Complete Noncompact Hermitian Manifolds

• Complete noncompact Hermitian manifolds are complex manifolds with Hermitian metrics ensuring geodesic completeness and noncompactness, enabling detailed curvature and torsion analyses.
• Analytic strategies like sub- and super-solution methods, monotone iteration, and Laplacian comparisons effectively address the prescribed Chern scalar curvature problem.
• Extending classical curvature prescription problems beyond the Kähler framework, these techniques facilitate the construction of Chern–Einstein metrics and robust geometric flow theories.

A complete noncompact Hermitian manifold is a complex manifold $(M^n, J)$ of complex dimension $n \geq 2$ equipped with a Hermitian metric $g$ (or its associated fundamental $(1,1)$-form $\omega$) such that the induced Riemannian metric is geodesically complete and $M$ is noncompact. The theory of such manifolds is central to complex differential geometry, particularly in the study of curvature prescription, fully nonlinear elliptic equations, and geometric flows beyond the Kähler category.

1. Fundamental Structures and Notions

Let $(M^n, J, g)$ denote a complex manifold with Hermitian metric $g$. The fundamental $(1,1)$-form is defined by $\omega(X, Y) = g(JX, Y)$. The unique Chern connection, denoted $\nabla^{\mathrm{Ch}}$, preserves both $g$ and $J$ and is characterized by its torsion tensor having vanishing $(1,1)$-component. The first Chern–Ricci form is expressed as

$$\mathrm{Ric}^{(1)}(\omega) = \sqrt{-1}\,\bar\partial \partial \log\omega^n,$$

and the Chern scalar curvature is

$$S^{\mathrm{Ch}}_g = \mathrm{tr}_{\,\omega}\bigl(\mathrm{Ric}^{(1)}(\omega)\bigr).$$

A Hermitian manifold is complete if the metric is geodesically complete, and noncompact if the underlying manifold is not compact. On such manifolds, the analytic and geometric behavior at infinity is governed by growth conditions on the Chern–Ricci curvature, torsion, and scalar curvature, frequently phrased in terms of the Riemannian distance $r(x)$ from a fixed basepoint (Yu, 17 Jun 2025, Yu, 5 Jan 2026, Lee, 2018).

2. Prescribing Chern Scalar Curvature

Prescribing the Chern scalar curvature on complete noncompact Hermitian manifolds centers on the conformal transformation law for scalar curvature. Given a conformal change $\tilde\omega=e^{\tfrac{2}{n}u}\omega$, the Chern scalar curvature transforms according to

$$-\Delta^{\mathrm{Ch}}_\omega u + S^{\mathrm{Ch}}_\omega = S^{\mathrm{Ch}}_{\tilde{\omega}}\,e^{\frac{2}{n}u},$$

with $\Delta^{\mathrm{Ch}}_\omega$ the Chern Laplacian. The central PDE for prescribing a target function $K(x)$ is

$$-\Delta^{\mathrm{Ch}}_\omega u + S^{\mathrm{Ch}}_\omega = K(x)\,e^{\frac{2}{n}u}. \tag{★}$$

When $K(x)$ is negative and the background geometry satisfies natural negativity and boundedness conditions on the scalar curvature, Ricci curvature, and torsion, results of Yu and others guarantee the existence of a smooth bounded solution $u$, yielding a new conformal metric with prescribed scalar curvature $K(x)$ (Yu, 17 Jun 2025, Yu, 5 Jan 2026).

The table below summarizes some central conditions and results:

Condition on $K(x)$, $S^{\mathrm{Ch}}_\omega$	Geometric Hypotheses at Infinity	Result
$K \leq 0$, uniformly negative outside compact set	Chern–Ricci curvature and torsion growth ≤ quadratic/linear; scalar curvature uniformly negative outside compact set	Existence and uniqueness of bounded smooth solution $u$, yielding complete metric with $S^{\mathrm{Ch}}_{\tilde{\omega}} = K(x)$
$K(x)$ sign-changing, small positive perturbations allowed	As above	Existence of bounded conformal metric with $S^{\mathrm{Ch}}_{\tilde{\omega}} = K(x)$
$S^{\mathrm{Ch}}_\omega \leq 0$ everywhere, $\leq -b^2<0$ off compact set	No upper bound needed on $K$	Existence of unbounded, but complete, conformal metric for constant negative curvature (Yu, 17 Jun 2025, Yu, 5 Jan 2026)

3. Analytic and Geometric Techniques

The existence and uniqueness of metrics with prescribed Chern scalar curvature rest on several analytic strategies:

• Sub- and super-solution method: Construct lower and upper solutions (constant or model functions) to trap any genuine solution between explicit barriers.
• Monotone iteration: Employ an exhaustion by compact sets with Dirichlet data imposed between lower and upper solutions; produce a sequence of approximate solutions converging $C^2_{\mathrm{loc}}$ to a global one.
• A priori $C^0$-estimates: Use the maximum principle (classical and Omori–Yau type) to show boundedness of solutions on compact exhaustion sets.
• Laplacian comparison and volume growth estimates: Analyze the behavior of the Chern Laplacian of the distance function using curvature and torsion growth constraints, yielding control over geometry at infinity.
• Uniqueness: Apply the Omori–Yau maximum principle to the difference of two bounded solutions, exploiting the sign structure of the target curvature outside compact sets (Yu, 17 Jun 2025, Yu, 5 Jan 2026).

