Applications of the Weil-Petersson and the Takhtajan-Zograf metrics

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The reseach with W.-K. To and L. Weng has been advanced. In the paper with them published in 2008, We gave some estimates for the asymptotic behaviors ofthe Takhtajan-Zograf metric. When we compare the Takhtajan-Zograf metric with the Weil-Petersson metric, we found that the order of the asymptotic behavior of the Takhtajan-Zograf metric is strictly smaller than the one of the Weil-Petersson metric. But the precise order of the asymptotics of the Takhtajan-Zograf metric are still unknown. Recently withW.-K. To and L. Weng, I succeeded in improving the upper bound estimate of the Takhtajan-Zograf metric which we could believe is the best upper estimate.