Existence and geometry of Hermitian metrics with constant second scalar curvature

Abstract: We study Hermitian metrics with constant second scalar curvature on compact manifolds. We first consider a Yamabe-type problem for the second Bismut scalar curvature under a natural topological condition, and then analyze elliptic equations arising from constant second Chern scalar curvature within a fixed Hermitian conformal class and derive geometric consequences. Finally, under an Einstein-type condition on the second Chern curvature, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, which in certain cases further implies the existence of a Kähler-Einstein metric.

• The paper analyzes Hermitian metrics with constant second scalar curvature, deriving existence, uniqueness, and constancy results for both Chern and Bismut connections on compact complex manifolds, through elliptic and conformality transformation.
• Transformations under conformal change of metrics reveal key metrics $S^{(2)}(\omega,G)$ particularly the conformal transformations, equations, identities and Kähler-types rigidity derived from them.
• The paper presents explicit non-Kähler examples, such as Hopf and Inoue surfaces, validating the theory and illustrating the geometry of second scalar curvature outside the Kähler category. It covers implications for Kodaira dimension and pseudo-effectiveness, and outlines open questions

This paper by Liangdi Zhang studies Hermitian metrics with constant second scalar curvature on compact complex manifolds, extending the Chern–Yamabe and Bismut–Yamabe programs from first scalar curvatures to second ones. The second scalar curvature $S^{(2)}(\omega,t)$ of a Gauduchon connection ${^t}\nabla$ is obtained by tracing the curvature tensor in the "crossed" pattern $h^{i\bar l}h^{k\bar j}R_{i\bar j k\bar l}$, which coincides with the usual trace only when torsion vanishes. The paper establishes existence results for both the Bismut and Chern connections, derives Kodaira-dimension and pseudo-effectiveness consequences, proves constancy results under an Einstein-type condition, and supplies explicit non-Kähler examples.

Conformal transformation formulas

The analytic foundation is Proposition 3.1, which computes how the third and fourth Ricci curvatures and the second scalar curvature of ${^t}\nabla$ transform under $\omega_f = e^f\omega$. The key formula is

$$S^{(2)}(\omega_f,t)=e^{-f}\Big(S^{(2)}(\omega,t)-(1+2(n-1)t)\Delta_\omega^{\mathbb C}f-(n^2-1)t^2|\partial f|^2_\omega+2(n+1)t^2\mathrm{Re}\langle\partial^*\omega,\sqrt{-1}\,\bar\partial f\rangle_\omega\Big).$$

Specializing to $t=0$ (Chern) yields the remarkably simple law

$$S_C^{(2)}(e^f\omega)=e^{-f}\big(S_C^{(2)}(\omega)-\Delta_\omega^{\mathbb C}f\big),$$

which is structurally identical to the conformal formula for the first Chern scalar curvature and makes the Chern case amenable to standard elliptic methods. For $t=1$ (Bismut) the equation acquires a gradient term $(n^2-1)|\partial f|^2$, leading to a genuinely semilinear problem. A companion identity compares the two traces pointwise:

${^t}\nabla$0

with the two-dimensional specialization ${^t}\nabla$1. From this the paper derives Kähler-type rigidity: on a complete Gauduchon manifold, if the total integral of ${^t}\nabla$2 dominates that of ${^t}\nabla$3, then for ${^t}\nabla$4 at ${^t}\nabla$5 the metric is balanced; for ${^t}\nabla$6 balanced metrics are Kähler; and for ${^t}\nabla$7 outside ${^t}\nabla$8 (or outside ${^t}\nabla$9 when $h^{i\bar l}h^{k\bar j}R_{i\bar j k\bar l}$0), the metric is necessarily Kähler. These are strong dichotomies: near-critical values of the Gauduchon parameter force torsion to vanish outright.

