Extremal metrics on blowups

By C. Arezzo, Frank Pacard, Michael F. Singer. 2006

75 citations recorded by the source corpus.

In this paper we provide conditions that are sufficient to guarantee the existence of extremal metrics on blowups at finitely many points of Kahler manifolds which already carry an extremal metric. As a particular case, we construct extremal metrics on $\mathbb {P}^2$ blown-up k points in general position, with $k \lt m+2$.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cea118fb061ec4eb008080e1d0eea21a33167cfa

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Relative K-stability of extremal metrics (2009, background): “om [3] Theorem 2.1. Indeed we can choose an extremal metric ω on M such that the isometry group of ω contains a compact maximal torus TR, which is contained in the complex torus T. In the notation of [3] we let K = TR, and let kbe its Lie algebra. Since K is a maximal torus, any K-invariant holomorp”
Extremal K\"ahler metrics (2014, background): “or ﬁelds, then the problem becomes more subtle, since then BlpMmay not admit a cscK (or even extremal) metric for every point p. The problem was addressed by Arezzo-Pacard [3] and ArezzoPacard-Singer [4] in the case of extremal metrics, as well as the author [51, 46]. For the case of cscK metrics th”
On blowing up extremal Kähler manifolds (2010, methodology): “in Section 4 we set up the equation that we want to solve and give the proof of Theorem 1 assuming that we can solve this equation. In Section 5 we solve the equation using a gluing method similar to [4], completing the proof of Theorem 1. Our approach is slightly diﬀerent than that of Arezzo-Pacard”
Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K\ (2010, background): “|∇f(μ0)|L2 ≥ C · |f(μ0)| for some α ∈ [12 , 1).”
Kähler metrics with constant weighted scalar curvature and weighted K‐stability (2018, methodology): “Using a blowup technique and the glueing theorem of Arezzo–Pacard–Singer [8], Stoppa [76] and Stoppa–Székelyhidi [80] have shown that the existence of a cscK or extremal Kähler metric in 2πc1(L) does actually imply (1,wex)-K-stability relative to T, that is, that the corresponding Futaki invariant o”