Volume comparison without curvature-assumption

Abstract

Let \(\, X^n ,\ M^n\,\) be any pair of compact connected Riemannian manifolds; the only assumption on \(\,M^n\,\) is that the Gromov–Hausdorff distance \(\,\varepsilon \,\) between \(\,M^n\,\) and \(\,X^n\,\) is smaller than the (normalized) injectivity radius of \(\, X^n\,\). Using a transport of measures and a new notion of barycentre, we construct a \(\,C^1\,\) Gromov–Hausdorff \(\,C_0\cdot \varepsilon ^{2/3}\)-approximation \(\,H : M^n \rightarrow X^n\,\), whose energy and Jacobian determinant are sharply bounded from above by \(\, 1 + C_1\cdot \varepsilon ^{1/3}\,\) (the constants \(\,C_i\,\) are explicit), implying a sharp lower bound for the ratio between the volumes of \(\,M^n\,\) and \(\,X^n\,\). This result extends to the non compact case, when \(\,X^n\,\) and \(\,M^n\,\) are only supposed to be quasi-isometric.