Systole of congruence coverings of arithmetic hyperbolic manifolds

Murillo, Plinio G. P.

doi:10.4171/ggd/515

Cited by 6 publications

(18 citation statements)

References 12 publications

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“…Note that when n = 1, H 1 H is isometric to the four dimensional real hyperbolic space, and the constant 2 5 agrees with that of [15]. In Section 6 we generalize the argument of Dória and Murillo to prove that the constant 4 (n+1)(2n+3) is optimal; see Theorem 6.1.…”

Section: Introductionmentioning

confidence: 74%

“…They also proved that for compact arithmetic hyperbolic 3-manifolds the constant C = 2 3 works. For real arithmetic hyperbolic n-manifold of the first type, these results were generalized by Murillo in [15], where he proved that the constant is equal to Recently, Lapan, Linowitz and Meyer obtained a value for the constant C for a large class of arithmetic locally symmetric spaces, including real, complex and quaternionic hyperbolic orbifolds [9]. However, the values of the constants obtained in [9] are not optimal, as the comparison with the results mentioned above shows.…”

Section: Introductionmentioning

confidence: 95%

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On the systole growth in congruence quaternionic hyperbolic manifolds

Emery¹,

Ким²,

Murillo³

2019

Preprint

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Section: Introductionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 95%

On the systole growth in congruence quaternionic hyperbolic manifolds

Emery¹,

Ким²,

Murillo³

2019

Preprint

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“…For us, of course, the interest is in n 4. This was treated in a recent article by Murillo [20], which shows that, with a couple of caveats, the optimal inequality has…”

Section: A Spin Version Of Gromov-thurston Manifoldsmentioning

confidence: 99%

“…We review this in the next Subsection 2.1. To bound the volume of Σ k we will make use of recent work of Murillo [20], which does not apply to all the sequences arising from Gromov-Thurston's original construction, but instead to a special subclass of them. Put loosely, we need the manifolds M k to be spin.…”

Section: A Spin Version Of Gromov-thurston Manifoldsmentioning

confidence: 99%

Negatively curved Einstein metrics on ramified covers of closed four-dimensional hyperbolic manifolds

Premoselli¹

2021

Séminaire De Théorie Spectrale Et Géométrie

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This paper is a shortened version of the recent article Examples of compact Einstein four-manifolds with negative curvature [11] written in collaboration with J. Fine (ULB). Its content was presented by the author at the Séminaire de Théorie Spectrale et Géométrie in Grenoble in December 2017. In [11], new examples of compact, negatively curved Einstein manifolds of dimension 4 have been obtained. These are seemingly the first such examples which are not locally homogeneous. The Einstein metrics we construct are carried by a sequence of 4-manifolds (X k ), previously considered by Gromov and Thurston [13], and obtained as ramified coverings of closed hyperbolic 4-manifolds. Our proof relies on a deformation procedure. We first find an approximate Einstein metric on X k by interpolating between a model Einstein metric near the branch locus and the pull-back of the hyperbolic metric from the base hyperbolic manifolds. We then perturb to a genuine solution to Einstein's equations, by a parameter dependent version of the inverse function theorem.

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“…For us, of course, the interest is in n ≥ 4. This was treated in a recent article by Murillo [29], which shows that, with a couple of caveats, the optimal inequality has C(n) = n(n + 1) 4…”

Section: Murillo's Volume Estimatementioning

confidence: 99%

Examples of compact Einstein four-manifolds with negative curvature

Fine

Premoselli

2020

J. Amer. Math. Soc.

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We give new examples of compact, negatively curved Einstein manifolds of dimension 4. These are seemingly the first such examples which are not locally homogeneous. Our metrics are carried by a sequence of 4-manifolds (X k ) previously considered by Gromov and Thurston [19]. The construction begins with a certain sequence (M k ) of hyperbolic 4-manifolds, each containing a totally geodesic surface Σ k which is nullhomologous and whose normal injectivity radius tends to infinity with k. For a fixed choice of natural number l, we consider the l-fold cover X k → M k branched along Σ k . We prove that for any choice of l and all large enough k (depending on l), X k carries an Einstein metric of negative sectional curvature. The first step in the proof is to find an approximate Einstein metric on X k , which is done by interpolating between a model Einstein metric near the branch locus and the pull-back of the hyperbolic metric from M k . The second step in the proof is to perturb this to a genuine solution to Einstein's equations, by a parameter dependent version of the inverse function theorem. The analysis relies on a delicate bootstrap procedure based on L 2 coercivity estimates.

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Systole of congruence coverings of arithmetic hyperbolic manifolds

Cited by 6 publications

References 12 publications

On the systole growth in congruence quaternionic hyperbolic manifolds

On the systole growth in congruence quaternionic hyperbolic manifolds

Negatively curved Einstein metrics on ramified covers of closed four-dimensional hyperbolic manifolds

Examples of compact Einstein four-manifolds with negative curvature

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