Title: The gap phenomenon for conformally related Einstein metrics Show Chinese title

Title: 共形相关的爱因斯坦度量的间隙现象

Abstract: We determine the submaximal dimensions of the spaces of almost Einstein scales and normal conformal Killing fields for connected conformal manifolds. The results depend on the signature and dimension $n$ of the conformally nonflat conformal manifold. In the Riemannian case, these two dimensions are at most $n-3$ and $\frac{(n-4)(n-3)}{2}$, respectively. In the Lorentzian case, these two dimensions are at most $n-2$ and $\frac{(n-3)(n-2)}{2}$, respectively. In the remaining signatures, these two dimensions are at most $n-1$ and $\frac{(n-2)(n-1)}{2}$, respectively. This upper bound is sharp and to realize examples of submaximal dimensions, we first provide them directly in dimension 4. In higher dimensions, we construct the submaximal examples as the (warped) product of the (pseudo)-Euclidean base of dimension $n-4$ with one of the 4-dimensional submaximal examples. Show Chinese abstract

Abstract: 我们确定了连通共形流形上几乎爱因斯坦尺度空间和正常共形 Killing 场的次最大维数。 结果取决于共形非平坦共形流形的度规符号和维数$n$。 在黎曼情况下，这两个维数分别最多为$n-3$和$\frac{(n-4)(n-3)}{2}$。 在洛伦兹情况下，这两个维数分别最多为$n-2$和$\frac{(n-3)(n-2)}{2}$。 在其余符号情况下，这两个维数分别最多为$n-1$和$\frac{(n-2)(n-1)}{2}$。 这个上限是紧的，为了实现次最大维数的例子，我们首先在维数 4 中直接提供它们。 在高维情况下，我们构造次最大例子为维度$n-4$的 (伪) 欧几里得基底与其中一个 4 维次最大例子的(扭曲)乘积。