On cohomogeneity one Hermitian non-K\"ahler metrics

We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following B\'erard-Bergery which includes, among the others, the holomorphic line bundles on $\mathbb C\mathbb P^{m-1}$, the linear Hopf manifolds and the Hirzebruch surfaces. We characterize their invariant special Hermitian metrics, such as balanced, K\"ahler-like, pluriclosed, locally conformally K\"ahler, Vaisman, Gauduchon. Furthermore, we construct new examples of cohomogeneity one Hermitian metrics solving the second-Chern-Einstein equation and the constant Chern-scalar curvature equation.

Theorem B (see Corollary 4.7, Corollary 4.9 and Proposition 4.6). Let (M (i,n) (G, K), J, g) be one of the Bérard-Bergery standard cohomogeneity one Hermitian manifolds. The following three conditions are equivalent: g is pluriclosed, g is balanced, g is Kähler. Moreover, if (M (i,n) (G, K), J) has singular orbits, then g is Gauduchon if and only if it is Kähler.
Finally, we investigate the second-Chern-Einstein and the constant Chern-scalar curvature problems (see Section 2.2) on this class of manifolds. Firstly, we prove a local existence and uniqueness result for second-Chern-Einstein metrics with prescribed Chern-scalar curvature (see Theorem 5.1), by using a method due to Malgrange [39] already exploited by Eschenburg and Wang [20] and by Böhm [14,15]. Then, concerning the existence of complete solutions to the second-Chern-Einstein equations, we prove the following Let us stress that the homogeneous second-Chern-Einstein metrics on (M (1,n) (G, K), J) and (M (3,n) (G, K), J), corresponding to case (a) in Theorem C, are clearly of constant Chern-scalar curvature. They include the classical example of the standard metric on the diagonal Hopf manifold, which corresponds in our notation to M (3,1) (G, K) with G = SU(m) and K = S(U(1) × U(m − 1)). On the other hand, the metrics that we constructed on manifolds with singular orbits, corresponding to cases (b) and (c) in Theorem C, are weakly second-Chern-Einstein, namely, they do not have constant Chern-scalar curvature. Therefore, this brings us to investigate the constant Chern-scalar curvature problem for such manifolds. In this direction, we obtain Theorem D (see Remark 5.2, Theorem 5.7, Theorem 5.8). All the Bérard-Bergery standard cohomogeneity one complex manifolds (M (i,n) (G, K), J) admit complete, non-Kähler metrics of cohomogeneity one with constant Chern-scalar curvature.
Notice that, in complex dimension m = 2, the compact case with two singular orbits (corresponding to i = 4) reduces to the Hirzebruch surfaces. This case has been treated by Koca and Lejmi [33], who proved the existence of positive constant Chern-scalar curvature of cohomogeneity one.
The paper is organized as follows. In Section 2, we recall some basics on the Chern connection and cohomogeneity one actions. In Section 3, we recall the construction of (M (i,n) (G, K), J, g) following Bérard-Bergery, and we compute the Hermitian curvatures and torsion of such manifolds. In Section 4, we investigate the existence of special non-Kähler Hermitian metrics, proving Theorem A and Theorem B. In Section 5, we prove Theorem C and Theorem D. Finally, in Appendix A, we collect the detailed computations needed in Section 3, precisely to prove Propositions 3.15 and 3.17, and in Section 4, namely to prove Equation (4.3).

Preliminaries
In this section, we briefly recall some facts on the Chern connection and on Hermitian manifolds with cohomogeneity one actions by holomorphic isometries.

The Chern connection.
Let (M, J, g) be a connected, complete, Hermitian manifold of real dimension dim R M = 2m. Let us denote by ω := g(J · , · ) its fundamental 2-form, by D its Levi-Civita connection and by ∇ its Chern connection, which is defined by  and, for any smooth function f : M → R, we denote by ∆ Ch g f := g(dd c f, ω) the Chern-Laplacian of f .

