Lower bounds on the Calabi functional

By Simon Donaldson. 2005

274 citations recorded by the source corpus.

The main result of this paper shows that "test configurations" give new lower bounds on the $L^{2}$ norm of the scalar curvature on a Kahler manifold. This is closely analogous to the analysis of the Yang-Mills functional over Riemann surfaces by Atiyah and Bott. The proof uses asymptotic approximation by finite-dimensional problems: the essential ingredient being the Tian-Zelditch-Lu expansion of the "density of states" function.

Subfields: Applied Mathematics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/3637e84a2add6dc4b6f3866d03095118e64af28d

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In Applied Mathematics

In Geometry and Topology

In Nuclear and High Energy Physics

Citations in the corpus

Relative K-stability of extremal metrics (2009, methodology): “Taking this information into account we now compute the Donaldson-Futaki invariant for the action α on the central fiber X̂0.”
Hamiltonian 2-forms in Kähler geometry, III extremal metrics and stability (2005, result): “Thomas [45] and G. Sz´ ekelyhidi [50, 51] independently obtained similar results for rational Kahler classes on ruled surfaces, by using a different approach relying on recent works of S. Donaldson [15, 17, 18 ].”
K-stability of constant scalar curvature polarization (2008, result): “This generalizes the results of ChenTian [1], Donaldson [5] and Stoppa [22].”
Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, methodology): “1 initially follows Donaldson’s beautiful lower bound on the Calabi functional [15] to prove semistability.”
A non-Archimedean approach to K-stability (2018, methodology): “When F is associated to a test conﬁguration ( X , L ) for ( X, L ), (cid:107)F(cid:107) p coincides with the L p -norm of the test conﬁguration as deﬁned in [Don05, WN12, His16].”
Kähler–Ricci flow, Kähler–Einstein metric, and K–stability (2015, methodology): “On the other hand, by [13], we also have”
The Kähler–Ricci flow and optimal degenerations (2016, background): “This makes little difference in the Donaldson-Futaki invariant, since it is linear, but the H-invariant is more sensitive to changes in such conventions.”
K-stability of constant scalar curvature Kähler manifolds (2008, background): “1 (Donaldson [8]) For a polarised manifold (X, L)”
Existence of approximate Hermitian-Einstein structures on semi-stable bundles (2010, background): “In [10], Donaldson recalls a result of Atiyah and Bott from [2] on the YangMills functional and discusses an analogous result for the Calabi functional.”
A generalization of the Ross--Thomas slope theory (2009, background): “Let us recall that a polarized manifold with a constant scalar curvature Kähler metric is K-polystable, due to the works of [5], [2], [22], [12] and [13].”
The complex Monge–Ampère equation on compact Kähler manifolds (2010, background): “Recently, Donaldson solved the constant scalar curvature equation on toric Kähler surfaces with K-stability in a series of paper [23, 25, 26, 27, 28].”