Centers of disks in Riemannian manifolds (Q2305497)

If \(G\) is a subgroup of the isometry group of a smooth connected Riemannian manifold \(M\), \(\mathcal{X}(M)\) is the space of compact connected \(C^1\) embedded submanifolds of \(M\) equipped with the \(C^1\) topology, and \(X\) is a subspace of \(\mathcal{X}(M)\), then a continuous map \(\mathfrak{c}:X\to M\) such that \(\mathfrak{c}(D)\) lies in the relative interior of \(D\) for every \(D\in X\) is called a center. If, in addition, \(X\) is \(G\)-invariant and \(\mathfrak{c}(gD)=g\mathfrak{c}(D)\) for all \(D\in X\), \(g\in G\), then \(\mathfrak{c}\) is said to be \(G\)-equivariant. If \(G_D\) is the subgroup of \(G\) which maps \(D\) to itself, and \(D^{G_D}\) is the set of points of \(D\) which are fixed by \(G_D\) for any \(D\in\mathcal{X}(M)\), and \(\mathfrak{c}\) is \(G\)-equivariant, then \(\mathfrak{c}(D)\in D^{G_D}\) and, in particular, no \(G\)-equivariant center exists if \(D^{G_D}=\varnothing\) for some \(D\in X\). In this paper, the authors investigate the existence of \(\mathfrak{c}\) for a given triple \((M, G,X)\). The examples of a center are the center of mass and the Steiner point, which are called the classical centers. They are \(\mathrm{Iso}(\mathbb{R}^n)\)-equivariant centers on the space of convex compacta in \(\mathcal{X}(\mathbb{R}^n)\), where \(\mathrm{Iso}(\mathbb{R}^n)\) is the group of Euclidean isometries. The main result of the paper states that the space of submanifolds \(D\in\mathcal{X}(M)\) such that \(D^{G_D}\) is contractible admits a \(G\)-equivariant center. Furthermore, this center can be chosen to agree with any given \(G\)-equivariant center defined on a closed subset \(A\) of \(\mathcal{X}(M)\). Next, the authors consider some natural conditions under which \(D^{G_D}\) is contractible. If \(G\) is trivial, then \(D^G=D\) and the space of contractible submanifolds \(D\in\mathcal{X}(M)\) admits a center. If \(\mathfrak{D}^k(M)\) is the subspace of \(D\in\mathcal{X}(M)\) such that \(D\) is homeomorphic to a disk of dimension \(\le k\), then the authors show that \(\mathfrak{D}^3(M)\) admits a \(G\)-equivariant center, and if \(\dim(M)=4\) and the \(G\)-action on \(M\) is orientation-preserving, then \(\mathfrak{D}^4(M)\) admits a \(G\)-equivariant center. A Hadamard manifold is a contractible complete Riemannian manifold of nonpositive sectional curvature, and the fixed point set of any isometric action on a Hadamard manifold is diffeomorphic to a Euclidean space. The authors show that, if either \(M=S^4\) with its standard action of \(G=O(5)\), or \(M\) is a Hadamard 4-manifold with isometry group \(G\), then \(\mathfrak{D}^4(M)\) admits a \(G\)-equivariant center. Furthermore, it is shown that, for any \(m, n\) such that either \(m\ge n+r-6\) or \(m\ge n\ge r\), there is an orthogonal \(A_5\)-action on \(\mathbb{R}^m\) that preserves a smoothly embedded \(n\)-disk on which \(A_5\) acts without a fixed point, where \(A_5\) is the alternating group on \(S^n\). In particular, \(\mathfrak{D}^n(\mathbb{R}^m)\setminus\mathfrak{D}^{n-1}(\mathbb{R}^m)\) does not admit an \(O(m)\)-equivariant center. Finally, the authors summarize their results for the Euclidean space in the statement that any \(\mathrm{Iso}(\mathbb{R}^n)\)-equivariant center on the set of convex compacta in \(\mathbb{R}^n\) extends to a center \(\mathfrak{c}:\mathfrak{D}^n(\mathbb{R}^n)\to\mathbb{R}^n\) that is \(\mathrm{Iso}(\mathbb{R}^n)\)-equivariant on \(\mathfrak{D}^3(\mathbb{R}^n)\). Moreover, if \(n=4\), then \(\mathfrak{c}\) is \(\mathrm{Iso}(\mathbb{R}^4)\)-equivariant on \(\mathfrak{D}^4(\mathbb{R}^4)\). If \(k\) is suficiently large, then \(\mathfrak{D}^k(\mathbb{R}^n)\) does not admit an \(\mathrm{Iso}(\mathbb{R}^n)\)-equivariant center.