Second-Chern-Einstein metrics.
Let (M, J, g) be a connected, complete, Hermitian manifold of real dimension dim R M = 2m. We recall that, by the lack of symmetries of the Chern-curvature we have (at least) two ways to trace the Ricci tensor. We call first Chern-Ricci curvature the tensor defined by where (e α , Je α ) is a (J, g)-unitary frame for the tangent space at x. Similarly, we call second Chern-Ricci curvature the tensor defined by Finally, the Chern-scalar curvature is the function given by We remark that, according to our notation, when g is Kähler it holds that Ric Ch [1] (g) = Ric Ch [2] (g) = Ric(g) and scal Ch (g) = scal(g), where Ric(g) and scal(g) denote the Riemannian Ricci tensor and the Riemannian scalar curvature of g, respectively.
This yields to the following 2m g . We stress that the first-Chern-Einstein problem is basically understood for compact complex manifolds X = (M, J), see [7,52]. Indeed, strongly first-Chern-Einstein metrics with non-zero Chern-scalar curvature are Kähler-Einstein. Moreover, by conformal methods, if a Hermitian metric g is weakly first-Chern-Einstein with non-identically-zero Chern-scalar curvature, then it is conformal to a Kähler metric in the class ±c 1 (X) (see [7,Theorem A]). Finally, compact complex manifolds with first Bott-Chern class c BC 1 (X) = 0 are the so-called non-Kähler Calabi-Yau manifolds [52] and always admit Chern-Ricci flat metrics [52,Theorem 1.2], see also [53,Corollary 2].
On the other hand, the second-Chern-Einstein problem seems to be geometrically more appealing, see e.g. [24,51,47,7]. Note that a Hermitian metric g on X = (M, J) is second-Chern-Einstein according to Definition 2.3 if and only if the induced Hermitian metric h(V, W ) := g(V, W ) on the holomorphic tangent bundle T 1,0 X is Hermite-Einstein by taking trace with itself [22]. Remark 2.4. We observe that the second-Chern-Einstein condition is satisfied by a Hermitian metric g if and only if it is satisfied by all the metrics in its conformal class (see [22]), since for any smooth function f : M → R it holds that Ric Ch [2] (e f g) = Ric Ch [2] (g) − (∆ Ch g f )g . We remark that this is strongly different from the Riemannian analogue, i.e. the Einstein condition. On the other hand, we stress that a Kähler metric is second-Chern-Einstein if and only if it is Einstein.
Note that, up to our knowledge, a non-Kähler example of second-Chern-Einstein metric on a compact complex manifold of complex dimension m > 2 with negative scalar curvature is still missing.
Then, for r ∈ I, the orbit S r := G · γ(r) is principal and can be identified with a fixed homogeneous space by means of the 1-parameter family of G-equivariant diffeomorphisms where H ⊂ G is a closed subgroup. For r ∈ ∂I, the following cases occur: (i) ∂I = ∅ and all the orbits are principal; (ii) ∂I = {0}, the orbit S 0 := G · γ(0) is non-principal and G-equivariantly diffeomorphic to a homogeneous space G/L, where H ⊂ L ⊂ G is an intermediate subgroup and L/H is a sphere, see e.g. [9, Section 2.12]; (iii) ∂I = {±π}, the orbits S ±π := G · γ(±π) are principal but in general γ(−π) = γ(π), see e.g. [9, Section 2.10]; (iv) ∂I = {0, π}, the orbits S 0 := G · γ(0), S π := G · γ(π) are non-principal and G-equivariantly diffeomorphic to two homogeneous space G/L ± , with H ⊂ L − ∩ L + and L ± /H are spheres, see e.g. [9, Section 2.13]. The subset M reg := r∈I S r of regular points is an open, dense submanifold of M which projects onto I by means of the canonical projection M → Ω and the restricted Riemannian metric splits as g| M reg = dr 2 + g| Sr , where r is the coordinate on I. We fix an Ad(G)-invariant inner product Q on the Lie algebra g := Lie(G) and a Q-orthogonal decomposition g = h + m, with h := Lie(H). We consider identified m ≃ T eH G/H by means of the evaluation map X → d ds exp(sX)H s=0 , where exp : g → G denotes the Lie exponential map of G. We also identify any G-invariant tensor field on G/H with the corresponding Ad(H)-invariant tensor on m in the usual way.
Firstly, we define the 1-parameter family (g r ) ⊂ Sym 2 (m * ) Ad(H) by We also set Then, for any r ∈ I, the complex structure J induces a linear complex structure J r on p r := ker(θ r ) ⊂ m by setting J r : p r → p r , J r := (dφ r ) −1 eH • J γ(r) • (dφ r ) eH | pr . The integrability of J implies that, for any X, Y ∈ p r , (2.8) Moreover, since the subgroup H ⊂ G leaves any point of the geodesic γ fixed and G acts by holomorphic isometries, it follows that [h, T r ] = 0 for any r ∈ I (2.9) and so (θ r , J r ) is a 1-parameter family of G-invariant CR structures on G/H, see e.g. [3].
We recall now the following definition [48]: G/H is said to be ordinary if G is semisimple, the normalizer K := N G (H o ) of the connected component H o of H is the centralizer of a torus and dim K = 1 + dim H. This implies that: · T ≡ T r and p ≡ p r do not depend on r ∈ I; · the Lie algebra k := Lie(K) splits Q-orthogonally as k = h ⊕ RT .
Moreover, the complex structure J is said to be projectable if each J r is Ad(K)-invariant. In this case, (J r ) r∈I is mapped onto a 1-parameter family of G-invariant complex structures on the flag manifold G/K. Since the set of invariant complex structures on a flag manifold is discrete [42,1], it follows that J ≡ J r is constant. · the principal orbits are ordinary and the complex structure J is projectable; · the non-principal orbits, if they exist, are flag manifolds with the induced complex structure.
In Section 3, a distinguished kind of cohomogeneity one standard Hermitian manifolds will be investigated.

