<abstract xmlns="http://www.w3.org/1999/xhtml">Let (<italic>M, g</italic>) be any compact, connected, Riemannian manifold of dimension <italic>n</italic>. We use a transport of measures and the barycentre to construct a map from (<italic>M, g</italic>) onto a Hyperbolic manifold (ℍ<italic>n/</italic>Λ, <italic>g</italic>0) (Λ is a torsionless subgroup of <italic>Isom</italic>(ℍ<italic>n,g</italic>0)), in such a way that its jacobian is sharply bounded from above. We make no assumptions on the topology of (<italic>M, g</italic>) and on its curvature and geometry, but we only assume the existence of a measurable Gromov-Hausdorff <italic>ε</italic>-approximation between (ℍ<italic>n/</italic>Λ, <italic>g</italic>0) and (<italic>M, g</italic>). When the Hausdorff approximation is continuous with non vanishing degree, this leads to a sharp volume comparison, if <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2018-0008_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>ɛ</mi><mo>&lt;</mo><mfrac><mn>1</mn><mrow><mn>64</mn><mi> </mi><msup><mrow><mi>n</mi></mrow><mn>2</mn></msup></mrow></mfrac><mtext>min</mtext><mrow><mo>(</mo><mrow><msub><mrow><mi>inj</mi></mrow><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>ℍ</mi></mrow><mi>n</mi></msup><mo>/</mo><mi>Λ</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow></msub><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math><tex-math>$\varepsilon &lt; {1 \over {64\,{n^2}}}\min \left( {in{j_{\left( {{{\Bbb H}^n}/\Lambda ,{g_0}} \right)}},1} \right)$</tex-math></alternatives></inline-formula>, then
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2018-0008_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd><mrow><mi>Vol</mi><mrow><mo>(</mo><mrow><msup><mrow><mi>M</mi></mrow><mi>n</mi></msup><mo>,</mo><mi>g</mi></mrow><mo>)</mo></mrow><mo>≥</mo></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mn>160</mn><mi>n</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><msqrt><mrow><mfrac><mi>ɛ</mi><mrow><mtext>min</mtext><mrow><mo>(</mo><mrow><msub><mrow><mi>inj</mi></mrow><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>H</mi></mrow><mi>n</mi></msup><mo>/</mo><mi>Λ</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow></msub><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mfrac><mi>n</mi><mn>2</mn></mfrac></mrow></msup><mrow><mo>|</mo><mrow><mi>deg</mi><mi> </mi><mi>h</mi></mrow><mo>|</mo></mrow><mo>⋅</mo><mi>Vol</mi><mrow><mo>(</mo><mrow><msup><mrow><mi>X</mi></mrow><mi>n</mi></msup><mo>,</mo><msub><mrow><mi>g</mi></mrow><mn>0</mn></msub></mrow><mo>)</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math><tex-math>$$\matrix{{Vol\left( {{M^n},g} \right) \ge }\cr {{{\left( {1 + 160n\left( {n + 1} \right)\sqrt {{\varepsilon \over {\min \left( {in{j_{\left( {{{\Bbb H}^n}/\Lambda ,{g_0}} \right)}},1} \right)}}} } \right)}^{{n \over 2}}}\left| {\deg \,h} \right| \cdot Vol\left( {{X^n},{g_0}} \right).} \cr }$$</tex-math></alternatives></disp-formula></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 Hyperbolic manifold (ℍn/Λ, g0) (Λ is a torsionless subgroup of Isom(ℍn,g0)), 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 ε-approximation between (ℍn/Λ, g0) and (M, g). When the Hausdorff approximation is continuous with non vanishing degree, this leads to a sharp volume comparison, if ɛ<164 n2min(inj(ℍn/Λ,g0),1)$\varepsilon < {1 \over {64\,{n^2}}}\min \left( {in{j_{\left( {{{\Bbb H}^n}/\Lambda ,{g_0}} \right)}},1} \right)$, then
Vol(Mn,g)≥(1+160n(n+1)ɛmin(inj(Hn/Λ,g0),1))n2|deg h|⋅Vol(Xn,g0).$$\matrix{{Vol\left( {{M^n},g} \right) \ge }\cr {{{\left( {1 + 160n\left( {n + 1} \right)\sqrt {{\varepsilon \over {\min \left( {in{j_{\left( {{{\Bbb H}^n}/\Lambda ,{g_0}} \right)}},1} \right)}}} } \right)}^{{n \over 2}}}\left| {\deg \,h} \right| \cdot Vol\left( {{X^n},{g_0}} \right).} \cr }$$

Volume Comparison in the presence of a Gromov-Hausdorff <italic>ε−</italic>approximation II

Dipartimento di Scienze di Base ed Applicate per l’Ingegneria

Università degli Studi di Roma “ La Sapienza “

Annals of West University of Timisoara - Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume Comparison in the presence of a Gromov-Hausdorff approximation II

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...

Volume Comparison in the presence of a Gromov-Hausdorff ε−approximation II

Published Online: Dec 07, 2018

Page range: 99 - 135

Received: Jul 21, 2017

Accepted: Jan 08, 2018

DOI: https://doi.org/10.2478/awutm-2018-0008

Keywords
Volumes comparison, Gromov-Hausdorff distance, energy bounds, barycentres

© 2018 Luca Sabatini, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume Comparison in the presence of a Gromov-Hausdorff ε−approximation II

Published Online: Dec 07, 2018

Page range: 99 - 135

Received: Jul 21, 2017

Accepted: Jan 08, 2018

DOI: https://doi.org/10.2478/awutm-2018-0008

KeywordsVolumes comparison, Gromov-Hausdorff distance, energy bounds, barycentres

© 2018 Luca Sabatini, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Volumes comparison, Gromov-Hausdorff distance, energy bounds, barycentres