Complex Finsler Metrics

• Complex Finsler metrics are functions defined on the holomorphic tangent bundle that generalize Hermitian metrics by relaxing quadratic dependence and retaining strong pseudoconvexity.
• They enable explicit computation of holomorphic sectional, bisectional, and Ricci curvatures using the Chern–Finsler connection, with applications in symmetry and invariant metric models.
• Their study underpins rigidity, classification, and projective flatness theorems that connect Kähler–Finsler, Berwald, and Randers-type structures in complex geometry.

A complex Finsler metric is a function defined on the holomorphic tangent bundle of a complex manifold, generalizing Hermitian metrics by relaxing the quadratic dependence on fiber coordinates while retaining homogeneity and strong (pseudo-)convexity. Complex Finsler structures provide a rich framework encompassing classical Kähler geometry, complex Minkowski spaces, Randers-type metrics, and a variety of explicit and invariant metrics with diverse curvature, symmetry, and rigidity properties. The theory synthesizes differential, complex, and metric geometry, and is central to the study of holomorphic mappings, geometric invariants, and the complex analogues of classical Finslerian and Riemannian questions.

1. Foundational Definitions and Structural Properties

Let $M$ be a complex manifold of dimension $n$ and denote by $T^{1,0}M$ its holomorphic tangent bundle. A complex Finsler metric is a function $F:T^{1,0}M\to[0,\infty)$ satisfying the following:

• Smoothness: $F$ is $C^\infty$ on $T^{1,0}M\setminus\{0\}$.
• 1-Homogeneity: $F(z, \lambda v) = |\lambda|F(z, v)$ for $\lambda\in\mathbb{C}$, $v\ne0$.
• Strong pseudoconvexity: The Levi form $$G_{\alpha\bar\beta} = \partial^2(F^2)/\partial v^\alpha\partial \bar v^\beta$$ is positive-definite for all $v\ne 0$ (Xia et al., 2017).

Additional convexity is sometimes required: strong convexity in the complex sense incorporates positivity of the real Hessian under identification with real Finsler metrics.

The Chern–Finsler connection is a canonical $(1,0)$-connection associated to $F$, with connection coefficients and curvature tensors extending the classical Chern connection in Hermitian geometry (Li et al., 2021).

The fundamental tensor $G_{\alpha\bar\beta}(z,v)$ plays an analogous role to the Hermitian metric tensor, and is used to define curvature notions, holomorphic sectional, and Ricci curvature.

2. Kähler, Berwald, and Projective Flatness Conditions

Multiple "Kähler-type" properties refine the geometry of complex Finsler spaces:

• Kähler–Finsler: The horizontal torsion of the Chern–Finsler connection vanishes ($$\Gamma^\alpha_{\beta\gamma} = \Gamma^\alpha_{\gamma\beta}$$), or, equivalently, $i\partial\bar\partial F^2 = 0$. This implies equality of the holomorphic sectional curvature tensors associated to canonical and Chern–Finsler connections (Li et al., 2021, Chen et al., 2019).
• Complex Berwald: The connection coefficients $\Gamma^k_{ij}(z,v)$ are independent of $v$. A metric that is both Kähler–Finsler and (complex) Berwald is termed Kähler–Berwald (Ge et al., 2022, Zhong, 2023).
• Strongly Kähler / Kähler / Weakly Kähler: These denote increasingly weaker vanishing conditions on various components or contractions of the Chern–Finsler torsion (Cui et al., 2021, Xu et al., 22 Dec 2025).
• Projectively Flat: The real geodesics of $F$ are straight lines. In the strongly convex complex Finsler category, the only projectively flat metrics are complex Minkowski norms—no richer local models arise (Xia et al., 2017).

Property	Defining Condition	Examples
Kähler–Finsler	$$\Gamma^\alpha_{\beta\gamma} = \Gamma^\alpha_{\gamma\beta}$$, $i\partial\bar\partial F^2 = 0$	Hermitian metrics, Bergman
Complex Berwald	$\Gamma^k_{ij}(z,v)$ independent of $v$	Bergman, explicit deformations
Kähler–Berwald	Both of the above	Bergman, certain invariants
Projectively Flat	Geodesics are straight lines	Only complex Minkowski norms

3. Invariant Metrics and Explicit Models

Complex Finsler metrics invariant under large symmetry groups are central to explicit constructions and classification results.

• On the complex unit ball $B_n$, any $\mathrm{Aut}(B_n)$-invariant, strongly pseudoconvex Finsler metric is a scalar multiple of the Poincaré–Bergman metric (Zhong, 2021, Zhong, 2023).
• For the unit polydisc $P_n$, there exist infinitely many $\mathrm{Aut}(P_n)$-invariant, strongly convex Kähler–Berwald metrics $F_{t,k}$, which are non-Hermitian quadratic for $t>0$ (Zhong, 2021).
• On classical domains of types I–IV (bounded symmetric domains), explicit non-Hermitian quadratic, Kähler–Berwald metrics $F^A$ are constructed via deformation of the Bergman metric by higher-order invariants; all such metrics inherit strong pseudoconvexity, completeness, and bisectional/sectional curvature bounds (Ge et al., 2022).
• On complex Lie groups, every left-invariant complex Finsler metric is complex Berwald, with vanishing holomorphic sectional curvature. Kähler-type properties coincide with the abelian property of the Lie algebra (Xu et al., 22 Dec 2025).