Bérard-Bergery standard cohomogeneity one Hermitian manifolds
In this section, we consider a special class of standard cohomogeneity one Hermitian manifolds, following the construction of Bérard-Bergery [9]. Then, we compute the Chern connection and the Chern-Ricci tensors of such manifolds.

Chern connection of standard cohomogeneity one Hermitian manifolds.
Let (M, J, g) be a standard cohomogeneity one Hermitian manifolds acted effectively by holomorphic isometries by a compact, connected real Lie group G.
Hereafter, we adopt the same notation introduced in Section 2. Notice that the complement m of h in g admits a Q-orthogonal, Ad(H)-invariant decomposition m = a + p, where a := RT is the trivial submodule. Since, by hypothesis, p does not contain any trivial Ad(H)-submodule, the metrics g r induced by g on m split uniquely as g r = F (r) 2 Q| a⊗a + g r | p⊗p by means of the Schur Lemma, where F : I → R is a smooth, positive function, possibly satisfying some appropriate boundary condition. From now on, given V ∈ g, we denote by V * ∈ Γ(T M ) the fundamental vector field on M associated to V , that is V * p := d ds exp(sV ) · p s=0 , and we also set N := F ∂ ∂r . (3.1) Notice that, by construction, and, for any V, W ∈ g, it holds We observe also that, for any X ∈ p and for any Y 1 , Y 2 ∈ m, we have By using Equations (2.2) and (2.3), one can directly obtain the explicit formulas for the Levi-Civita connection and the 3-form dω. Here and in the following statements, we consider the context and assume the notation as above.
Proposition 3.1. Let X, Y, Z ∈ p. Then, the non-vanishing components of the Levi-Civita connection are given by Proposition 3.2. Let X, Y, Z ∈ p. Then, the 3-form dω is given by By combining Proposition 3.1 and Proposition 3.2, and by Equation (2.1), we obtain explicit formulas for the Chern connection along the geodesic γ(r). More precisely Proposition 3.3. Let X, Y, Z ∈ p. Then the non-vanishing components of the Chern connection are As a direct consequence of Proposition 3.3 and Equation (2.6), we get We can also characterize the Kähler metrics as follows. By Proposition 3.2 it follows that dω(X * , Y * , T * ) γ(r) = 0 for any X, Y ∈ p if and only if the restriction g r | p⊗p is Ad(K)-invariant for any r ∈ I. This is equivalent to say that the metrics (g r ) on G/H are of submersion-type with respect to the homogeneous circle bundle K/H → G/H → G/K, namely, they induce metrics on the base such that the projection is a Riemannian submersion. Moreover, if g r | p⊗p is Ad(K)-invariant, then the condition dω(X * , Y * , Z * ) γ(r) = 0 for any X, Y, Z ∈ p holds true if and only if the induced metrics on the flag manifold G/K are Kähler. Let us fix now a Q-orthogonal, Ad(K)-invariant, irreducible decomposition Since flag manifolds are equal rank homogeneous spaces, namely rank(K) = rank(G), it follows that their isotropy representations are always monotypic, namely p i ≃ p j for any 1 ≤ i < j ≤ ℓ. Hence, the decomposition (3.6) is unique (up to order) and, by the Schur Lemma, the metrics g r splits uniquely as Then, the condition dω(X * , Y * , N ) γ(r) = 0 for any X, Y ∈ p holds true if and only if (3.7)

