Convexity of the K-energy on the space of K\"ahler metrics

By Robert J. Berman, B. Berndtsson. 2014

122 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/43f2626c0282195801037ea955ad460e82cfefe8

How researchers see this paper

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

1. Robert J. Berman
2. Chi Li
3. Tam'as Darvas
4. Jiyuan Han
5. B. Berndtsson
6. R. Dervan
7. Xiuxiong Chen
8. S. Boucksom

Citations in the corpus

• G-uniform stability and Kähler–Einstein metrics on Fano varieties (2019, unknown): “2] (see also [7, 13]), M satisfies the following convexity property along our geodesic segments: D(φj(s)) ≤ M(φj(s)) ≤ Sj − s Sj M(ψ) + s Sj M(φj) ≤ C + s Sj (γjJ(φj) − j) ≤ C + s Sj γjJ(φj).”
• On the Yau‐Tian‐Donaldson Conjecture for Generalized Kähler‐Ricci Soliton Equations (2020, methodology): “2(Γ(u1)), which means that Γ is a closed 1-form and so the integral does not depend on the choice of paths. 4.3. Convexity of M when Xis smooth. In thissubsection, we will use Berman-Berndsson’s work [5] to show that M is convex along a weak geodesic in case Xis smooth. This is a prelude to the gene”
• Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, background): “One important feature of twisted cscK metrics is that they are unique, when they exist [3].”
• The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties (2019, unknown): “Moreover by [3] M̂ (φ (s)) is convex in s ∈ [0, 1].”
• SECOND-ORDER ESTIMATES AND REGULARITY FOR FULLY NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS (2012, unknown): “Their results and techniques have found useful applications in some important problems; see e.g. the work of P.-F. Guan [26], [27] and papers of Chen [9], Blocki [2], and Phong and Sturm [45] on the Donaldson conjectures [11] in K¨ahler geometry.”
• On the constant scalar curvature Kähler metrics (II)—Existence results (2020, methodology): “This conjecture was verified by the fundamental work of Berman and Berndtsson [2] (c.f. Chen-Li-Paun [23] also).”
• Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics (2014, background): “It has been conjectured by Chen that M(φt) is convex along any weak geodesic as above [19] (the case when c1(X) ≤ 0 was settled by Chen).”
• Geometric pluripotential theory on Kähler manifolds (2019, methodology): “Inspired from this, a careful analysis of infinite dimensional spaces led Bando–Mabuchi to prove uniqueness of KE metrics [2], and later Berman–Berndtsson to discover even more general uniqueness results [8].”
• On the constant scalar curvature Kähler metrics (I)—A priori estimates (2017, background): “It is well known that in a given Kähler class, cscK metrics is global minimizer of the K-energy functional, by the main result of [2].”
• On the constant scalar curvature Kähler metrics, general automorphism group (2018, unknown): “Then the uniqueness result of [1](Theorem 1.”