Upper and lower bounds on the filling radius (Q6631918)

The notion of filling radius of Riemannian manifolds was introduced by \textit{M. Gromov} in an ample article, [J. Differ. Geom. 18, 1--147 (1983; Zbl 0515.53037)]. In the same year, with the same theme and in the same journal [J. Differ. Geom. 18, 505--511 (1983; Zbl 0529.53032)], an article by \textit{M. Katz} appeared, and then by this author, another related article in [\textit{M. Katz}, Topology Appl. 42, No. 3, 201--215 (1991; Zbl 0749.53032)]. In the same period, an article by \textit{F. H. Wilhelm jun.} [Invent. Math. 107, No. 3, 653--668 (1992; Zbl 0729.53044)] appeared. After 2000, the subject returned to the attention of some authors: \textit{L. Liu} [Differ. Geom. Appl. 22, No. 1, 69--79 (2005; Zbl 1073.53056)], \textit{L. Guth} [Geom. Dedicata 123, 113--129 (2006; Zbl 1125.53028)]. And more recently, \textit{T. Yokota} [Math. Z. 273, No. 1--2, 161--171 (2013; Zbl 1266.53045)], and very recently, \textit{A. Nabutovsky} et al. [Geom. Funct. Anal. 31, No. 3, 721--766 (2021; Zbl 1486.53054)], and obviously the present article. \N\NLet \(M\) be a closed connected \(n\)-manifold with a random measure \(\mu\) and a distance function such that \(\operatorname{dist}_p(.)\in L^{\infty}(M)\). Then the Kuratowski embedding, \(\Phi:M\rightarrow L^{\infty}(M)\), \(\Phi(p)=\operatorname{dist}_p\), is an isometric embdedding since \(\operatorname{dist}_M(p,q)=\|\operatorname{dist}_ p-\operatorname{dist}_q\|_{L^\infty(M)}\). For \(r>0\) denote by \(U_r(M)\) the \(r\)-neighborhood of \(\Phi(M)\) in \(L^{\infty}(M)\) and by \(\iota:M\rightarrow U_r(M)\) the inclusion. For a given coefficient group \(\mathbb{F}\), the homology induced by \(\iota_r\) is denoted \(\iota_{r,\ast}:H_n(M,\mathbb{F})\rightarrow H_n(U_r(M),\mathbb{F})\). Then assume that \(M\) is equipped with a Riemannian metric inducing in \(M\) a distance and a Riemannian volume. Regarding \(\mathbb{F}\), this is considered \(\mathbb{F}=\mathbb{Z}\) when \(M\) is orientable and \(\mathbb{F}=\mathbb{Z}_2\) otherwise.\N\NDefinition (Gromov): The filling radius of \(M\) , denoted by \(\mathrm{FillRad}(M)\), is the infimum of those \(r>0\) for which \(i_{r,\ast}([M])=0\) , where \([M]\) is the fundamental class of \(M\). ``Gromov introduced the filling radius in [\textit{M. Gromov}, J. Differ. Geom. 18, 1--147 (1983; Zbl 0515.53037)] as a tool to obtain his systolic inequality, proving along the way that \(\mathrm{FillRad}(M)\) is bounded above by the \(n\)-th root of the volume times a constant depending only on the dimension of the manifold. Upper bounds of the filling radius in terms of the diameter were obtained by Katz in [\textit{M. Katz}, J. Differ. Geom. 18, 505--511 (1983; Zbl 0529.53032)]; Wilhelm studied it for manifolds with positive sectional curvature in [\textit{F. H. Wilhelm jun.}, Invent. Math. 107, No. 3, 653--668 (1992; Zbl 0729.53044)]. Later works include [\textit{L. Liu}, Differ. Geom. Appl. 22, No. 1, 69--79 (2005; Zbl 1073.53056)], [\textit{L. Guth}, Geom. Dedicata 123, 113--129 (2006; Zbl 1125.53028)], [\textit{A. Nabutovsky} et al., Geom. Funct. Anal. 31, No. 3, 721--766 (2021; Zbl 1486.53054)], [\textit{S. Lim} et al., Algebr. Geom. Topol. 24, No. 2, 1019--1100 (2024; Zbl 1545.55006)]. But this should not be considered an extaustive list''. \N\NFrom the abstract: ``We give a curvature dependent lower bound for the filling radius of all closed Riemannian manifolds as well as an upper one for manifolds which are total space of a Riemannian submanifolds. The latter applies also the case submetries. We also see that the reach (in the sense of Federer) of the image of the Kuratowski embedding vanishes, and we finish by giving some inequalities involving the \(k\)-intermediate filling radius''. In relation to those summarized in the abstract, the authors prove four theorems and a proposition. In order to help the reader of this review, we only give the corollaries of these.\N\NCorollary 1.3. Let \(M\) be a closed Riemannian manifold. Then \(\mathrm{FillRad}IM)\) is strictly positive.\N\NCorollary 1.5. Let \(\pi:M\rightarrow B\) be a Riemannian submersion between closed manifolds. Then \(\frac{1}{2}\operatorname{min}\{\operatorname{inj} M,\frac{\pi}{\sqrt{K}}\}\leq \max_{b\in B}\{\operatorname{diam} \pi^{-1}(b)\}\), where \(\operatorname{inj} M\) is the injective radius, \(K\) is an upper positive bound for the sectional curvature of \(M\), and once again, the diameter of each fiber is considered in the extrensic metric.\N\NCorollary 3.1. For \(B,F\) closed Riemanninan manifolds, let \(f:B\rightarrow (0,\infty)\) be a smooth function, and \(M=B\times_f F\) the warped product over \(B\) with fiber \(F\). Then\N\N\[\mathrm{FillRad}(M)\leq \min\{\mathrm{FillRad}(B),\frac{1}{2}\max f.\operatorname{diam} F\}.\]\N\NCorollary 4.1.Let \((X,\widehat{\operatorname{diam}}_X)\) be a metric manifold (i.e., a closed manifold with a distance), \((Y,\operatorname{dist}_Y)\) a metric space and \(\pi:X\rightarrow Y \) a submetry between them. Then\N\N\[\mathrm{FillRad}(X)\leq \frac{1}{2} \max_{y\in Y}\{\operatorname{diam} \pi^{-1}(y)\}.\]\N\NCorollary 4.2. Suppose \(M\) is a Riemannian manifold admitting a singular Riemannian foliation \(\mathfrak{F}\) with closed leaves. Then \(\mathrm{FillRad}(M)\leq \frac{1}{2}\max_{N\in \mathfrak{F}}\{\operatorname{diam} N\}\).\N\NDefinition 6.1 (Intermediate Fillig Radius): For any integer \(k\geq 1\), any abelian group \(\mathbb{F}\), and any homology class \(\omega\in H_k(N;\mathbb{}F)\), we define the \(k\)-intermediate filling radius of \(\omega\) as \(\mathrm{FillRad}_k(M,F,\omega)=inf\{r>0;\iota_{r,\ast}(\omega)=0\}\), where \(\iota_r:M\hookrightarrow U_r(M)\) is the isometric embedding. This gives us a map \(\mathrm{FillRad}_k(M,\mathbb{F},.):H_k(M;\mathbb{F})\rightarrow \mathbb{R}_{\geq 0}\).\N\NProposition 6.2. For \(M\) a closed Riemannian manifold with injectivity radius \(\operatorname{inj} M\) and with sectional curvature \(sec\leq K\), where \(K\geq 0\), we have \(\mathrm{FillRad}_k(M)\geq \frac{1}{4}\{\operatorname{inj} M,\frac{\pi}{\sqrt{K}}\}\).