abstract:

On a bounded strictly pseudoconvex domain in $\Bbb{C}^n$, $n>1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local curvature invariant of the boundary. For bounded strictly pseudoconvex domains in $\Bbb{C}^2$ which are diffeomorphic to the ball, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball. In particular we observe that, up to biholomorphism, the unit ball in $\Bbb{C}^2$ is rigid with respect to deformations in the class of strictly pseudoconvex domains with obstruction flat boundary. We further show that for more general deformations of the unit ball, the order of vanishing of the obstruction equals the order of vanishing of the CR curvature. Finally, we give a generalization of the recent result of the second author that for an abstract CR manifold with transverse symmetry, obstruction flatness implies local equivalence to the CR $3$-sphere.

摘要：

在$ \ Bbb {C} ^ n $，$ n> 1 $中的有界严格伪凸域上，费弗曼复蒙格-安培方程对边界的成渝解的光滑度受到的局部曲率不变的阻碍边界。对于$ \ Bbb {C} ^ 2 $中有界的严格伪伪凸域，它们对球是微分形的，我们激发并考虑确定该障碍物的整体消失是否暗指单位球的等胚性等价问题。特别地，我们观察到，直到双全同构性为止，$ \ Bbb {C} ^ 2 $中的单位球对于具有阻塞平面边界的严格伪凸域中的变形是刚性的。我们进一步表明，对于单位球更普遍的变形，障碍物消失的顺序等于CR曲率消失的顺序。最后，