Strongly Convex Kähler-Berwald Metrics

• Strongly convex Kähler-Berwald metrics are complex Finsler metrics combining strong convexity with both Kähler and Berwald properties, bridging Hermitian and Finsler geometry.
• They are explicitly constructed on classical symmetric domains, offering non-Hermitian deformations of the Bergman metric via curvature-invariant quadratic and higher order forms.
• These metrics exhibit rigidity, invariant geodesic and curvature properties, and precise curvature bounds, serving as key models for advanced studies in complex geometry.

A strongly convex Kähler–Berwald metric is a distinguished class of complex Finsler metrics exhibiting both strong convexity and simultaneous Kähler and complex Berwald structure. These metrics unify core properties from classical Hermitian geometry and complex Finsler geometry, and their classification, explicit construction on symmetric domains, and rigidity theorems reveal rich geometric structures underpinning complex analysis, several complex variables, and differential geometry. Recent work has produced explicit families of non-Hermitian, holomorphically invariant strongly convex Kähler–Berwald metrics on classical domains, and characterized their geometric and curvature properties (Zhong, 2021, Ge et al., 2022, Xia et al., 8 Jan 2026, Wang et al., 2020).

1. Foundational Definitions and Characterizations

Let $M$ be a complex manifold of dimension $n\geq 1$. The holomorphic tangent bundle is $T^{1,0}M \to M$. A function $F:T^{1,0}M\rightarrow [0,+\infty)$ is a strongly pseudoconvex complex Finsler metric if:

• $F(z,v)>0$ for $v\ne 0$ and $F(z,0)=0$;
• $G(z,v):= F(z,v)^2$ is $C^\infty$ on $T^{1,0}M\setminus\{0\}$;
• $G(z, \lambda v) = |\lambda|^2 G(z, v)$ for all $\lambda\in \mathbb{C}$;
• The Hermitian matrix $$G_{i\bar j}(z, v) := \partial^2 G/\partial v^i \partial \bar v^j$$ is positive definite for $(z, v)\ne 0$.

$F$ is strongly convex if, under realification of $T^{1,0}M$, the real Hessian $$g_{AB}(x,u) = \frac12 \partial^2 F^2/\partial u^A \partial u^B$$ is positive-definite on $TM\setminus\{0\}$.

A Kähler–Finsler metric satisfies symmetry of the horizontal Chern–Finsler connection coefficients: $\Gamma^i_{jk}(z, v)$ symmetric in $j,k$. $F$ is a complex Berwald metric if $\Gamma^i_{jk}(z, v)$ are independent of $v$. A Kähler–Berwald metric satisfies both (i.e., the horizontal connection coefficients are $v$-independent and symmetric), reducing to Kähler geometry on the Hermitian side but admitting non-quadratic examples in the Finsler field (Zhong, 2021, Wang et al., 2020, Xia et al., 8 Jan 2026).

The geometric content is as follows:

• Parallelism Condition: $F$ is Kähler–Berwald if and only if the canonical almost complex structure $\mathbf{J}$ on $T^{1,0}M$ is horizontally parallel with respect to the real Cartan connection, i.e., $\nabla_X \mathbf{J} = 0$ for all horizontal vectors $X$ (Xia et al., 8 Jan 2026).
• Connection Coincidence: The Cartan and Chern–Finsler connections coincide if and only if $\mathbf{J}$ is both horizontally and vertically parallel with respect to the Cartan connection.

2. Invariant Strongly Convex Kähler–Berwald Metrics on Classical Domains

Rigidity and Explicit Constructions

• On the unit ball $B_n\subset\mathbb{C}^n$, the only $\mathrm{Aut}(B_n)$-invariant strongly pseudoconvex complex Finsler metric is a constant multiple of the Poincaré–Bergman metric:

$$F_P(z;v)=\sqrt{\frac{|v|^2}{(1-|z|^2)^2}+\frac{|\langle z,v\rangle|^2}{(1-|z|^2)^2}},$$

precluding any genuinely non-quadratic Berwald examples (Zhong, 2021, Wang et al., 2020).

• On the polydisk $P_n = \Delta^n$, $\mathrm{Aut}(P_n)$-invariant strongly convex Kähler–Berwald metrics exist in infinite families. For parameters $t\geq 0$ and $k\geq 2$,

$$F_{t,k}(z;v) = \frac{1}{\sqrt{1+t}} \left( \sum_{j=1}^n \frac{|v_j|^2}{(1 - |z_j|^2)^2} + t \sum_{j=1}^n \frac{|v_j|^{2k}}{(1 - |z_j|^2)^{2k}} \right)^{1/2}$$

is a complete, strongly convex Kähler–Berwald metric. The terms with $k>1$ introduce non-Hermitian, non-quadratic holomorphic deformations of the classical product Bergman metric (Zhong, 2021). On each product factor, strong convexity is verified blockwise, and the Kähler–Berwald property is inherited from the $v$-independence and symmetry of the Christoffel symbols.

• Analogous constructions extend to all four irreducible Cartan classical domains of types I–IV, via explicit deformations $F^2 = (1+t)^{-1} (\mathcal{B}_1 + t\mathcal{B}_k)$, where $\mathcal{B}_1$ and $\mathcal{B}_k$ are curvature-invariant quadratic and higher order forms, respectively (Ge et al., 2022). The automorphism group invariance and the completeness follow from the preservation of geodesic structure and the explicit form of the connections.
• On Siegel domains of type I or II biholomorphic to the polydisk, suitable pullbacks of $F_{t,k}$ yield complete, automorphism-invariant strongly convex Kähler–Berwald metrics (Zhong, 2021).

