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Register a new userRemarks on the existence of bilaterally symmetric extremal Kahler metrics on $mathbb{CP}^2sharp 2bar{mathbb{CP}^2}$
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Authors
Weiyong He
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In this short note we show that the existence of bilaterally symmetric extremal Kahler metrics on $mathbb{CP}^2sharp 2bar{mathbb{CP}^2}$.
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In this article, we show the existence of a nontrivial Riemann surface lamination embedded in $mathbb{CP}^2$ by using Donaldsons construction of asymptotically holomorphic submanifolds. Further, the lamination we obtain has the property that each leaf is a totally geodesic submanifold of $mathbb{CP}^2 $ with respect to the Fubini-Study metric. This may constitute a step in understanding the conjecture on the existence of minimal exceptional sets in $mathbb{CP}^2$.
The scalar curvature equation for rotation invariant Kahler metrics on $mathbb{C}^n backslash {0}$ is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on $mathbb{C}^n backslash {0}$ in lower dimensions. We also prove that there does not exist negative csck on $mathbb{C}^n backslash {0}$ for $n=2,3$.
We construct a toric generalised Kahler structure on $mathbb{C}P^2$ and show that the various structures such as the complex structure, metric etc are expressed in terms of certain elliptic functions. We also compute the generalised Kahler potential in terms of integrals of elliptic functions.
We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature Kahler metrics to the case of extremal metrics. Namely, we show that a given extremal metric is the limit of some specific relatively balanced metrics. As a corollary, we recover uniqueness and splitting results for extremal metrics in the polarized case.
We prove the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings.
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