复Monge-Ampère方程的不变解到非不变解的提升
Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations
摘要：
我们展示了椭圆和双曲复Monge-Ampère方程（CMA和HCMA）的伙伴对称如何提供三维和二维简化方程的非不变解的提升，即，提升原来的CMA和HCMA方程的不变解，后者四维方程的非不变解。将提升应用于二维Helmholtz方程的非不变解以产生CMA的非不变解，并且应用于三维波动方程和三维双曲Boyer-Finley方程的非不变解以产生HCMA的非不变解。通过将这些解作为度量势，可以构造具有非零曲率张量和无Killing向量的四维欧氏和超双曲签名的Ricci平坦度量。
英文摘要：
We show how partner symmetries of the elliptic and hyperbolic complex Monge-Ampère equations (CMA and HCMA) provide a lift of non-invariant solutions of three- and two-dimensional reduced equations, i.e., a lift of invariant solutions of the original CMA and HCMA equations, to non-invariant solutions of the latter four-dimensional equations. The lift is applied to non-invariant solutions of the two-dimensional Helmholtz equation to yield non-invariant solutions of CMA, and to non-invariant solutions of three-dimensional wave equation and three-dimensional hyperbolic Boyer-Finley equation to yield non-invariant solutions of HCMA. By using these solutions as metric potentials, it is possible to construct four-dimensional Ricci-flat metrics of Euclidean and ultra-hyperbolic signatures that have non-zero curvature tensors and no Killing vectors.
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