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- Title
Volume comparison in the presence of a Gromov-Hausdorff $${\varepsilon}$$ -approximation I.
- Authors
Sabatini, Luca
- Abstract
Let ( M, g) be any compact, connected, Riemannian manifold of dimension n. We use a transport of measures and the barycentre to construct a map from ( M, g) onto a flat torus $${ (\mathbb R^n/\Lambda, g_0)}$$ , in such a way that its jacobian is sharply bounded from above. We make no assumptions on the topology of ( M, g) and on its curvature and geometry, but we only assume the existence of a measurable Gromov-Hausdorff $${\varepsilon}$$ -approximation between $${(\mathbb R^n/\Lambda,g_0)}$$ and ( M, g). When the Hausdorff approximation is continuous with non vanishing degree, this leads to a sharp volume comparison,where $${{\rm inj}(\mathbb R^n/\Lambda,g_0)}$$ is the injectivity radius of the flat torus. Notice that the above inequality is valid for any $${\varepsilon < \frac{{\rm inj}(\mathbb R^n/\Lambda,g_0)}{37(n+2)^2}}$$ .
- Subjects
RIEMANNIAN manifolds; MATHEMATICAL mappings; APPROXIMATION theory; TOPOLOGY; JACOBIAN matrices; JACOBIAN determinants
- Publication
Mathematische Zeitschrift, 2013, Vol 274, Issue 1/2, p1
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-012-1054-4