Determine existence of extremal Kähler metrics in explicit Kähler classes

Determine whether an extremal Kähler metric exists in a given Kähler class on a general compact Kähler manifold, thereby addressing the widely open problem of testing relative uniform K-stability in explicit examples.

Background

The paper places its results in the context of the Yau–Tian–Donaldson correspondence, which characterizes the existence of extremal Kähler metrics through relative uniform K-stability. Although this characterization is established in general, verifying relative uniform K-stability for concrete Kähler manifolds and classes remains difficult because it requires testing positivity against degenerations such as test configurations or more general models.

The authors identify determining existence in a specified Kähler class as a broadly unresolved problem. Their contribution addresses a special family—projectivizations of holomorphic vector bundles over compact complex curves—by translating the relevant stability condition into a weighted stability problem for a moment polytope and relating it to a second-order linear PDE for compatible extremal almost Kähler metrics.

References

It is expected that the set \mathcal{S}_{\mathrm{ext}} is also connected.

— On the existence of coupled extremal Kähler metrics on ruled surfaces  (2609.18418 - Mete, 16 Sep 2026) in Section 1, subsection “Main results,” immediately after Theorem 1.1

It is an interesting question whether the pair of normalized Kähler classes (\Omega_a, \Omega_b) admits a coupled extremal Kähler metric.

— On the existence of coupled extremal Kähler metrics on ruled surfaces  (2609.18418 - Mete, 16 Sep 2026) in Section 1, immediately before “Organization of the paper”

For a general Kähler manifold, the notion of relative uniform K-stability is sophisticated and remains hard to test on explicit examples. In particular, the problem of determining whether an extremal metric exists in a given Kähler class remains widely open.

— New extremal Kähler metrics on projective bundles  (2609.01094 - Jubert, 1 Sep 2026) in Section 1, Introduction

The relationship between relative uniform K-stability and the existence of extremal almost Kähler metrics on a manifold admitting a Kähler structure remains a largely open question.

— New extremal Kähler metrics on projective bundles  (2609.01094 - Jubert, 1 Sep 2026) in Section 1, Introduction

In particular, we were not able to replicate Wang and Zhou's octagon examples.

— The Elusive Relatively K-Unstable Delzant Octagon: A Numerical Search  (2609.10019 - Delcroix et al., 9 Sep 2026) in Section 1, Introduction; Section 4, “The tentative examples of Wang and Zhou: a dead end?”