An \(\ell_1\)-norm-mass inequality for complete manifolds (Q6567981)

The authors prove an inequality that compares the \(\ell_1\)-norm and the mass of homology classes of complete Riemannian manifolds. The critical exponent of the fundamental group is involved in the inequality.\N\NFor a manifold \(M\) and a homology class \(\alpha\in H_k(M,\mathbb{R})\), the Gromov norm of \(\alpha\) (denoted by \(\|\alpha\|_1\)) measures the minimal \(\ell_1\)-norm of cycles realizing \(\alpha\). In [Publ. Math., Inst. Hautes Étud. Sci. 56, 5--99 (1982; Zbl 0516.53046)], \textit{M. Gromov} proved the following results for a complete connected Riemannian \(n\)-manifold \(M\) with Ricci curvature bounded below by \(-(n-1)\):\N\begin{itemize}\N\item[(1)] If \(M\) is closed and oriented, then \(\|[M]\|_1\leq (n-1)^nn!\cdot \text{vol}(M)\).\N\item[(2)] For any \(\alpha\in H_k(M,\mathbb{R})\), \(\|\alpha\|_1\leq (n-1)^kk!\cdot \text{mass}(\alpha).\)\N\end{itemize}\NHere \(\text{mass}(\alpha)\) denotes the supremum of pairings of \(\alpha\) and closed differential \(k\)-forms with a certain norm bounded by \(1\).\N\NIn [Invent. Math. 103, No. 2, 417--445 (1991; Zbl 0723.53029)], \textit{G. Besson} et al. strengthened item (1) to \(\|[M]\|_1\leq \frac{(n-1)^nn!}{n^{n/2}}\text{vol}(M)\), under the assumption that \(M\) has sectional curvature \(|K|\leq 1\). The authors strengthen item (2) to \N\[\N\|\alpha\|_1\leq \frac{\delta^kk!}{k^{k/2}}\cdot \text{mass}(\alpha)\N\]\Nfor any \(\alpha\in H_k(M,\mathbb{R})\), where \(\delta\) denotes the critical exponent of \(M\) (Theorem 1.5 of this paper).\N\NTo prove Theorem 1.5, the authors first prove an inequality between the \(\ell_{\infty}\)-norm and the comass of cohomology classes of \(M\), by choosing a particular family of smoothing operators \(\widetilde{M}\to \mathcal{M}_1\) (the space of probability measures on \(\widetilde{M}\)) depending on the critical exponent of \(M\). Then Theorem 1.5 is proved by (more or less) dualizing the previous proof.\N\NIn Section 2, the authors give a few applications of Theorem 1.5.