The Bismut Yamabe problem

Theorem 4.1 solves a Yamabe-type problem for the second Bismut scalar curvature under the topological hypothesis $h^{i\bar l}h^{k\bar j}R_{i\bar j k\bar l}$1: every Hermitian conformal class contains a metric of constant $h^{i\bar l}h^{k\bar j}R_{i\bar j k\bar l}$2. The proof exploits the fact that $h^{i\bar l}h^{k\bar j}R_{i\bar j k\bar l}$3 forces the Lee form of the Gauduchon representative to be exact, so a conformal rescaling produces a balanced metric $h^{i\bar l}h^{k\bar j}R_{i\bar j k\bar l}$4; on balanced metrics the Bismut equation reduces, via the substitution $h^{i\bar l}h^{k\bar j}R_{i\bar j k\bar l}$5, to the semilinear eigenvalue-type equation

$h^{i\bar l}h^{k\bar j}R_{i\bar j k\bar l}$6

where $h^{i\bar l}h^{k\bar j}R_{i\bar j k\bar l}$7 and $h^{i\bar l}h^{k\bar j}R_{i\bar j k\bar l}$8. Direct minimization of the Rayleigh-type quotient $h^{i\bar l}h^{k\bar j}R_{i\bar j k\bar l}$9 for ${^t}\nabla$0 yields a smooth strictly positive minimizer by Rellich–Kondrachov compactness and the strong maximum principle; no Sobolev critical-exponent analysis is required because the exponent is subcritical. A corollary weakens the hypothesis to the cohomological condition ${^t}\nabla$1. Note that in dimension two, ${^t}\nabla$2 already implies Kählerness, so the theorem is only geometrically new for ${^t}\nabla$3, where Calabi–Eckmann manifolds provide non-Kähler instances.

Constant second Chern scalar curvature

The sign of the conformal invariant

${^t}\nabla$4

called the second Gauduchon degree in analogy with Gauduchon's classical invariant, governs existence within the class:

• Zero case: if ${^t}\nabla$5, there is a unique (up to scaling) metric with ${^t}\nabla$6; moreover either ${^t}\nabla$7 or ${^t}\nabla$8 with ${^t}\nabla$9 holomorphically torsion. In a balanced class one also obtains a unique metric with vanishing first Chern scalar curvature.
• Negative case: if $\omega_f = e^f\omega$0, there is a unique (up to scaling) metric with constant negative value $\omega_f = e^f\omega$1; in a balanced class, $\omega_f = e^f\omega$2 is not pseudo-effective and there is additionally a unique metric with constant negative first Chern scalar curvature.
• Positive case: if $\omega_f = e^f\omega$3, then $\omega_f = e^f\omega$4, and $\omega_f = e^f\omega$5 (for any curve $\omega_f = e^f\omega$6 of genus $\omega_f = e^f\omega$7) carries Hermitian metrics with positive constant first and second Chern scalar curvatures.

