CR 3-Sphere的阻塞平面刚度
Obstruction Flat Rigidity of the CR 3-Sphere
摘要：
在$\mathbb{C}^n$中的有界严格拟凸区域$n >1$上，February man复蒙日-安培方程的Cheng-Yau解到边界的光滑性被边界的局部曲率不变量CR阻塞密度$\mathcal{O}$所阻塞.虽然阻塞平坦CR流形的局部例子很多，但唯一已知的紧致例子是球面CR流形。我们考虑标准CR 3-球小变形的障碍平坦性问题。对于可嵌入的CR结构的情况，这种刚性对于CR球体是先前已知的（在所有维度上），其中它也在无穷小水平上成立。然而，在三维情况下，CR结构不需要是可嵌入的。虽然在不可嵌入的情况下，我们可能不再根据February man方程的边界规则性（或Bergman核的对数奇异性）来解释阻塞密度$\mathcal{O}$，但方程$\mathcal{O}\equiv 0$仍然很有意义，例如，因为它对应于CR结构（共形洛伦兹4-流形）的费曼空间的共形引力的巴赫平坦方程。与嵌入的情况下，它原来是在nonembeddable的情况下，有一个无限维空间的解决方案的线性化障碍平坦性方程的标准CR 3球和这个空间定义了一个自然的补充切空间的嵌入变形。尽管如此，我们表明，CR 3-球不承认非平凡的障碍平坦变形，嵌入或nonembeddable。
英文摘要：
On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n >1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampère equation up to the boundary is obstructed by a local curvature invariant of the boundary, the CR obstruction density $\mathcal{O}$. While local examples of obstruction flat CR manifolds are plentiful, the only known compact examples are the spherical CR manifolds. We consider the obstruction flatness problem for small deformations of the standard CR 3-sphere. That rigidity holds for the CR sphere was previously known (in all dimensions) for the case of embeddable CR structures, where it also holds at the infinitesimal level. In the 3-dimensional case, however, a CR structure need not be embeddable. While in the nonembeddable case we may no longer interpret the obstruction density $\mathcal{O}$ in terms of the boundary regularity of Fefferman's equation (or the logarithmic singularity of the Bergman kernel) the equation $\mathcal{O}\equiv 0$ is still of great interest, e.g., since it corresponds to the Bach flat equation of conformal gravity for the Fefferman space of the CR structure (a conformal Lorentzian 4-manifold). Unlike in the embeddable case, it turns out that in the nonembeddable case there is an infinite dimensional space of solutions to the linearized obstruction flatness equation on the standard CR 3-sphere and this space defines a natural complement to the tangent space of the embeddable deformations. In spite of this, we show that the CR 3-sphere does not admit nontrivial obstruction flat deformations, embeddable or nonembeddable.
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