Beyond these, unitary-invariant and Randers-type metrics provide infinite families in domains in $\mathbb{C}^n$; explicit formulas for the fundamental tensor, determinant, and inverse are available, especially for infinite-series $(\alpha, \beta)$-type metrics (Shanker et al., 2018).

4. Holomorphic Sectional, Bisectional, and Ricci Curvature

Curvature in complex Finsler geometry generalizes Hermitian and Riemannian settings via the Chern–Finsler connection:

• Holomorphic sectional curvature along a direction $v$ is defined as

$$K(z,v) = \frac{2R_{i\bar{j}k\bar{\ell}}(z,v)\,v^i\bar v^j v^k\bar v^\ell}{F^4(z,v)}$$

where $R_{i\bar{j}k\bar{\ell}}$ denotes the curvature tensor of the Chern–Finsler connection (Wan, 2017).

• For invariant Kähler–Berwald metrics on classical domains, the holomorphic sectional curvature is strictly negative and bounded between two constants; the bisectional curvature is nonpositive and similarly bounded (Zhong, 2023, Ge et al., 2022).
• The Ricci and scalar curvatures can be defined for both the Chern–Finsler and Munteanu's canonical connections; their equivalence under the Kähler–Finsler condition and precise difference formulas in terms of torsion are established (Li et al., 2021).
• The study of conformal classes, extremal metrics, and Yamabe-type problems yields existence and uniqueness results for metrics with prescribed mean Ricci or sectional curvature in the complex Finsler setting (Chen et al., 2019).

5. Rigidity, Classification, and Uniformization

Several rigidity and classification theorems delineate the landscape of complex Finsler geometries:

• Rigidity on projective flatness: Strongly convex, projectively flat (or dually flat) complex Finsler metrics are necessarily Minkowski—no other local models arise (Xia et al., 2017).
• Invariance and uniqueness: Any $\mathrm{Aut}(B_n)$-invariant strongly pseudoconvex metric is Poincaré–Bergman; on $P_n$ and domains biholomorphic to it, a rich moduli of Kähler–Berwald metrics exist (Zhong, 2021).
• Holomorphic invariant metrics on classical domains: All such strongly pseudoconvex metrics are Kähler–Berwald; Hermitian-quadratic ones are (up to scaling) Bergman (Zhong, 2023, Ge et al., 2022).
• Uniformization of weakly Kähler Randers metrics: Unitary-invariant, non-Hermitian Randers metrics exist with constant curvature $k\in\{4, 0, -4\}$, up to homothety and domain geometry (Cui et al., 2021).
• Decomposition theorems: Complete, simply connected, strongly convex Kähler–Berwald manifolds decompose uniquely (up to order) as products of complex Minkowski spaces and irreducible symmetric factors (Zhong, 2021).

6. Interplay with Holomorphic Mappings and Schwarz Lemmas

Schwarz–Pick type results extend to mappings between complex Finsler manifolds equipped with invariant metrics.

• For holomorphic maps between domains with invariant Kähler–Berwald metrics $F_1, F_2$, the Schwarz lemma asserts

$$F_2(f(z);f_*(v)) \le \sqrt{\frac{K_1^{(\max)}}{K_2^{(\min)}}}\;F_1(z;v),$$

where $K_i^{(\max)}, K_i^{(\min)}$ encode sharp curvature bounds (Zhong, 2023).

• The Lu constant, defined as $\ell_0(D,F) = \sqrt{K_1^{(\max)}/K_2^{(\min)}}$, is both an analytic and geometric invariant, optimizing metric contraction under holomorphic maps.
• For bounded strongly convex domains, the Kobayashi and Carathéodory metrics are $C^\infty$-smooth, coincide, have constant holomorphic sectional curvature $-4$, and are pairwise uniformly equivalent with classical invariant metrics (Nie, 2023).
• Applications include uniform equivalence of complex Finsler and Kähler metrics, underlying the geometric hyperbolicity and large-scale structure of complex domains.

7. Projective Relations, Berwald Property, and Randers Metrics

The projective geometry of complex Finsler spaces closely parallels the real theory, with complexities arising from the holomorphic setting:

• Two metrics are projectively related if their unparameterized geodesics coincide. In the weakly Kähler or generalized Berwald case, such relations are governed by explicit coordinate identities (complex Rapcsák conditions), and the principal projective change lies in a scalar function times the fiber direction (Aldea et al., 2011).
• Hilbert's Fourth Problem (Complex Version): Only complex Minkowski metrics (i.e., constant-coefficient Hermitian metrics) yield straight complex geodesics under strong convexity; the corresponding projective class is extremely rigid (Xia et al., 2017, Aldea et al., 2011).
• For complex Randers metrics, projectivity and the Berwald property hinge on closedness and parallelism of the additional one-form data; explicit formulas describe metric and curvature tensors for infinite-series ($\alpha, \beta$)-metrics (Shanker et al., 2018, Aldea et al., 2011).

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This framework positions complex Finsler geometry as a mature theory encompassing canonical examples, explicit non-Hermitian models, robust rigidity results, and a rich interplay with holomorphic mappings, curvature, and global geometric properties. The structure is flexible yet constrained, supporting both concrete calculations and far-reaching generalizations in several complex variables and differential geometry.