Construction of Bérard-Bergery manifolds.
Let us assume that P = P (G, K) := G/K is a simply-connected, irreducible compact Hermitian symmetric space, i.e. G is a connected, compact, simple Lie group and K ⊂ G is a maximal, connected, compact subgroup with center isomorphic to the circle group [30, Theorem 6.1]. Set dim R P = 2(m−1), with m ≥ 2, and let p ∈ N be the unique positive integer such that p −1 c 1 (P ) is an indivisible (positive) class in the cohomology group H 2 (P ; Z), see [13,Chapter 5,Section 16]. In the following, we list the possibilities for P , following [10, Section 7.102 and Section 9.124]. Then, consider the associated bundle with the projection π n : Σ n → P given by π n ([a, z]) := aK. Notice that G acts transitively on the left on Σ n byã · [a, z] := [ãa, z] and the stabilizer of [e, 1] is the subgroup ker(ρ n ) = H×Z n ⊂ K, where We are ready to construct the family M (i,n) (G, K), with 1 ≤ i ≤ 4 and n ∈ N, of standard cohomogeneity one Hermitian manifolds in the following way.
C be the homogeneous complex line bundle over P associated to π n : Σ n → P by means of the standard action of U(1) on C. Then, M (2,n) (G, K) is equivariantly diffeomorphic to the quotient of Σ n × [0, +∞) by the fibration Σ n × {0} → P , on which G acts by left multiplication on the first factor. iii) We set M (3,n) (G, K) := Σ n × S 1 and we let G act on M (3,n) (G, K) via a · (x, z) := (a · x, z). The orbit space is Ω = S 1 . iv) We let M (4,n) (G, K) := Σ n × U(1) CP 1 be the homogeneous CP 1 -bundle over P associated to π n : Σ n → P by means of the standard action of U(1) on CP 1 . Then, M (4,n) (G, K) is equivariantly diffeomorphic to the quotient of Σ n × [0, π] under the identification of the two boundaries by means of the fibrations Σ n × {0} → P and Σ n × {π} → P , on which G acts by left multiplication on the first factor. We observe that the manifolds in families M (3,n) (G, K) and M (4,n) (G, K) are compact, and the manifolds in families M (2,n) (G, K) and M (4,n) (G, K) are simply connected. Moreover, manifolds in family M (4,n) (G, K) are almost-homogeneous in the sense of [31], i.e. the complexified Lie group G C acts on them by biholomorphisms with one open orbit, see [49].
Let M (i,n) (G, K) be as above. We denote by B be the Cartan-Killing form of g := Lie(G) and we set k := Lie(K), h := Lie(H), a := z(k) = Lie(Z(K)). Then, the positive definite Ad(G)-invariant scalar product Q : We fix a vector T ∈ a with Q(T, T ) = 1 and we pick the only λ > 0 such that is a linear complex structure on p [32, Chapter XI, Theorem 9.6]. Notice that a direct computation and so Moreover, we stress that: · the restriction Q p := Q| p⊗p induces a G-invariant Kähler-Einstein metric on the base space P satisfying the equation Ric(Q p ) = 2mQ p · the scalar product 2m(m−1)n 2 p 2 Q a , with Q a := Q| a⊗a , corresponds to the standard metric of radius 1 on the fibres of π n : Σ n → P Then, being P irreducible, any G-invariant Hermitian metric g on (M (i,n) (G, K), J) which is of submersiontype with respect to M (i,n) (G, K) → P is completely determined by two positive, smooth functions f, h : I → R, satisfying some appropriate smoothness conditions, by means of the splitting Remark 3.6. We notice that the smoothness condition in (iii) and Equation (3.11) imply immediately that the complex manifolds (M (3,n) (G, K), J) cannot admit cohomogeneity one, submersion-type Kähler metrics (see also [4,Corollary 20]). Actually, it holds more: it can be easily check that π 1 (Σ n ) is finite, and so its first Betti number is b 1 (Σ n ) = 0. In particular, this implies that (M (3,n) (G, K), J) do not admit Kähler metrics at all.
From now on, we will adopt the following Definition 3.7. A Bérard-Bergery standard cohomogeneity one Hermitian manifold is a triple (M (i,n) (G, K), J, g), where M (i,n) (G, K) is the bundle over P = G/K constructed as above, J is the unique G-invariant projectable complex structure on M (i,n) (G, K) as in Remark 3.5 and g = g(f, h) is the Riemannian metric described in Equation (3.10). Accordingly, any pair (M (i,n) (G, K), J) will be called Bérard-Bergery standard cohomogeneity one complex manifold.
Let us point out that the above construction can be performed in a more general setting, i.e. by requiring that the base space P = G/K is a Kähler C-space, see [9]. However, in this work, we will just focus in the case of P being symmetric and irreducible.
Example 3.8. Consider the Hermitian symmetric space P = CP m−1 , corresponding to G = SU(m) and K = S(U(1)×U(m−1)). Here, the Ad(G)-invariant scalar product Q(A 1 , A 2 ) := − 1 2 Tr(A 1 A 2 ) on the Lie algebra g = su(m), defined following the above normalization, induces on P the Fubini-Study metric with sectional curvature satisfying 1 ≤ sec ≤ 4. In this case, the principal orbits are equivariantly diffeomorphic to the lens space n z, and   iii) if i = 3, ϕ is smooth and S 1 -periodic; iv) if i = 4, I = (0, π) and ϕ is the restriction of a smooth even function on R satisfying ϕ(π+r) = ϕ(π−r). From now on, any function ϕ : I → R satisfying the appropriate smoothness condition will be called admissible.
We begin this section by listing the Levi-Civita connection and the Riemannian Ricci tensor of the manifolds (M (i,n) (G, K), J, g). By straightforward computations, from Proposition 3.1 and Equations (3.10), (3.8) we get . Moreover, from [27, Proposition 1.14], we directly obtain Proposition 3.13. Let X ∈ p with Q(X, X) = 1. Then the Riemannian Ricci tensor is given by 4 . We compute now the Chern connection and the Chern-Ricci tensors of the manifolds (M (i,n) (G, K), J, g). From Proposition 3.3 and Equations (3.10), (3.8) we get Proposition 3.14. Let X, Y, Z ∈ p. Then . We are ready to state the following proposition, whose proof will be given in Appendix A.

