• Analysing the Wu metric on a class of eggs in $\mathbb{C}^{n} – \rm{I}$

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    • Keywords


      Wu metric; Kobayashi metric; negative holomorphic curvature

    • Abstract


      We study the Wu metric on convex egg domains of the form $E_{2m}= \{z \in \mathbb{C}^n : \mid z_1\mid ^{2m} + \mid z_2\mid ^2 +\cdots +\mid z_{n−1}\mid^2 + \mid z_n \mid^2$ < $1\}$, where $m \geq 1/2$, $m \neq 1$. The Wu metric is shown to be real analytic everywhere except on a lower dimensional subvariety where it fails to be $C^2$-smooth. Overall however, the Wu metric is shown to be continuous when $m = 1/2$ and even $C^1$-smooth for each $m > 1/2$, and in all cases, a non-K$\ddot{a}$hler Hermitian metric with its holomorphic curvature strongly negative in the sense of currents. This gives a natural answer to a conjecture of S. Kobayashi and H. Wu for such $E_{2m}$.

    • Author Affiliations



      1. Indian Statistical Institute, Chennai 600 113, India.
      2. Department of Mathematics, Indian Institute of Technology – Bombay, Mumbai 400 076, India.
    • Dates

  • Proceedings – Mathematical Sciences | News

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