Prescribing Chern Scalar Curvatures

• Prescribing Chern scalar curvatures is the study of designing Hermitian metrics via solving nonlinear elliptic and parabolic PDEs on complex manifolds.
• The approach leverages conformal invariants like the Gauduchon degree to classify existence scenarios based on negative, zero, or positive curvature integrals.
• Analytic techniques such as the maximum principle, parabolic flows, and barrier methods extend classical Yamabe theory to complex, non-Kähler settings.

Prescribing Chern scalar curvatures concerns the existence, uniqueness, and structure of Hermitian metrics on complex manifolds with prescribed values of the Chern scalar curvature, particularly within a conformal class. The problem generalizes the classical prescribed scalar curvature and Yamabe problems from Riemannian to Hermitian, including non-Kähler, settings, introducing new phenomena due to torsion and structure of the Chern connection. This topic underpins the study of geometric flows, variational problems, existence theory on compact and noncompact manifolds, and links to broader complex differential geometry.

1. Geometric Background and Conformal Structure

Let $(M^{2n},J)$ be a complex manifold of complex dimension $n$, endowed with a Hermitian metric $\omega$ (or $g$). The Chern connection $\nabla^{\mathrm{Ch}}$ is the unique connection preserving $g$ and $J$ with torsion of type $(2,0)+(0,2)$ only. Its curvature yields the first Chern–Ricci form $\mathrm{Ric}^{(1)}(\omega)$ and the Chern scalar curvature $$S^{\mathrm{Ch}}(\omega) = \operatorname{tr}_\omega(\mathrm{Ric}^{(1)}(\omega)) = \operatorname{tr}_\omega(i\partial\bar\partial\log\omega^n)$$.

For any smooth function $u$, a conformal change $\widetilde\omega = e^{2u}\omega$ induces a transformation law for the Chern scalar curvature: $$S^{\mathrm{Ch}}(\widetilde\omega) = e^{-2u}( \Delta^{\mathrm{Ch}} u + S^{\mathrm{Ch}}(\omega) )$$ where $$\Delta^{\mathrm{Ch}}u = 2\,\operatorname{tr}_\omega(\partial\bar\partial u) = \Delta^d u + \langle du, \theta \rangle_\omega$$, with $\theta$ the Lee form ($d\omega^{n-1} = \theta \wedge \omega^{n-1}$) (Fusi, 2021).

The prescribed Chern scalar curvature problem thus reduces to solving nonlinear elliptic or parabolic PDEs for $u$ such that $S^{\mathrm{Ch}}(\widetilde\omega) = g$ for a given function $g:M \to \mathbb{R}$.

2. The Gauduchon Degree and Conformal Classes

The Gauduchon degree $I([\omega])$ is a conformal invariant defined for a Hermitian conformal class $[\omega]$ by

$$I([\omega]) = \int_M S^{\mathrm{Ch}}(\gamma) \, d\mu_\gamma$$

where $\gamma$ is the unique volume-one Gauduchon metric in $[\omega]$, characterized by $d^*\theta_\gamma = 0$ (Yu, 2022, Angella et al., 2021, Fusi, 2021).

This degree plays a fundamental role, paralleling the Yamabe invariant in Riemannian geometry, in dictating the existence theory for the prescribed Chern scalar curvature problem. The sign of $I([\omega])$ leads to a tripartite classification:

• Negative ($I([\omega])<0$): favorable existence theory for nonpositive, nontrivial $g$;
• Zero ($I([\omega])=0$): balanced metrics admit solutions for sign-changing $g$ with negative mean;
• Positive ($I([\omega])>0$): only local or perturbative existence in general (Fusi, 2021).

3. Compact Manifolds: Existence, Flows, and Techniques

On compact Hermitian manifolds with $I([\omega])<0$, given $g\leq 0$ and $g\not\equiv 0$, the equation

$-\,\Delta_{\rm Ch} u + S_{\rm Ch}(w) = g\,e^{2u}$

admits a unique smooth solution $u$, yielding a conformal metric with $S_{\rm Ch}(\widetilde w)=g$. The analytic techniques involve:

• Maximum principle: Ensures $C^0$ a priori bounds, controlling the elliptic PDE or geometric flow (Yu, 2022).
• Parabolic flows: The Chern–Yamabe-type flow evolves $u$ by

$$\partial_t u = \Delta_{\rm Ch}u - S_{\rm Ch}(n) + g\,e^{2u}$$

and global existence and long-time convergence are guaranteed under the negativity of the Gauduchon degree, sometimes requiring a balanced metric in the conformal class (Yu, 7 Jan 2025, Yu, 2022).

• Sub/supersolution methods: Construction of explicit upper and lower barriers for monotone iteration schemes and variational minimization, especially useful for sign-changing $g$ (Yu, 2022, Fusi, 2021).

For sign-changing $g = g_0 + \lambda$ with $\max g_0=0$, there exists a threshold $\lambda^*\in(0, -\min g_0]$ such that:

• $0<\lambda<\lambda^*$: Problem is solvable;
• $\lambda>\lambda^*$: No solution exists;
• In complex dimension $n=2$ with balanced backgrounds, solutions exist even at the limiting value $\lambda=\lambda^*$ (Yu, 2022).

