G P BALAKUMAR
Articles written in Proceedings – Mathematical Sciences
Volume 127 Issue 2 April 2017 pp 323-335 Research Article
Analysing the Wu metric on a class of eggs in $\mathbb{C}^{n} – \rm{I}$
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 $E_{2m}$.
Volume 127 Issue 3 June 2017 pp 463-470 Research Article
Analysing the Wu metric on a class of eggs in $\mathbb{C}^{n} – \rm{II}$
We study the Wu metric for the non-convex 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 0 < m < 1/2. We give explicit expressions for the Kobayashi metric and the Wu metric on such pseudo-eggs $E_{2m}$. We verify that the Wu metric is a continuous Hermitian metric on $E_{2m}$, real analytic everywhere except along the complex hypersurface $Z = \{(0, z_{2}, . . . , z_{n}) \in E_{2m}\}$. We also show that the holomorphic curvature of the Wu metric for this noncompactfamily of pseudoconvex domains is bounded above in the sense of currents by a negative constant independent of $m$. This verifies a conjecture of S.Kobayashi and H.Wu for such $E_{2m}$.
Volume 130, 2020
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