4. Curvature Flows and Chern–Einstein Metrics

On complete noncompact Hermitian manifolds, fully nonlinear PDE and geometric flows are used to construct Chern–Einstein metrics, i.e., Hermitian metrics for which the Chern–Ricci curvature satisfies $R_{i\bar{\jmath}}(g) = \lambda g_{i\bar{\jmath}}$. The second Ricci flow evolves the Hermitian metric by

$$\frac{\partial}{\partial t}g_{i\bar{j}}(t) = -2\,\mathrm{Ric}^{(2)}_{i\bar{j}}(t)$$

and preserves nonpositive curvature bounds under suitable initial data. If the initial Hermitian metric has nonpositive bisectional curvature and the first Chern–Ricci form is negative outside a compact set, this flow yields (after suitable regularity and limiting procedures involving the complex Monge–Ampère equation) a complete Kähler–Einstein metric with bounded geometry (Lee, 2018).

Similarly, solving complex Monge–Ampère–type equations directly (cf. Yin (Yin, 17 Dec 2025)) produces complete Chern–Einstein metrics, provided the background geometry is controlled (bounded geometry, proper exhaustion function with bounded Hessian). These solutions are obtained via bootstrapped a priori estimates and the Evans–Krylov theory, with completeness inherited from the background metric and the decay properties of the solution at infinity.

5. Geometric and Analytic Assumptions

The analytic framework on complete noncompact Hermitian manifolds depends crucially on the following technical assumptions:

• Completeness of the background Hermitian metric.
• Noncompactness of the manifold.
• Curvature control: Polynomial bounds on the negative part of the second Chern–Ricci curvature and torsion tensor, parameterized by an exponent $\alpha$.
• Volume growth/Laplacian comparison: The Chern Laplacian of the distance function satisfies

$$\Delta_\omega^{\mathrm{Ch}} r \leq \frac{2n(\alpha+2)}{r} + 4n \sqrt{C_3}(1+r)^{\alpha/2},$$

facilitating maximum principle arguments and barrier constructions.

• Negativity and decay of Chern scalar curvature: Typically $S^{\mathrm{Ch}}(\omega)\leq 0$ on $M$, with $S^{\mathrm{Ch}}(\omega)\leq -b^2 r^{-l}$ outside a compact set (Yu, 5 Jan 2026, Yu, 17 Jun 2025, Yin, 17 Dec 2025).

These hypotheses allow the extension of classical compactness-dependent elliptic and parabolic PDE techniques to the noncompact Hermitian context.

6. Connections to Classical Problems and Extensions

The prescription of Chern scalar curvature in the complete noncompact Hermitian setting is closely analogous to the Kazdan–Warner and Yamabe problems in conformal Riemannian geometry, with the principal distinction that torsion effects and complex geometric structures play an essential role. These frameworks generalize the classical work of Aviles–McOwen on Poincaré disks to higher dimensions and non-Kähler geometry, granting much greater flexibility in the choice of prescribed curvature functions, including sign-changing and vanishing-at-infinity targets (Yu, 17 Jun 2025).

Techniques developed in this context are not restricted to Chern scalar curvature. Fully nonlinear elliptic equations of Monge–Ampère type, crucial in constructing metrics with prescribed Ricci curvature or volume form, are also solvable on noncompact Hermitian manifolds under bounded geometry, with geometric results extending Kähler to general Hermitian metrics (Yin, 17 Dec 2025).

7. Representative Results and Further Directions

Recent results can be distilled as follows:

• General prescription of negative or sign-changing Chern scalar curvature is possible under curvature decay and geometric boundedness assumptions, with geometric completeness and uniform equivalence of metrics preserved (Yu, 17 Jun 2025, Yu, 5 Jan 2026).
• Constant negative Chern scalar curvature and Chern–Einstein metrics are obtainable via sub- and supersolution methods, monotone iteration, or geometric flows coupled to complex Monge–Ampère equations, in arbitrary dimension and for general (possibly non-Kähler) settings (Yin, 17 Dec 2025, Lee, 2018).
• Flexibility in the prescription problem arises from the interplay between global geometric control and the analytic properties of the Chern Laplacian and curvature tensors at infinity, offering robust analogues of classical uniformization phenomena in higher dimensions and noncompact settings.

These results underscore the maturity of analytic methods on complete noncompact Hermitian manifolds and their pivotal role in modern complex and differential geometry.