The zero-case proof rests on the fact that the kernel of the formal adjoint of $\omega_f = e^f\omega$8 consists exactly of constants when $\omega_f = e^f\omega$9 is Gauduchon — a computation using $$S^{(2)}(\omega_f,t)=e^{-f}\Big(S^{(2)}(\omega,t)-(1+2(n-1)t)\Delta_\omega^{\mathbb C}f-(n^2-1)t^2|\partial f|^2_\omega+2(n+1)t^2\mathrm{Re}\langle\partial^*\omega,\sqrt{-1}\,\bar\partial f\rangle_\omega\Big).$$0 that parallels the Chern–Yamabe argument of Angella–Calamai–Spotti. The negative case is handled by continuity method: after normalizing to a pointwise-negative representative, openness follows from injectivity of the linearization (maximum principle with $$S^{(2)}(\omega_f,t)=e^{-f}\Big(S^{(2)}(\omega,t)-(1+2(n-1)t)\Delta_\omega^{\mathbb C}f-(n^2-1)t^2|\partial f|^2_\omega+2(n+1)t^2\mathrm{Re}\langle\partial^*\omega,\sqrt{-1}\,\bar\partial f\rangle_\omega\Big).$$1), and closedness from uniform $$S^{(2)}(\omega_f,t)=e^{-f}\Big(S^{(2)}(\omega,t)-(1+2(n-1)t)\Delta_\omega^{\mathbb C}f-(n^2-1)t^2|\partial f|^2_\omega+2(n+1)t^2\mathrm{Re}\langle\partial^*\omega,\sqrt{-1}\,\bar\partial f\rangle_\omega\Big).$$2 bounds $$S^{(2)}(\omega_f,t)=e^{-f}\Big(S^{(2)}(\omega,t)-(1+2(n-1)t)\Delta_\omega^{\mathbb C}f-(n^2-1)t^2|\partial f|^2_\omega+2(n+1)t^2\mathrm{Re}\langle\partial^*\omega,\sqrt{-1}\,\bar\partial f\rangle_\omega\Big).$$3 together with Calderón–Zygmund and Schauder estimates. Uniqueness again uses the maximum principle. These results imply that the sign of the second Gauduchon degree constrains the birational geometry of $$S^{(2)}(\omega_f,t)=e^{-f}\Big(S^{(2)}(\omega,t)-(1+2(n-1)t)\Delta_\omega^{\mathbb C}f-(n^2-1)t^2|\partial f|^2_\omega+2(n+1)t^2\mathrm{Re}\langle\partial^*\omega,\sqrt{-1}\,\bar\partial f\rangle_\omega\Big).$$4 through Yang's theorems relating total first Chern scalar curvature to pseudo-effectiveness of $$S^{(2)}(\omega_f,t)=e^{-f}\Big(S^{(2)}(\omega,t)-(1+2(n-1)t)\Delta_\omega^{\mathbb C}f-(n^2-1)t^2|\partial f|^2_\omega+2(n+1)t^2\mathrm{Re}\langle\partial^*\omega,\sqrt{-1}\,\bar\partial f\rangle_\omega\Big).$$5.

Weak second Hermitian–Einstein metrics

Since neither $$S^{(2)}(\omega_f,t)=e^{-f}\Big(S^{(2)}(\omega,t)-(1+2(n-1)t)\Delta_\omega^{\mathbb C}f-(n^2-1)t^2|\partial f|^2_\omega+2(n+1)t^2\mathrm{Re}\langle\partial^*\omega,\sqrt{-1}\,\bar\partial f\rangle_\omega\Big).$$6 nor $$S^{(2)}(\omega_f,t)=e^{-f}\Big(S^{(2)}(\omega,t)-(1+2(n-1)t)\Delta_\omega^{\mathbb C}f-(n^2-1)t^2|\partial f|^2_\omega+2(n+1)t^2\mathrm{Re}\langle\partial^*\omega,\sqrt{-1}\,\bar\partial f\rangle_\omega\Big).$$7 is Hermitian symmetric in general, the paper introduces the condition

$$S^{(2)}(\omega_f,t)=e^{-f}\Big(S^{(2)}(\omega,t)-(1+2(n-1)t)\Delta_\omega^{\mathbb C}f-(n^2-1)t^2|\partial f|^2_\omega+2(n+1)t^2\mathrm{Re}\langle\partial^*\omega,\sqrt{-1}\,\bar\partial f\rangle_\omega\Big).$$8

termed weak second Hermitian–Einstein. On pluriclosed Gauduchon manifolds this forces $$S^{(2)}(\omega_f,t)=e^{-f}\Big(S^{(2)}(\omega,t)-(1+2(n-1)t)\Delta_\omega^{\mathbb C}f-(n^2-1)t^2|\partial f|^2_\omega+2(n+1)t^2\mathrm{Re}\langle\partial^*\omega,\sqrt{-1}\,\bar\partial f\rangle_\omega\Big).$$9 to satisfy the linear elliptic equation $t=0$0, whose kernel is trivial precisely by the Gauduchon condition; hence $t=0$1 is constant and $t=0$2 is constant. This is the paper's most rigid structural result: an Einstein-type condition alone, without any variational framework, pins down the scalar curvature.

Two corollaries sharpen this into Kähler–Einstein alternatives. On a compact Hermitian surface ($t=0$3, where pluriclosed equals Gauduchon), if $t=0$4 then either $t=0$5 or $t=0$6 is Kähler–Einstein with negative scalar curvature — the dichotomy following from the identity $t=0$7. For $t=0$8, pluriclosed balanced metrics satisfying the Einstein condition are either flat in scalar curvature or Kähler–Einstein with nonzero scalar curvature. Combining these with the existence theorems gives uniqueness (up to scaling) of constant-second-Chern-scalar-curvature representatives whenever the Einstein condition holds with $t=0$9.

