Gromov's minimal volume conjecture for finite-volume hyperbolic manifolds

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Let (Mn,g0)(M^n,g_0) be a complete hyperbolic manifold with finite volume, where g0g_0 is a hyperbolic metric on MnM^n. The minimal volume of MM is denoted by

Min Vol(M)=inf⁡{Vol⁡(M,g):g is a Riemannian metric on M with ∣Kg∣⩽1}.{\rm Min\,Vol}(M)=\inf\{\operatorname{Vol}(M,g): g\text{ is a Riemannian metric on }M\text{ with }|K_g|\leqslant 1\}.

Gromov's conjecture. For every complete hyperbolic manifold of finite volume,

Min Vol(M)=Vol⁡(M,g0).{\rm Min\,Vol}(M)=\operatorname{Vol}(M,g_0).

This conjecture proposes that the hyperbolic metric realizes the minimal volume among metrics with sectional curvature bounded in absolute value by 11. The supplied source does not state whether the conjecture has been resolved.

References

Primary source

E. Costa, R. Diógenes and E. Ribeiro, “Estimates for Minimal Volume and Minimal Curvature on 4-dimensional compact manifolds”, arXiv:1407.8137 (2015).

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