On Weighted Twisted K-Energy and Its Applications

Abstract: We establish the convexity of the weighted twisted Mabuchi K-energy functional along geodesics in the finite energy space $\mathcal{E}<sup>{1,T}(X,ω)$, covering the case of divisors with mixed cusp and conic singularities. We then prove that coercivity (relative to the complex torus) of this functional is an open condition under cone angle perturbations. This is obtained from a general result of independent interest, which shows the stability of the coercivity under perturbations by certain twist currents. In particular, this yields the openness for the existence for cscK cone metrics and proves that coercivity at the cusp limit implies existence of cscK cone metrics for small cone angles.

• The paper establishes a variational framework extending weighted twisted Mabuchi energy to finite-energy spaces, where it is geodesically convex and its critical points solve weighted twisted scalar-curvature equations.
• The paper proves that relative coercivity remains stable under singular-data, twist-class, and cone-angle perturbations, with coercivity constants losing at most linearly in the perturbation size.
• The paper applies these results to show openness of cscK cone-metric existence and derive small-angle existence from coercivity at the cusp limit, while identifying automorphism and correspondence limitations.

This paper develops a variational framework for weighted twisted constant scalar curvature Kähler (cscK) metrics on compact Kähler manifolds, with twisting by positive $(1,1)$-currents that may carry divisorial singularities. The author establishes geodesic convexity of the weighted twisted Mabuchi functional in the finite energy space $\mathcal{E}^{1,T}(X,\omega)$ of $T$-invariant potentials, and proves that coercivity of the relative functional is an open condition under perturbations of cone angles. These results yield openness of existence for cscK cone metrics and a route from coercivity at the cusp limit to existence of conic metrics with small angles.

The weighted twisted variational framework

The setting is a compact Kähler manifold $(X,\omega)$ with a compact torus $T \subset \mathrm{Aut}_{red}(X,\chi)$ preserving a $T$-invariant positive $(1,1)$-current $\chi$ satisfying the decomposition condition

$\chi = \beta + \tfrac{1}{2}dd^c f + dd^c \hat f,$

where $\beta$ is smooth, $\mathcal{E}^{1,T}(X,\omega)$0, and $\mathcal{E}^{1,T}(X,\omega)$1. For smooth weights $\mathcal{E}^{1,T}(X,\omega)$2 and $\mathcal{E}^{1,T}(X,\omega)$3 on the moment polytope $\mathcal{E}^{1,T}(X,\omega)$4, following Lahdili's weighted formalism, the central object is the weighted twisted Mabuchi energy

$\mathcal{E}^{1,T}(X,\omega)$5

the sum of the weighted entropy, the twisted Ricci energy, and the weighted energy. Its Euler–Lagrange operator is $\mathcal{E}^{1,T}(X,\omega)$6, so critical points solve the weighted twisted scalar curvature equation $\mathcal{E}^{1,T}(X,\omega)$7.

A notable feature of this framework is its breadth: it recovers classical cscK ($\mathcal{E}^{1,T}(X,\omega)$8, $\mathcal{E}^{1,T}(X,\omega)$9, constant $T$0), extremal Kähler metrics ($T$1 affine-linear), generalized Kähler–Ricci solitons / $T$2-cscK metrics via $T$3, $T$4, and scalar-flat metrics on line bundle total spaces via $T$5. In each case, allowing $T$6 to be a current of integration along a divisor incorporates prescribed cone singularities.

The Futaki obstruction is handled by Calabi's extremal modification: since the weighted twisted Futaki pairing is positive definite (proved here by a moment-map argument), there is a unique affine extremal function $T$7 making the relative functional $T$8 invariant under the complexified torus $T$9. A useful structural fact is that $(X,\omega)$0 depends affinely on the twist $(X,\omega)$1, which underpins the quantitative openness estimates later.

Convexity along weak geodesics

The first main theorem asserts that $(X,\omega)$2 admits a unique greatest lower semicontinuous extension to $(X,\omega)$3 that is convex and continuous along weak geodesics. The proof strategy combines three ingredients:

• Convexity of the twist term: using $(X,\omega)$4-geodesic approximations of the degenerate Monge–Ampère boundary value problem on the annulus $(X,\omega)$5, the author shows that $(X,\omega)$6, which is a positive current; hence $(X,\omega)$7 is convex.
• Convexity of the weighted part: inherited from Apostolov–Jubert–Lahdili's theorem for $(X,\omega)$8.
• Extension to finite energy: the entropy term extends as a greatest lsc functional via Berman–Darvas–Lu-type approximation arguments, while the Ricci and weighted energy terms extend continuously with explicit Hölder/Lipschitz estimates (exponent $(X,\omega)$9).

An important technical remark concerns the decomposition condition: the auxiliary potential $T \subset \mathrm{Aut}_{red}(X,\chi)$0 is essential when handling Poincaré-type singularities. Without it, $T \subset \mathrm{Aut}_{red}(X,\chi)$1 is not smooth and the Berman–Darvas–Lu machinery does not apply; alternatively dropping $T \subset \mathrm{Aut}_{red}(X,\chi)$2 changes the Euler–Lagrange equation to an undesired form.

Compared with prior work of Xu and Xu–Zheng, which established uniqueness of Poincaré-type extremal and cscK metrics only for smooth divisors, this framework handles arbitrary simple normal crossing divisors with mixed cusp (cone angle $T \subset \mathrm{Aut}_{red}(X,\chi)$3) and conic singularities, and works in the full finite energy space rather than the space of Poincaré-type potentials. As a corollary, the first variation formula for $T \subset \mathrm{Aut}_{red}(X,\chi)$4 holds on Poincaré-type potentials, so critical points satisfy $T \subset \mathrm{Aut}_{red}(X,\chi)$5; this relies on a Ricci decomposition $T \subset \mathrm{Aut}_{red}(X,\chi)$6 proved in the appendix via a variant of the Poincaré–Lelong formula.

