Constant Negative Chern Scalar Curvature Metric

• Constant negative Chern scalar curvature metrics are Hermitian metrics on complex manifolds with a uniform prescribed negative curvature, generalizing classical uniformization and the Yamabe problem.
• They are shown to exist uniquely via nonlinear elliptic PDEs, continuity methods, and curvature flows under negative Gauduchon degree conditions.
• Explicit constructions in symmetric settings, variational formulations, and asymptotic analyses on noncompact manifolds illustrate practical applications and deeper geometric insights.

A constant negative Chern scalar curvature metric is a Hermitian metric on a complex manifold for which the Chern scalar curvature is a prescribed negative constant throughout the manifold. This problem—prescribing constant negative Chern scalar curvature in a given conformal class—lies at the intersection of complex differential geometry and nonlinear elliptic PDE theory, generalizing the uniformization of Riemann surfaces and the Yamabe problem to the Hermitian non-Kählerian setting. The existence, uniqueness, and structure of such metrics depend crucially on the interplay between the underlying complex geometry, the analytic properties of the relevant PDEs, and invariants such as the Gauduchon degree.

1. Definitions and Fundamental Structures

The Chern connection $\nabla^{\mathrm{Ch}}$ on a Hermitian manifold $(M,J,g)$ is the unique affine connection preserving both the metric $g$ and the complex structure $J$, with the $J$-torsion restricted to types $(2,0)+(0,2)$. The associated Chern scalar curvature is given, in local holomorphic coordinates, by

$$S_C(\omega) = -g^{i\overline{j}}\partial_i \overline{\partial}_j \log \det(g_{k\overline{\ell}}) = -\Delta_{\mathrm{Ch}}(\log \omega^n),$$

where $\Delta_{\mathrm{Ch}}$ is the Chern-Laplacian.

For a Hermitian conformal class $$\{\omega_0\}_{\mathrm{conf}}=\{e^u \omega_0 : u\in C^\infty(M)\}$$, the unique Gauduchon metric $\eta=e^\phi \omega_0$ is characterized (up to scaling) by the condition $d(\theta\wedge\eta^{n-1})=0$, where $\theta$ is the Lee form. The Gauduchon degree of a conformal class is

$\Gamma(\{\omega_0\}) := \int_M S_C(\eta)\,\eta^n.$

If $\Gamma(\{\omega_0\})<0$, the conformal class admits special analytic and geometric properties for the existence of metrics with constant negative Chern scalar curvature (Angella et al., 2015, Fusi, 2021, Yu, 7 Jan 2025).

2. Existence and Uniqueness on Compact Manifolds

For compact complex manifolds, a canonical approach to constructing constant negative Chern scalar curvature metrics is to reduce to a second-order nonlinear elliptic PDE. Given a unit-volume Gauduchon representative $g_G \in [g_0]$ with $S^{\mathrm{Ch}}(g_G) = I < 0$, the metric $g = e^{2u/n}g_G$ has

$$S^{\mathrm{Ch}}(g) = e^{-2u/n}\left( \Delta^{\mathrm{Ch}}_{g_G}u + S^{\mathrm{Ch}}(g_G) \right),$$

so prescribing $S^{\mathrm{Ch}}(g) \equiv K < 0$ reduces to solving

$$-\Delta^{\mathrm{Ch}}_{g_G} u + I = K e^{2u/n}, \quad \int_M e^{2u/n} \, dV_{g_G} = 1.$$

Existence and uniqueness are established for all $K < 0$, provided $I<0$, by the continuity method, exploiting elliptic regularity, a priori $C^0$ bounds from the maximum principle, and volume normalization (Angella et al., 2015, Fusi, 2021, Li et al., 2022). When $\Gamma(\{\omega_0\})=0$, the analogous problem becomes linear and is always solvable, while for positive Gauduchon degree obstructions arise.

3. Analytic Methods and Prescribed Curvature Flows

The Chern scalar curvature flow offers a powerful dynamical framework for realizing constant negative curvature metrics. For compact Hermitian manifolds with negative Gauduchon degree and a balanced metric in the conformal class, the evolution equation

$$\partial_t u = \Delta_{\mathrm{Ch}} u - S_0 + c e^u, \quad c<0,$$

is globally well-posed for all initial data and converges smoothly as $t \to \infty$ to the unique solution $u_\infty$ of

$$- \Delta_{\mathrm{Ch}} u_\infty + S_0 = c e^{u_\infty}, \quad \int_M e^{u_\infty} \eta^n = 1,$$

ensuring existence and uniqueness of a conformal Hermitian metric with Chern scalar curvature $c<0$ (Yu, 7 Jan 2025). The proof leverages a strictly decreasing energy functional, $C^0$ maximum-principle bounds, parabolic Schauder estimates, and Łojasiewicz–Simon theory to control convergence.

