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Examples of compact Einstein four-manifolds with negative curvature. (arXiv:1802.00608v1 [math.DG])
来源于:arXiv
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. The construction begins
with a certain sequence $(M_k)$ of hyperbolic 4-manifolds, each containing a
totally geodesic surface $\Sigma_k$ 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 $X_k \to M_k$ branched along
$\Sigma_k$. 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 t 查看全文>>