On systolic growth-type (Q1917359)

Let \(f,g:\mathbb{R}^+\to \mathbb{R}^+\) be two not necessarily continuous functions. The function \(g\) is said to be dominated by \(f\) if \(f(t)\geq \alpha g(\beta t+\gamma)\) for certain constants \(\alpha> 0\), \(\beta>0\) and \(\gamma\). Then \(f\) and \(g\) are said to be of the same growth type if \(g\) is dominated by \(f\) and vice versa. The growth type of a complete Riemannian manifold is defined to be the growth type of the volume function \(v_p(r)=\text{vol}(B(p;r))\), where \(B(p;r)\) denotes the closed geodesic ball of radius \(r\) centered at \(p\). In this paper, the author introduces the \((n-1)\)-dimensional systolic growth type of an \(n\)-dimensional complete Riemannian manifold and shows the following result. Let \((W^{n-1},g_1)\) be a noncompact complete Riemannian manifold and consider the standard 1-dimensional unit sphere \((S^1,g_2)\). Then there exist a metric \(g'\) and a differentiable function \(\varphi\) on \(W^{n-1}\) such that the warped product \((W^{n-1}\times S^1,g'\times_\varphi g_2)\) has the same growth type as \((W^{n-1}\times S^1,g_1\times g_2)\) but with arbitrarily high \((n-1)\)-dimensional systolic growth type.