4. Variational and Extremal Metric Perspective

Constant negative Chern scalar curvature metrics also arise as minimizers of the $n$-Calabi functional,

$$C_n(\omega_f) = \int_M |s_f|^n\, \frac{\omega_f^n}{n!},$$

where $s_f$ is the Chern scalar curvature of $\omega_f=e^f \omega$. The associated Euler-Lagrange equation is the fourth-order nonlinear PDE

$\square_f^*\left( s_f |s_f|^{n-2} \right) = 0,$

which, when restricted to the Gauduchon representative, reduces to a second-order Poisson equation whose solution gives the unique extremal metric in the conformal class with constant negative Chern scalar curvature precisely when the total Chern scalar curvature of the Gauduchon metric is negative (Yang et al., 21 May 2025). The sign of the constant is completely determined by the sign of $\int_M S_G \omega_G^n$.

5. Noncompact Manifolds and Asymptotics

For complete noncompact Hermitian manifolds, the existence of complete conformal metrics with prescribed constant negative Chern scalar curvature is established under geometric hypotheses: pointwise nonpositivity (and controlled decay at infinity) of the background Chern scalar curvature,

$$S^{\mathrm{Ch}}(\omega)(x) \leq -\frac{b^2}{r(x)^l} \qquad (r(x)\gg1),$$

and lower bounds for the second Chern–Ricci tensor and controlled torsion growth. The PDE

$$-\Delta^{\mathrm{Ch}}_\omega u + S^{\mathrm{Ch}}(\omega) = \lambda\, e^{2u/n}, \quad \lambda<0,$$

is solved using the construction of global barriers (lower solutions), monotone iteration, $C^0$ and local regularity estimates, and a proof of geodesic completeness of the metric $\tilde\omega = e^{2u/n} \omega$ (Yu, 5 Jan 2026).

6. Explicit Examples and Geometric Constructions

Explicit families of constant negative Chern scalar curvature metrics have been constructed in high symmetry settings. For small resolutions of orbifolds such as $C^2/\mathbb{Z}_n$, metric ansätze of Eguchi-Hanson type provide both Kähler (ALE) and non-Kähler (ALH) solutions. For instance, cohomogeneity-one metrics of the form

$$g_I = f(r)^{-2} dr^2 + r^2 (\sigma_1^2 + \sigma_2^2 + f(r)^2 \sigma_3^2),$$

can yield, after solving an explicit ODE for $f(r)^2$, a complete Kähler metric with $S = R = -24B < 0$ (Chen et al., 2020). Non-Kähler ALH metrics, with asymptotically hyperbolic geometry and potentially negative total energy, are similarly constructed in the noncompact setting.

7. Broader Setting: Almost Hermitian Manifolds and Related Problems

The prescribed Chern scalar curvature problem is subsumed within the broader context of prescribed Gauduchon scalar curvature and the generalized Yamabe problem for almost Hermitian structures. The solvability criteria remain tied to a sign condition on an invariant (the Gauduchon degree) of the conformal class, and the exact structure of the nonlinearity in the governing equation. When the integral of the Chern scalar (or Gauduchon scalar) curvature is negative, the solution to the corresponding Kazdan–Warner-type equation exists and is unique (Li et al., 2022). When generalized to other connections (Bismut, Lichnerowicz), the sign of associated degree constants remains decisive.

Summary Table: Key Results on Constant Negative Chern Scalar Curvature Metrics

Context	Existence Criteria	Uniqueness
Compact, Hermitian	$\Gamma(\{\omega_0\})<0$	Unique up to scaling/volume (Angella et al., 2015, Fusi, 2021)
Complete noncompact	Background $S^{\mathrm{Ch}}\le0$, decay	Unique given background, under hypotheses (Yu, 5 Jan 2026)
Variational problem	$\int_M S_G \omega_G^n<0$	Unique extremal/minimizing metric (Yang et al., 21 May 2025)

Existence, uniqueness, and the explicit form of constant negative Chern scalar curvature metrics are thus shaped by topological invariants (Gauduchon degree), analytic methods (continuity path, flow, monotone iteration), and geometric structure (balanced metrics, torsion, completeness). These results collectively extend the classical uniformization and Yamabe paradigms to the general Hermitian framework.