Table: Existence Results for Negative Gauduchon Degree (Compact Case)

Curvature function $g$	Balanced in $[\omega]$?	Existence of Solution
$g<0$ everywhere	Not required	Yes, unique/global
$g\le0$, $g\not\equiv0$	Yes	Yes, unique/global
$g=g_0+\lambda$ (sign-changing)	Sometimes needed	If $\lambda<\lambda^*$
$g=g_0+\lambda^*$, $n=2$	Yes	Yes

4. Noncompact Manifolds and Generalizations

On complete noncompact Hermitian manifolds with $S^{\mathrm{Ch}}(\omega)\leq 0$, the existence theory extends via:

• Analytic sub/supersolution and barrier methods, leveraging sharp Laplacian comparison and local $C^0$-estimates (Yu, 17 Jun 2025, Yu, 5 Jan 2026, Wu et al., 2023).
• Monotone iteration provides global smooth solutions provided explicit lower solutions can be constructed, combining Dirichlet problems on compact exhaustion domains and global barriers (Yu, 17 Jun 2025, Yu, 5 Jan 2026).
• Weak Omori–Yau maximum principle, Laplacian comparison theorems, and Sobolev techniques are used to handle the noncompactness and torsion contributions in the underlying geometry (Yu, 17 Jun 2025).

Sufficient conditions for the solvability involve decay of the background Chern scalar curvature, uniform control on the Chern–Ricci tensor and torsion, and the existence of appropriately bounded subsolutions. For instance, if $S^{\mathrm{Ch}}(\omega)\leq -b^2 r^{-\ell}$ outside a compact set, and the prescribed $S$ satisfies $-\hat c^2(1+r^k) \leq S<0$, then a complete conformal metric with $S^{\mathrm{Ch}}(\widetilde\omega)=S$ exists (Yu, 5 Jan 2026, Yu, 17 Jun 2025).

5. Linearization, Stability, and Local Theory

The mapping from Hermitian metrics to their Chern scalar curvature,

$$\mathrm{Scal}^{\mathrm{Ch}}: \mathcal{M}_H \to C^\infty(M, \mathbb{R}),$$

is smooth, and its linearization at a background metric $g$ is an elliptic operator involving the trace of the variation, the Lee form, and the Chern–Ricci endomorphism: $$L_g(h) = \Delta_g(\mathrm{tr}_g h) + g(d(\mathrm{tr}_g h), \theta_g) - g(h, S^{\mathrm{Ch}}(g))$$ (Angella et al., 2021).

Fredholm theory applies, and under suitable nondegeneracy (e.g., $g$ not first-Chern-Einstein, or with eigenvalue below the divergence of the Lee form), the linearization is surjective, giving local submersion and thus local existence of metrics with prescribed Chern scalar curvature near a given background. The solution space near such $g$ forms a smooth submanifold of metrics, modeled on the kernel of $L_g$ (Angella et al., 2021).

6. Special Cases, Examples, and Comparative Features

Examples:

• On flat complex tori, all small perturbations of the flat metric with zero-average functions as target curvature are admissible (Angella et al., 2021).
• On primary Hopf surfaces (positive Gauduchon degree), the problem is locally solvable for target functions near the background curvature (Angella et al., 2021).
• For surfaces, the theory generalizes classical Kazdan–Warner and multiplicity theorems, including nonuniqueness and infinite solutions for the prescribed Chern scalar curvature on noncompact 2-manifolds (Wu et al., 2023).

Comparative Remarks:

• The Chern–scalar prescription problem reduces to the Riemannian case when the Hermitian metric is Kähler; the problem then coincides with the classical Yamabe problem (Fusi, 2021, Yu, 17 Jun 2025).
• Outside the balanced case, the presence of torsion (Lee form) destroys the self-adjointness and variational structure of the scalar curvature operator, necessitating more refined analytic techniques.
• Volume obstructions arise via the integral identity enforced by the Gauduchon degree, as in the zero-degree, balanced case, where the prescribed function must have zero mean with respect to the conformal measure (Fusi, 2021).
• On noncompact manifolds, existence results rely on weighted Sobolev/Poincaré inequalities, exhaustion, and barrier methods, paralleling Aviles–McOwen theory for the Riemannian setting (Wu et al., 2023, Yu, 17 Jun 2025).

7. Open Directions and Further Developments

Recent works suggest extensions to:

• Singular Chern–Yamabe problems on divisorially singular complex spaces (Wu et al., 2023).
• Flow approaches in noncompact or balance-lacking settings (Yu, 7 Jan 2025, Wu et al., 2023).
• Generalizations to higher-rank settings, such as Hermitian–Yang–Mills–Higgs metrics on vector bundles (Wu et al., 2023).
• The identification of sharp analytic (weighted Sobolev-type) conditions to replace the various main assumptions for curvature control at infinity and existence in the noncompact case.

Taken together, prescribing Chern scalar curvature merges geometric analysis, parabolic and elliptic PDE theory, and the specifics of Hermitian and complex geometric structures, with key advances focusing on the interaction of conformal invariants, torsion, and global geometric/topological obstructions (Yu, 2022, Yu, 5 Jan 2026, Yu, 17 Jun 2025, Yu, 7 Jan 2025, Fusi, 2021, Angella et al., 2021, Wu et al., 2023).