Volume growth of submanifolds and the Cheeger isoperimetric constant (Q2845439)

Let \(M\) be a non-compact Riemannian manifold of dimension \(n\geq 2\). Let \( U\left( M\right) \) denote the set of all open submanifolds \(\Omega \) of \(M\) possessing compact closure and smooth boundary. Then the isoperimetric constant \(I_{\infty }\left( M\right) \) is defined by \(I_{\infty }\left( M\right) =\inf \left\{ \text{Vol}\left( \partial \Omega \right) /\text{Vol} \left( \Omega \right) \text{ : }\Omega \in U\left( M\right) \right\} \), where \(\text{Vol}\left( \partial \Omega \right) \) is the \(\left( n-1\right) \)-dimensional volume of \(\partial \Omega \) and \(\text{Vol}\left( \Omega \right) \) is the \(n\)-dimensional volume of \(\Omega \) [\textit{J. Cheeger}, in: Probl. Analysis, Sympos. in Honor of Salomon Bochner, Princeton Univ. 1969, 195--199 (1970; Zbl 0212.44903)]. NEWLINENEWLINENEWLINEThe authors obtain sharp upper and lower estimates of the isoperimetric constant \(I_{\infty }\left( P\right) \) in terms of the volume growth of a complete properly immersed submanifold \(P\) with controlled mean curvature in a Riemannian manifold \(N\) with sectional curvatures bounded above and which has at least one pole. Their main results, Theorems 3.2 and 3.3, are formulated under a more general hypothesis involving the constellations \( \left\{ N,P^{m},M_{W}^{m}\right\} \) (where \(W\left( r\right) \) is a wrapping function and \(M_{W}^{m}\) is a specially constructed model space, see [\textit{S. Markvorsen} and \textit{V. Palmer}, J. Geom. Anal. 20, No. 2, 388--421 (2010; Zbl 1185.53067)]). Applications to minimal submanifolds are given.