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edited Apr 20, 2016 at 3:58

Igor Belegradek

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I remembered another reason why closed surfaces of negative Euler characteristic cannot collapse under a lower bound on sectional curvature.

Much more is true: if a sequence of $n$-dimensional closed manifolds $M_i$ of Ricci curvature $\ge -k^2$ Gromov-Hausdorff converges to a compact space of (Hausdorff) dimension $<n$, then the simplical volume of $M_i$ is zero for large $i$.

Indeed, on the bottom of p.244 of of Gromov's Volume and bounded cohomology he shows that a lower Ricci curvature bound implies that the simplicial volume is bounded above in terms of volume and the dimension. If a sequence of manifolds converges to a space of dimension $<n$ under the lower Ricci bound, then their volume goes to zero; this is due to Colding, as mentioned e.g. on p.91 of Aspects of Ricci Curvature.

The simplicial volume of a closed surface of negative Euler charactersistic $\chi$ is $2|\chi|$, see p.217 in Gromov's paper, so it cannot collapse under a lower curvature bound. There are of course many high-dimensional manifolds of nonzero simplicial volume.

I remembered another reason why closed surfaces of negative Euler characteristic cannot collapse under a lower bound on sectional curvature.

Much more is true: if a sequence of $n$-dimensional closed manifolds $M_i$ of Ricci curvature $\ge -k^2$ Gromov-Hausdorff converges to a space of (Hausdorff) dimension $<n$, then the simplical volume of $M_i$ is zero for large $i$.

Indeed, on the bottom of p.244 of of Gromov's Volume and bounded cohomology he shows that a lower Ricci curvature bound implies that the simplicial volume is bounded above in terms of volume and the dimension. If a sequence of manifolds converges to a space of dimension $<n$ under the lower Ricci bound, then their volume goes to zero; this is due to Colding, as mentioned e.g. on p.91 of Aspects of Ricci Curvature.

The simplicial volume of a closed surface of negative Euler charactersistic $\chi$ is $2|\chi|$, see p.217 in Gromov's paper, so it cannot collapse under a lower curvature bound. There are of course many high-dimensional manifolds of nonzero simplicial volume.

I remembered another reason why closed surfaces of negative Euler characteristic cannot collapse under a lower bound on sectional curvature.

Much more is true: if a sequence of $n$-dimensional closed manifolds $M_i$ of Ricci curvature $\ge -k^2$ Gromov-Hausdorff converges to a compact space of (Hausdorff) dimension $<n$, then the simplical volume of $M_i$ is zero for large $i$.

Indeed, on the bottom of p.244 of of Gromov's Volume and bounded cohomology he shows that a lower Ricci curvature bound implies that the simplicial volume is bounded above in terms of volume and the dimension. If a sequence of manifolds converges to a space of dimension $<n$ under the lower Ricci bound, then their volume goes to zero; this is due to Colding, as mentioned e.g. on p.91 of Aspects of Ricci Curvature.

The simplicial volume of a closed surface of negative Euler charactersistic $\chi$ is $2|\chi|$, see p.217 in Gromov's paper, so it cannot collapse under a lower curvature bound. There are of course many high-dimensional manifolds of nonzero simplicial volume.

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created Apr 20, 2016 at 3:43

Igor Belegradek

29.1k
2
80
176

I remembered another reason why closed surfaces of negative Euler characteristic cannot collapse under a lower bound on sectional curvature.

Much more is true: if a sequence of $n$-dimensional closed manifolds $M_i$ of Ricci curvature $\ge -k^2$ Gromov-Hausdorff converges to a space of (Hausdorff) dimension $<n$, then the simplical volume of $M_i$ is zero for large $i$.

Indeed, on the bottom of p.244 of of Gromov's Volume and bounded cohomology he shows that a lower Ricci curvature bound implies that the simplicial volume is bounded above in terms of volume and the dimension. If a sequence of manifolds converges to a space of dimension $<n$ under the lower Ricci bound, then their volume goes to zero; this is due to Colding, as mentioned e.g. on p.91 of Aspects of Ricci Curvature.

The simplicial volume of a closed surface of negative Euler charactersistic $\chi$ is $2|\chi|$, see p.217 in Gromov's paper, so it cannot collapse under a lower curvature bound. There are of course many high-dimensional manifolds of nonzero simplicial volume.