On sharp lower bounds for Calabi-type functionals and destabilizing properties of gradient flows

Xia, Mingchen

doi:10.2140/apde.2021.14.1951

Cited by 13 publications

(11 citation statements)

References 48 publications

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“…Calabi flow) and minimize an L 2 -normalized non-Archimedean Ding-invariant (resp. L 2 -normalized radial Calabi-functional) (see [42,83]). In this paper we are interested in optimal degenerations that arise in the study of Hamilton-Tian conjecture about the long time behavior of Kähler-Ricci flows.…”

Section: Introductionmentioning

confidence: 99%

Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties

Han,

2020

Preprint

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We prove that for any Q-Fano variety X, the special R-test configuration that minimizes the H NA -functional is unique and has a K-semistable Q-Fano central fibre (W, ξ). Moreover there is a unique K-polystable degeneration of (W, ξ). As an application, we confirm the conjecture of Chen-Sun-Wang about the algebraic-uniqueness for Kähler-Ricci flow limits on Fano manifolds which implies that the Gromov-Hausdorff limit of the flow does not depend on the choice of initial Kähler metrics. The results are achieved by studying algebraic optimal degeneration problems via new functionals for real valuations, which are analogous to the minimization problem for normalized volumes.

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Section: Introductionmentioning

confidence: 99%

Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties

Han,

2020

Preprint

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“…[Ino2] of µ λ -cscK metrics, we encounter extremal metrics in the limit λ → −∞. Here we observe its non-archimedean counterpart: in the limit λ → −∞ the nonarchimedean µ-entropy is related to normalized Donaldson-Futaki invariant which appears in Donaldson-Xia's minimax principle [Don1,Xia1]. See also [Don2,Sze1], [Der1,Der2,Ino4] and section 4.3.3.…”

Section: It Follows From What We Proved That μλmentioning

confidence: 66%

“…[He,CSW,DS,HL2,BLXZ]). We also refer to analytic result for Calabi flow [Xia1]. We will discuss these works from our non-archimedean perspective in section 4.3.1 and section 4.3.3.…”

Section: −2πmentioning

confidence: 99%

“…for smooth (X, L), which can be made into the equality by completing the domain of the functionals [Xia1] in a suitable way. Now as in [Ino2,Ino4], let us observe the extremal limit λ → −∞ of the nonarchimedean µ-entropy.…”

Section: It Follows From What We Proved That μλmentioning

confidence: 99%

“…Here we discuss the maximization problem for C NA , referring to various fundamental conjectures. It is studied in [Xia1] that for smooth (X, L) the normalized Donaldson-Futaki invariant extends to the space R 2 (X, ω) of geodesic rays in E 2 (X, L) in an (archimedean) analytic way: we put…”

Section: It Follows From What We Proved That μλmentioning

confidence: 99%

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Entropies in $μ$-framework of canonical metrics and K-stability, II -- Non-archimedean aspect: non-archimedean $μ$-entropy and $μ$K-semistability

Inoue¹

2022

Preprint

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This is the second in a series of two papers studying µ-cscK metrics and µK-stability from a new perspective, inspired by observations on µcharacter in [Ino3] and on Perelman's W -entropy in the first paper [Ino4].This second paper is devoted to studying a non-archimedean counterpart of Perelman's µ-entropy. The concept originally appeared as µ-character of polarized family in the previous research [Ino3], where we used it to introduce an analogue of CM line bundle adapted to µK-stability.We firstly show some differential of the characteristic µ-entropy μλ ch is the minus of µ λ -Futaki invariant, which connects µ λ K-semistability to the maximization of characteristic µ λ -entropy. It in particular provides us a criterion for µ λ K-semistability working without detecting the vector ξ involved in the µ λ ξ -Futaki invariant. We observe a family of filtrations {F ξ+τ (X ,L) } τ ∈[0,∞) associated to a test configuration (X , L) and a vector field ξ acting on (X, L) to consider the differential. We conceptualize such family of filtrations as polyhedral configuration and study its generalities. The concept implicitly appeared in many literatures involved in R-test configuration.In the latter part, we propose a non-archimedean pluripotential approach to the maximization problem. In order to adjust the characteristic µ-entropy μλ ch to Boucksom-Jonsson's non-archimedean framework, we introduce a natural modification μλ NA which we call non-archimedean µ-entropy. We extend the non-archimedean µ-entropy from the set of test configurations to a space E exp NA (X, L) of non-archimedean psh metrics on the Berkovich space X NA , which is endowed with a complete metric structure. We introduce moment measure χDϕ on Berkovich space for this sake, which can be considered as a hybrid of Monge-Ampère measure and Duistermaat-Heckman measure.We also compare our µ-framework with other frameworks: H-entropy framework in the context of Kähler-Ricci flow and Calabi energy framework in the context of Calabi flow. Some illustrations by toric examples are attached in Appendix.

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Calabi type functionals for coupled Kähler–Einstein metrics

Nakamura

2023

Ann Glob Anal Geom

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We introduce the coupled Ricci–Calabi functional and the coupled H-functional which measure how far a Kähler metric is from a coupled Kähler–Einstein metric in the sense of Hultgren–Witt Nyström. We first give corresponding moment weight type inequalities which estimate each functional in terms of algebraic invariants. Secondly, we give corresponding Hessian formulas for these functionals at each critical point, which have an application to a Matsushima type obstruction theorem for the existence of a coupled Kähler–Einstein metric.

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On sharp lower bounds for Calabi-type functionals and destabilizing properties of gradient flows

Cited by 13 publications

References 48 publications

Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties

Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties

Entropies in $μ$-framework of canonical metrics and K-stability, II -- Non-archimedean aspect: non-archimedean $μ$-entropy and $μ$K-semistability

Calabi type functionals for coupled Kähler–Einstein metrics

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