Critical embeddings for tangent-point energies in arbitrary dimensions

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 $\mathfrak{g}\in \{5,7,11\}.$