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 . In [11], new examples of compact, negatively curved Einstein manifolds of dimension 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 , 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 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.
@article{TSG_2017-2019__35__129_0, author = {Premoselli, Bruno}, title = {Negatively curved {Einstein} metrics on ramified covers of closed four-dimensional hyperbolic manifolds}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {129--161}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {35}, year = {2017-2019}, doi = {10.5802/tsg.367}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/tsg.367/} }
TY - JOUR AU - Premoselli, Bruno TI - Negatively curved Einstein metrics on ramified covers of closed four-dimensional hyperbolic manifolds JO - Séminaire de théorie spectrale et géométrie PY - 2017-2019 SP - 129 EP - 161 VL - 35 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/tsg.367/ DO - 10.5802/tsg.367 LA - en ID - TSG_2017-2019__35__129_0 ER -
%0 Journal Article %A Premoselli, Bruno %T Negatively curved Einstein metrics on ramified covers of closed four-dimensional hyperbolic manifolds %J Séminaire de théorie spectrale et géométrie %D 2017-2019 %P 129-161 %V 35 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/tsg.367/ %R 10.5802/tsg.367 %G en %F TSG_2017-2019__35__129_0
Premoselli, Bruno. Negatively curved Einstein metrics on ramified covers of closed four-dimensional hyperbolic manifolds. Séminaire de théorie spectrale et géométrie, Tome 35 (2017-2019), pp. 129-161. doi : 10.5802/tsg.367. http://archive.numdam.org/articles/10.5802/tsg.367/
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