Abstract
Let 5 be a Hirzebruch surface, n > 1 . Using the family of ex- tremal metrics on these surfaces constructed by Calabi (1), we study a closely related scale-invariant variational problem, and show that only Si admits an extremal Kahler metric which is critical for this new functional. Applying a result of Derdzinski (3), we prove that this metric cannot be conformally equiv- alent to an Einstein metric on SiWhen n = 2, we show there is a critical orbifold metric on the space obtained from $2 by blowing down the negative section.
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