Non-Kähler examples

The final section verifies the theory on explicit manifolds. On the Hopf manifold $$S_C^{(2)}(e^f\omega)=e^{-f}\big(S_C^{(2)}(\omega)-\Delta_\omega^{\mathbb C}f\big),$$0 with its standard metric, direct computation gives $$S_C^{(2)}(e^f\omega)=e^{-f}\big(S_C^{(2)}(\omega)-\Delta_\omega^{\mathbb C}f\big),$$1 while $$S_C^{(2)}(e^f\omega)=e^{-f}\big(S_C^{(2)}(\omega)-\Delta_\omega^{\mathbb C}f\big),$$2, illustrating that the two scalar curvatures differ substantially off the Kähler locus. On Tosatti–Weinkove's properly elliptic surface, the Vaisman metric satisfies $$S_C^{(2)}(e^f\omega)=e^{-f}\big(S_C^{(2)}(\omega)-\Delta_\omega^{\mathbb C}f\big),$$3 and $$S_C^{(2)}(e^f\omega)=e^{-f}\big(S_C^{(2)}(\omega)-\Delta_\omega^{\mathbb C}f\big),$$4. On Inoue surfaces with Tricerri metrics, $$S_C^{(2)}(e^f\omega)=e^{-f}\big(S_C^{(2)}(\omega)-\Delta_\omega^{\mathbb C}f\big),$$5 and $$S_C^{(2)}(e^f\omega)=e^{-f}\big(S_C^{(2)}(\omega)-\Delta_\omega^{\mathbb C}f\big),$$6; for the Vaisman family on the second Inoue type, $$S_C^{(2)}(e^f\omega)=e^{-f}\big(S_C^{(2)}(\omega)-\Delta_\omega^{\mathbb C}f\big),$$7, showing continuous dependence on the parameter $$S_C^{(2)}(e^f\omega)=e^{-f}\big(S_C^{(2)}(\omega)-\Delta_\omega^{\mathbb C}f\big),$$8. All these examples have constant (indeed constant-pointwise) second Chern scalar curvature without being Kähler, confirming that the existence theory is not vacuous outside the Kähler category.

Limitations and open questions

Several restrictions bound the scope of the results. The Bismut existence theorem requires $$S_C^{(2)}(e^f\omega)=e^{-f}\big(S_C^{(2)}(\omega)-\Delta_\omega^{\mathbb C}f\big),$$9 (or exactness of the Lee form); whether the second Bismut Yamabe problem is solvable on arbitrary compact Hermitian manifolds remains open, as does uniqueness in the Bismut case. The Chern-side theorems address only the sign-definite and zero cases of $t=1$0; the positive case yields existence only on products with high-genus curves rather than on $t=1$1 itself, leaving open whether a general positive-degree existence theorem holds. The Einstein-type constancy result requires the simultaneous pluriclosed and Gauduchon hypotheses, and the resulting Kähler–Einstein alternatives depend on the sign assumption $t=1$2 in dimension two. Finally, the paper does not address stability analogues of the Yau–Tian–Donaldson correspondence for the second Gauduchon degree.

Conclusion

The paper extends the conformal geometry of Hermitian scalar curvatures to the second traces of the Chern and Bismut connections. Its main contributions are: a complete solution of the second Bismut Yamabe problem under $t=1$3; sign-determined existence and uniqueness for constant second Chern scalar curvature governed by a new conformal invariant, the second Gauduchon degree, with Kodaira-dimension and pseudo-effectiveness corollaries; a rigidity theorem showing that a weak second Hermitian–Einstein condition on pluriclosed Gauduchon manifolds forces constant second Chern scalar curvature and, in low dimension or balanced settings, collapses to a Kähler–Einstein alternative; and explicit Hopf, elliptic, and Inoue examples demonstrating non-Kähler realizations throughout.