Extremal Kähler metrics

By Gang Tian. 2000

408 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/bb4c140ec8759e8feeb6c6ab4be469a39432df0d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

Citations in the corpus

Hamiltonian 2-forms in Kähler geometry, II Global Classification (2004, methodology): “If the Kähler structure given by (24) (which pulls back to the fibrewise metric on the fibres of M̂ → S) extends smoothly toM , then we say that this Kähler manifold (M,g, J, ω) is given by the generalized Calabi construction (with blow-downs).”
Hamiltonian 2-forms in Kähler geometry, III extremal metrics and stability (2005, background): “Suppose there is some admissible class Ω0 (with parameters x 0 a) which contains an inadmissible extremal metric g0; by the Calabi Theorem, we can assume that g0 is G-invariant, i.e., Ω0 ∈ U .”
Geometry of Kähler metrics and foliations by holomorphic discs (2005, methodology): “Following [4], we call a Kähler metric extremal if the complex gradient of its scalar curvature is a holomorphic vector field.”
On the constant scalar curvature Kähler metrics, general automorphism group (2018, background): “Note that if ω0 is a cscK metric, then it is symmetric with respect to a maximal compact subgroup ([8], [9]).”
Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K\ (2010, background): “Such examples were already implicit in Calabi’s seminal paper [6].”
Extremal Kähler metrics on projective bundles over a curve (2009, background): “As shown in [13], g is extremal if and only if the symplectic gradient K := gradωsg = J gradgsg of sg is a Killing vector field (i.”
Notes on GIT and symplectic reduction for bundles and varieties (2005, methodology): “It was Calabi [Ca] who initiated the study of cscK and extremal metrics: those which extremise the Calabi functional ∫ X s 2ωn over cohomologous Kähler forms; they are the metrics with ∇s a holomorphic vector field.”
Extremal metrics and K-stability (PhD thesis) (2006, background): “These will then be used in Section 5.4 to determine the infimum of the Calabi functional for the unstable polarisations.”
The Kähler–Ricci flow on Hirzebruch surfaces (2009, methodology): “We see from the above that, assuming the Calabi symmetry, thee ar three distinct behaviors of the Kähler-Ricci flow on a Hirzebruch surface, dependingo k and the initial Kähler class: • TheP1 fiber collapses (cases (a) and (b).”
Smooth and singular Kahler-Einstein metrics (2014, background): “The zeros of the vector field s are the constant scalar curvature (csc) metrics inH, and its integral curves are the trajectories of the Calabi flow [48].”
Calabi flow and projective embeddings (2008, background): “3 Additional Context Calabi suggested in [4] that, when one exists, a constant scalar curvature Kähler metric should be considered a “canonical” representative of a Kähler class.”
Combinatorial Calabi flows on surfaces (2012, background): “Calabi studied the variational problem of minimizing the so-called “Calabi energy” in any fixed cohomology class of Kähler metrics and proposed the Calabi flow ([2], [3]).”
Canonical Sasakian Metrics (2006, background): “We describe explicitly those that are of constant scalar curvature, and show that the standard metric is the only one of these that is Sasaki-Einstein.”
Blowing up Kähler manifolds with constant scalar curvature, II (2005, methodology): “This metric g0 is associated to a Kähler form 0 and has been discovered, inspired by previous work of Calabi [6], by Burns when mD 2 (and first described by Lebrun in [13]) and by Simanca [25] when m 3.”
A New Parabolic Flow in Kahler Manifolds (2000, methodology): “Calabi first proved in [3] that critical point of Calabi energy minimizes Calabi energy locally, one should compare this to proposition 3 aforementioned.”