Displaying similar documents to “Extremal Kähler metrics on blow-ups of parabolic ruled surfaces”

Toric Hermitian surfaces and almost Kähler structures

Włodzimierz Jelonek (2007)

Annales Polonici Mathematici

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The aim of this paper is to investigate the class of compact Hermitian surfaces (M,g,J) admitting an action of the 2-torus T² by holomorphic isometries. We prove that if b₁(M) is even and (M,g,J) is locally conformally Kähler and χ(M) ≠ 0 then there exists an open and dense subset U ⊂ M such that ( U , g | U ) is conformally equivalent to a 4-manifold which is almost Kähler in both orientations. We also prove that the class of Calabi Ricci flat Kähler metrics related with the real Monge-Ampère equation...

Some critical almost Kähler structures

Takashi Oguro, Kouei Sekigawa (2008)

Colloquium Mathematicae

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We consider the set of all almost Kähler structures (g,J) on a 2n-dimensional compact orientable manifold M and study a critical point of the functional ℱ λ , μ ( J , g ) = ∫ M ( λ τ + μ τ * ) d M g with respect to the scalar curvature τ and the *-scalar curvature τ*. We show that an almost Kähler structure (J,g) is a critical point of ℱ - 1 , 1 if and only if (J,g) is a Kähler structure on M.

Canonical metrics on some domains of ℂ n

Fabio Zuddas (2008-2009)

Séminaire de théorie spectrale et géométrie

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The study of the existence and uniqueness of a preferred Kähler metric on a given complex manifold M is a very important area of research. In this talk we recall the main results and open questions for the most important canonical metrics (Einstein, constant scalar curvature, extremal, Kähler-Ricci solitons) in the compact and the non-compact case, then we consider a particular class of complex domains D in ℂ n , the so-called Hartogs domains, which can be equipped with a natural Kaehler...

On the Kähler Rank of Compact Complex Surfaces

Matei Toma (2008)

Bulletin de la Société Mathématique de France

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Harvey and Lawson introduced the Kähler rank and computed it in connection to the cone of positive exact currents of bidimension ( 1 , 1 ) for many classes of compact complex surfaces. In this paper we extend these computations to the only further known class of surfaces not considered by them, that of Kato surfaces. Our main tool is the reduction to the dynamics of associated holomorphic contractions ( ℂ 2 , 0 ) → ( ℂ 2 , 0 ) .

Stability under deformations of Hermite-Einstein almost Kähler metrics

Mehdi Lejmi (2014)

Annales de l’institut Fourier

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On a 4 -dimensional compact symplectic manifold, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermite-Einstein almost-Kähler metric with zero or negative Hermitian scalar curvature. We prove, under certain hypothesis, the existence of a smooth family of compatible almost-complex structures, diffeomorphic at each time to the initial one, and inducing constant Hermitian scalar curvature metrics.

Locally conformally Kähler metrics on Hopf surfaces

Paul Gauduchon, Liviu Ornea (1998)

Annales de l'institut Fourier

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A primary Hopf surface is a compact complex surface with universal cover ℂ 2 - { ( 0 , 0 ) } and cyclic fundamental group generated by the transformation ( u , v ) ↦ ( α u + λ v m , β v ) , m ∈ ℤ , and α , β , λ ∈ ℂ such that ∣ α ∣ ≥ ∣ β ∣ > 1 and ( α - β m ) λ = 0 . Being diffeomorphic with S 3 × S 1 Hopf surfaces cannot admit any Kähler metric. However, it was known that for λ = 0 and ∣ α ∣ = ∣ β ∣ they admit a locally conformally Kähler metric with parallel Lee form. We here provide the construction of a locally conformally Kähler metric with parallel Lee form for primary Hopf surfaces of class 1 ( λ = 0 )....

Extremal metrics and lower bound of the modified K-energy

Yuji Sano, Carl Tipler (2015)

Journal of the European Mathematical Society

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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.