E}_p^q(e)=\mathrm{TP}^{(p,q)}(e(M))$ whenever one
                      is finite. First, we study the energy space of
                      $\mathcal{E}_p^q$. To this end, we consider spaces of
                      functions that map from $M$ into $\mathbb{R}^n.$ We
                      introduce a $\mathcal{C}^1(M,\mathbb{R}^n)$ space with a
                      norm depending on one fixed $\mathcal{C}^1$-embedding
                      $e:M\to \mathbb{R}^n$ and a fractional Sobolev space
                      $W^{s,q}(M,\mathbb{R}^n).$ We show that both spaces are
                      Banach spaces and that the fractional Sobolev space is a
                      subspace of the $\mathcal{C}^1$ space. This allows us to
                      characterize the energy space in terms of this fractional
                      Sobolev regularity. On the one hand, the generalized
                      tangent-point energy is finite for each $W^{s,q}$-embedding
                      of $M$ into $\mathbb{R}^n.$ On the other hand, for each
                      $\mathcal{C}^1$-embedding $e:M\to \mathbb{R}^n$ with finite
                      generalized tangent-point energy $\mathcal{E}_p^q,$ there is
                      a $W^{s,q}$-embedding $\hat{e}:M\to \mathbb{R}^n$ satisfying
                      $e(M)=\hat{e}(M)$ and, hence,
                      $\mathcal{E}_p^q(e)=\mathcal{E}_p^q(\hat{e}).$Second, we
                      compute the Fréchet derivative of the generalized
                      tangent-point energy $\mathcal{E}_p^q$ on the open set of
                      $W^{s,q}$-embeddings and prove that the Fréchet derivative
                      is even continuous, i.e., the generalized tangent-point
                      energy is $\mathcal{C}^1.$ Consequently, we have a
                      $\mathcal{C}^1$ functional on an open subset of a Banach
                      space. This situation allows us to define critical points in
                      the usual manner as points at which the derivative
                      vanishes.Next, we define the scale-invariant generalized
                      tangent-point energy $\mathcal{SE}_p^q$ and transfer the
                      above results to this version, i.e., it has the same energy
                      space and is also continuously Fréchet differentiable on
                      the same open subset of the Banach space
                      $W^{s,q}(M,\mathbb{R}^n).$Now, we turn to the
                      scale-invariant tangent-point energy with only one parameter
                      $\mathcal{SE}_{2q}^{q},$ which compares to the classical
                      tangent-point energy $\mathrm{TP}_q.$ In this special case,
                      we investigate the energy landscape. We establish the
                      existence of global minimizers. Then, we show the existence
                      of minimizers within non-empty symmetry classes, which,
                      thanks to Palais' principle of symmetric criticality, turn
                      out to be critical points of $\mathcal{SE}_{2q}^q.$Finally,
                      we focus on embeddings of a $2$-dimensional manifold into
                      $\mathbb{R}^3,$ i.e., on surfaces. We use a symmetry
                      argument to conclude that, for each $\mathfrak{g}\ge 4,$
                      there are several distinct surfaces of given genus
                      $\mathfrak{g}$ that are critical for
                      $\mathcal{SE}_{2q}^{q}.$ More precisely, we get at least $3$
                      critical surfaces of given genus $\mathfrak{g}$ if
                      $\mathfrak{g}\ge 4$ and $\mathfrak{g}\notin \{5,7,11\},$ and
                      we get at least $2$ if $\mathfrak{g}\in \{5,7,11\}.$},
      cin          = {111810 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)111810_20140620$ / $I:(DE-82)110000_20140620$},
      pnm          = {GRK 2326 - GRK 2326: Energie, Entropie und Dissipative
                      Dynamik (320021702)},
      pid          = {G:(GEPRIS)320021702},
      typ          = {PUB:(DE-HGF)11},
      doi          = {10.18154/RWTH-2025-05718},
      url          = {https://publications.rwth-aachen.de/record/1013814},
}