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Gromov and Thurston proved in "Pinching constants for hyperbolic manifolds" that any finite ramified covering of a compact hyperbolic manifold, along a codimension $2$ totally geodesic submanifold, can be endowed with a riemannian metric of negative sectional curvature. Pansu mentions in a survey (4/TSG_1985-1986_4_101_0/TSG_1985-1986_4_101_0.pdf"> that he doesn't know how to make Gromov and Thurston's calculation in the general case. Is it possible to generalize ?

Namely let $M$ be a compact manifold endowed with a riemannian metric of negative sectional curvature. Suppose $X \longrightarrow M$ is a ramified covering of finite degree, along a compact submanifold of codimension $2$. Can $X$ be endowed with a riemannian metric of negative sectional curvature ? Can a counterexample be found when the locus of ramification is not totally geodesic ? Can one hope a pinching of the curvature like Gromov and Thurston ? Is the covering, in any reasonable sense, "more negatively" than the base ?

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To my knowledge it is unknown how to generalize Gromov-Thurston computation to arbitrary variable curvature. Even the complex hyperbolic case is tricky. On the othe hand, it is not clear this is worth the effort. What would be the pay off? – Igor Belegradek Nov 18 '13 at 23:49
@IgorBelegradek: Igor, in general, I do not think there will be any payoff, but it would be good to settle the complex-hyperbolic case since the construction is likely to produce Kahler manifolds and the situation is very much unclear here since Kahler manifolds tend to be much more rigid. In Kahler setting, pretty much the only known negatively curved (nonsymmetric) example is Mostow-Siu. – Misha Nov 19 '13 at 9:10
Staying in the Kahler category is even harder. Incidentally, Mostow-Siu is not the state of the art anymore: see e.g. Zheng "Hirzebruch-Kato surfaces, Deligne-Mostow's construction, and new examples of negatively curved compact Kähler surfaces" and Deraux "A negatively curved Kähler threefold not covered by the ball". – Igor Belegradek Nov 19 '13 at 12:49
up vote 7 down vote accepted

Suppose that $M$ is a closed hyperbolic 3-manifold and $K$ is a trivial knot contained in a ball $B\subset M$. Then $M$ admits a 2-fold covering $M'\to M$ ramified over $K$ so that $M'$ is not even aspherical (it is homeomorphic to the connected sum of two copies of $M$). Therefore, $M'$ cannot admit a negatively curved metric.

Do you want the ramification locus to be totally-geodesic? If the ramification locus is totally geodesic, then the covering space $M'$ has a locally $CAT(-k)$-metric where $k>0$ and the original manifold $M$ has lower curvature bound $-k$, so in this sense, it is more negatively curved than the original manifold $M$. In general, I do not think there is a (known) smoothing procedure for this metric among negatively curved metrics. Note that in the context of nonpositively curved manifolds, a branched covering ramified over a totally geodesic submanifold may not admit a smooth metric of nonpositive curvature (cf. Exercise 1 in Gromov-Ballmann-Schroeder, which was finally solved this year, almost 30 years after the book was published).

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Misha, where I find the solution of the exercise? (I assume you mean the one on pp 2-3 about a 4-manifold that is a branched cover and admits no Riemannian metric of nonpositive curvature.) – Igor Belegradek Nov 18 '13 at 23:46
@IgorBelegradek: It was proven by a student of Bernhard Leeb. I am not sure if he already posted his solution. – Misha Nov 19 '13 at 5:44
The name of the student is Stephan Stadler, – Anton Petrunin Jan 6 '14 at 23:29

In the case of 3-dimensional negatively curved manifolds, this result was proved by Yong Hou (at least when the submanifold is totally geodesic; as Misha says, this is the most natural interpretation of your question). The statement is a generalization of Theorem 3.2 of his paper, which applies as stated only to cyclic covers branched over a null-homologous geodesic; however it's clear that the proof generalizes to arbitrary branched covers since it is purely local. The idea of his proof is to show that in a tubular neighborhood of the metric, one may make a deformation in which the core geodesic has a neighborhood with constant negative sectional curvature (this sort of argument goes back to Gao). Then one may perform the change in curvature exactly as in the case of Gromov-Thurston.

So one could try to emulate Hou's proof in higher dimensions, but I don't know if this has been carried out.

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Ian: I would add that post-Perelman, the existence of a negatively curved metric in 3d case is a corollary of the Geometrization Theorem. – Misha Nov 19 '13 at 9:25
@ Misha: good point, although Hou's proof works for non-compact manifolds too. I think there's some chance his argument might extend to higher dimensions, so I think it is worth mention, even though it is superseded by your answer and geometrization. – Ian Agol Nov 19 '13 at 18:15

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