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Extremal Kähler metrics on blow-ups of parabolic ruled surfaces

Carl Tipler — 2013

Bulletin de la Société Mathématique de France

New examples of extremal Kähler metrics are given on blow-ups of parabolic ruled surfaces. The method used is based on the gluing construction of Arezzo, Pacard and Singer [5]. This enables to endow ruled surfaces of the form ( 𝒪 ⊕ L ) with special parabolic structures such that the associated iterated blow-up admits an extremal metric of non-constant scalar curvature.

Extremal metrics and lower bound of the modified K-energy

Yuji Sano;  Carl Tipler — 2015

Journal of the European Mathematical Society

We provide a new proof of a result of X.X. Chen and G.Tian [5]: for a polarized extremal Kähler manifold, the minimum of the modified K-energy is attained at an extremal metric. The proof uses an idea of C. Li [16] adapted to the extremal metrics using some weighted balanced metrics.

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