Openness of coercivity

The second main theorem states that coercivity of $T \subset \mathrm{Aut}_{red}(X,\chi)$7 relative to $T \subset \mathrm{Aut}_{red}(X,\chi)$8 is stable under perturbations of the twist. Two general perturbation results are proven:

• Perturbations of the singular data ($T \subset \mathrm{Aut}_{red}(X,\chi)$9, $T$0 with $T$1): coercivity persists with constants $T$2 and $T$3. The key estimate is an entropy comparison lemma showing $T$4, derived from the Legendre transform characterization of relative entropy.
• Smooth perturbations of the twist class: coercivity degrades at most linearly, $T$5, using the Lipschitz estimates for weighted energies and the affine dependence of $T$6 on $T$7.

Both estimates are quantitative: the loss in the coercivity constant is linearly controlled by the perturbation parameter, and the argument extends by induction to multi-parameter perturbations. Specializing to a simple normal crossing divisor $T$8 with angle vector $T$9, the main openness theorem states that if $(1,1)$0 satisfies a coercivity bound with constant $(1,1)$1, then for any $(1,1)$2 there is a neighborhood $(1,1)$3 of $(1,1)$4 such that the same bound holds with constant $(1,1)$5 for all angles in $(1,1)$6, with $(1,1)$7 linearly controlled by $(1,1)$8. The proof splits into the cases $(1,1)$9 (handled directly by the smooth perturbation estimate) and $\chi$0 (handled by interpolating through the reference angle $\chi$1 and applying the entropy comparison with exponent $\chi$2).

Applications to conic cscK metrics

Combining the openness theorem with Zheng's characterization of cscK cone metrics via coercivity yields two existence results, both assuming $\chi$3 is smooth and that $\chi$4 carries no nonzero holomorphic vector field preserving $\chi$5:

• Openness of existence: if $\chi$6 admits a cscK cone metric with cone angle $\chi$7, then it admits one for all angles in a neighborhood of $\chi$8. This recovers, by variational means, the phenomenon observed by Keller–Zheng via invertibility of the Lichnerowicz operator.
• From cusp coercivity to small-angle existence: if the twisted Mabuchi functional $\chi$9 is coercive on $\chi = \beta + \tfrac{1}{2}dd^c f + dd^c \hat f,$0, then $\chi = \beta + \tfrac{1}{2}dd^c f + dd^c \hat f,$1 admits a cscK cone metric with cone angle $\chi = \beta + \tfrac{1}{2}dd^c f + dd^c \hat f,$2 for all sufficiently small $\chi = \beta + \tfrac{1}{2}dd^c f + dd^c \hat f,$3. Compared with Aoi's deformation-theoretic result, which requires $\chi = \beta + \tfrac{1}{2}dd^c f + dd^c \hat f,$4 and triviality of automorphisms of both $\chi = \beta + \tfrac{1}{2}dd^c f + dd^c \hat f,$5 and $\chi = \beta + \tfrac{1}{2}dd^c f + dd^c \hat f,$6, this variational criterion assumes only coercivity—a strictly weaker hypothesis than existence.

The restriction to smooth divisors and the unweighted setting in these corollaries is not intrinsic to the method but reflects the current scope of Zheng's coercivity-to-existence correspondence.

Limitations and open questions

Several restrictions are acknowledged explicitly. The coercivity-to-existence corollaries require the absence of holomorphic vector fields preserving $\chi = \beta + \tfrac{1}{2}dd^c f + dd^c \hat f,$7; with nontrivial automorphisms, deformations do not preserve the cscK condition even at fixed polarization, as noted already by LeBrun–Simanca. The correspondence between coercivity and existence used in the applications is available only for smooth divisors and unweighted functionals, so the mixed cusp/conic and weighted versions of the openness theorem do not yet translate into existence statements. The Ricci decomposition for Poincaré-type metrics requires boundedness of the first four derivatives of the potential relative to the model metric, though the author notes only four derivatives are actually needed.

The paper closes with a conjecture characterizing existence of $\chi = \beta + \tfrac{1}{2}dd^c f + dd^c \hat f,$8-extremal Poincaré-type Kähler metrics by three conditions: global coercivity of the twisted K-energy on $\chi = \beta + \tfrac{1}{2}dd^c f + dd^c \hat f,$9, coercivity of the extremal K-energy on $\beta$0, and a boundary stability condition requiring $\beta$1. This refines Székelyhidi's relative K-stability notion and Auvray's slope inequality, and generalizes to all strata of a simple normal crossing divisor. Whether this conjecture holds—and whether the coercivity-existence correspondence can be extended to the weighted, singular settings treated here—remains open.

Conclusion

The paper provides a unified pluripotential treatment of weighted twisted K-energy, establishing its greatest lower semicontinuous extension and geodesic convexity on finite energy spaces for twists satisfying a natural decomposition condition, including divisors with mixed cusp and conic singularities. Building on this, it proves quantitative openness of coercivity under cone-angle perturbations, yielding openness of cscK cone metric existence and a coercivity-based small-angle existence criterion that weakens the hypotheses of earlier deformation arguments. The remaining gap between the generality of the analytic results and the currently available existence correspondences defines the natural next target for this program.