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 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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具有负曲率的紧凑爱因斯坦四流形示例

我们给出了尺寸为 $4$ 的紧凑、负弯曲的爱因斯坦流形的新例子。这些似乎是第一个这样的例子，它们不是局部同质的。我们的指标由一系列 4 流形 $(X_k)$ 先前由 Gromov 和 Thurston 考虑。构造以双曲 4-流形的特定序列 $(M_k)$ 开始，每个流形包含一个完全测地表面 $\Sigma_k$，它是零同源的，其法向注入半径趋向于无穷大 $k$。对于固定选择的自然数 $l$，我们考虑沿 $\Sigma_k$ 分支的 $l$-fold 覆盖 $X_k \to M_k$。我们证明，对于 $l$ 的任何选择和所有足够大的 $k$（取决于 $l$），$X_k$ 带有负截面曲率的爱因斯坦度量。证明的第一步是在 $X_k$ 上找到一个近似的爱因斯坦度量，这是通过在分支轨迹附近的模型爱因斯坦度量和从 $M_k$ 回拉的双曲线度量之间进行插值来完成的。证明的第二步是通过反函数定理的参数相关版本，将其扰乱为爱因斯坦方程的真正解。该分析依赖于基于 $L^2$ 矫顽力估计的精细引导程序。