3. Geometric and Curvature Properties

A central feature of the holomorphically invariant strongly convex Kähler–Berwald deformations is their congruence with the curvature properties of the classical Bergman metric, up to explicit parameters.

• The holomorphic sectional curvature $K_{t,k}(z, v)$ of $F_{t,k}$ on $\Delta^n$ satisfies

$-4(1+t) \leq K_{t,k}(z, v) \leq -4/(1+t)$

for all $(z, v)$, where $-4$ is the curvature of the disk factor (Zhong, 2021).

• More generally, for any Cartan domain deformation, the holomorphic sectional curvature takes the form

$$K(z; v) = -\frac{1}{F^4(z; v)} \left\{4\mu \mathcal{B}_2(z; v) + 2k t \mathcal{B}_{k+1}(z; v)\right\} / (1+t),$$

for a domain-specific constant $\mu$, and is strictly negative and bounded (Ge et al., 2022).

• Holomorphic bisectional curvature is also uniformly nonpositive and is bounded below in terms of explicit parameters and the deformation function.
• For $F_{t,k}$ and its analogs, geodesics coincide as point sets with the underlying Bergman metric, guaranteeing completeness.
• The explicit distance formula for $F_{t,k}$ on $\Delta^n$ is

$$d_{t,k}((z^{(1)},z^{(2)})) = \frac{1}{2\sqrt{1+t}}\sum_{j=1}^n \log \frac{1+\left| \frac{z_j^{(2)} - z_j^{(1)}}{1 - \bar z_j^{(1)} z_j^{(2)}} \right|}{1-\left| \frac{z_j^{(2)} - z_j^{(1)}}{1 - \bar z_j^{(1)} z_j^{(2)}} \right|}$$

as in the Bergman metric (Zhong, 2021).

• The curvature formulas and geodesic behavior generalize seamlessly to the constructed metrics on other Cartan domains (Ge et al., 2022).

4. Classification and Rigidity Theorems

The geometric structure of strongly convex Kähler–Berwald metrics is controlled by a series of rigidity and decomposition results:

• Rigidity: On a simply connected, complete manifold with a strongly convex Kähler–Berwald metric and constant holomorphic sectional curvature $c$, the metric must be (up to isometry) one of:
  • The Bergman-type metric on the unit ball for $c<0$,
  • A (flat) complex Minkowski space for $c=0$,
  • The Fubini–Study metric on $\mathbb{C}P^n$ for $c>0$ (Xia et al., 8 Jan 2026).
• Decomposition: Every connected, simply-connected, complete, strongly convex Kähler–Berwald manifold $(M, F)$ admits a unique (up to order) de Rham-type product

$$(M, F) \cong (\mathbb{C}^m, F_0) \times \prod_{a=1}^p (M_a, F_a) \times \prod_{b=1}^q (N_b, G_b)$$

where $(\mathbb{C}^m, F_0)$ is (flat) complex Minkowski, each $(M_a, F_a)$ is irreducible complete Hermitian (quadratic), and each $(N_b, G_b)$ is irreducible, complete, symmetric, non-Hermitian Kähler–Berwald, rank $\geq 2$ (Zhong, 2021).

This classification extends the Cartan–Ambrose–Hicks and de Rham decompositions from Riemannian and Hermitian contexts into the Finslerian complex analytic setting.

5. Symmetry, Invariant Models, and Non-Existence Results

• On $\mathbb{C}^n$, any $U(n)$-invariant, strongly convex complex Finsler metric which is real Berwald or complex Berwald must be Hermitian quadratic, enforcing a rigidity within the fully symmetric class (Wang et al., 2020). Explicit criteria for strong convexity in the $U(n)$-invariant ansatz are given by inequalities on deformation functions.
• On higher-rank reducible symmetric domains (e.g., polydisks), the product structure enables the construction of infinitely many non-quadratic but holomorphically invariant strongly convex Kähler–Berwald metrics, via independent deformation on each factor.
• The absence of non-quadratic, automorphism-invariant, strongly pseudoconvex Finsler metrics on the ball $B_n$ and other irreducible bounded symmetric domains of rank one is firmly established as a rigidity phenomenon (Zhong, 2021, Wang et al., 2020).

6. Significance and Theoretical Implications

Strongly convex Kähler–Berwald metrics represent a natural generalization of classical Kähler geometry to complex Finsler settings while retaining key analytic and geometric structures, especially on homogeneous and symmetric domains (Zhong, 2021, Ge et al., 2022). The construction of explicit, invariant, non-Hermitian but Kähler–Berwald metrics demonstrates:

• The possibility of genuinely Finslerian, yet holomorphically natural, invariants on higher rank and reducible spaces.
• Direct analogues to Bergman metrics in curvature behavior, completeness, and geodesic structure, despite the non-quadratic norm.
• The existence of infinite-dimensional deformation families parameterized by curvature-like invariants, extending classical moduli of Hermitian symmetric and Kähler–Einstein structures to a broader context.

These developments deepen the interface between several complex variables, geometric analysis, and global differential geometry. They also supply canonical models for further exploration of curvature, automorphism invariance, and complex Finsler geometry on general complex manifolds.

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Key References:

• "Holomorphic invariant strongly pseudoconvex complex Finsler metrics" (Zhong, 2021)
• "Geometry of holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains" (Ge et al., 2022)
• "Characterization of strongly convex Kähler-Berwald metrics" (Xia et al., 8 Jan 2026)
• "On $U(n)$-invariant strongly convex complex Finsler metrics" (Wang et al., 2020)