Growth of a primitive of a differential form (Q2773543)

Given a complete, non-compact manifold \(M\), an exact \(q\) form \(\omega\), and a continuous function \(f:M\to \mathbb R_+\), what are sufficient conditions for the existence of a primitive \(q-1\) form \(\tau\) with \(|\tau|\leq f\)? Clearly, from Stokes theorem, we must have NEWLINE\[NEWLINE\left|\int_T \omega\right|\leq \int_{|\partial T|} fNEWLINE\]NEWLINE for all singular, \(\mathcal C^1\), \(q\)-chains \(T\). The author proves the following theorem: Let \(M\) be a Riemannian manifold, with a triangulation \(K\) of bounded geometry. Let \(\omega \in \Omega^q(M)\) be a closed \(q\)-form, and let \(f \in C^0(M,\mathbb R_+)\) be such that NEWLINE\[NEWLINE \left|\int_T \omega\right|\leq \int_{|\partial T|} fNEWLINE\]NEWLINE for every simplicial chain \(T \in C_q(K)\). Then \(\omega\) has a primitive \(\tau\) such that for some constants \(C_1(M,K)\) and \(C_2(M,K)\), one has NEWLINE\[NEWLINE|\tau|_x \leq C_1 \max_{B(x,C_2)}\left(|\omega|+f\right).NEWLINE\]NEWLINE The proof follows a partial unpublished proof due to R. Souam, and constructions by \textit{F. Laudenbach} [Bull. Soc. Math. Fr. 111, 147-150 (1983; Zbl 0524.58003)] and \textit{I. Singer} and \textit{J. Thorpe} [Lectures on elementary topology and geometry, Springer-Verlag, Heidelberg (1976; Zbl 0342.54002)]. Using this, together with a recent result of \textit{A. Żuk} [Topology 39, No. 5, 947-956 (2000; Zbl 0988.20028)], the author obtains the following: NEWLINENEWLINENEWLINECorollary. Let \(X\) be a compact oriented Riemannian manifold with infinite fundamental group. Then the volume form on the universal covering \(M=\widetilde{X}\) has a primitive \(\tau\) with at most linear growth.