b) The second Chern-Ricci tensor verifies
Ric Ch [2] (g)(T * , T * ) γ(r) = 2m(m−1)n 2 (3.13) c) Both the Chern-Ricci tensors satisfy (3.14) d) The Chern-scalar curvature is given by Given Proposition 3.15, we are now able to study the second-Chern-Einstein equations and the constant Chern-scalar curvature equation for this special class of Hermitian cohomogeneity one manifolds. This will be done in Section 5.
As a direct consequence of Proposition 3.17, whose proof will be given in Appendix A, we get the following

Special Hermitian metrics on Bérard-Bergery manifolds
In this section, we investigate the existence of special non-Kähler Hermitian metrics, such as balanced, pluriclosed, locally conformally Kähler, Vaisman, and Gauduchon, on the Bérard-Bergery standard cohomogeneity one Hermitian manifolds. In particular, we prove Theorem A and Theorem B.

Proof of Theorem A.
We begin by pointing out the following Proposition 4.1. All the cohomogeneity one Hermitian metrics g of submersion-type on the complex manifold (M (i,n) (G, K), J) are locally conformally Kähler.
Proof. By [19, Corollary 1.1], we know that g is locally conformally Kähler if and only if the complex structure J is parallel with respect to the Weyl connection associated to (g, 1 m−1 ϑ), equivalently, the following is satisfied:  Let now g be a cohomogeneity one, submersion-type Hermitian metric on a complex manifold (M (i,n) (G, K), J). By Proposition 4.1, it follows that dω = 1 m−1 ϑ ∧ ω which in turn implies that L A ϑ = 0 for any holomorphic Killing vector field A ∈ Γ(T M ). Hence, by Equations (3.1), (3.10) and Proposition 3.12, the non-vanishing components of the Levi-Civita covariant derivative of ϑ are  Conversely, assume that Dϑ = 0. Notice that either ϑ(N ) γ(ro) = 0 for some r o ∈ I, or ϑ(N ) γ(r) is nowhere vanishing. In the former case, the first equation in (4.1) yields that ϑ(N ) γ(r) is constantly zero. In fact, one gets that ϑ = 0, that is, g is Kähler. In particular, if g is Vaisman, then the above observation implies that Dϑ = 0 and ϑ(N ) γ(r) is nowhere vanishing. Hence, Equations (4.1) immediately imply that f and h are constant.  admit cohomogeneity one, strictly locally conformally Kähler metrics that are non-Vaisman. Remarkably, in the homogeneous case, this is excluded by [25,29]. We stress that the Hopf manifold, which is the main example of Vaisman manifold [44], corresponds, in our notation, to (M (3,1) (G, K), J) with G = SU(m) and K = S(U(1) × U(m−1)), see Example 3.8.
Remark 4.5. The locally conformally Kähler metrics by Proposition 4.1 include the globally conformally Kähler, Einstein metrics by Bérard-Bergery [9, Théorème 1.10]. We also recall that Einstein, locally conformally Käher, non-Kähler metrics are completed classified by [36,18,38], and they are either the Einstein, globally conformally Kähler metrics by [9], or they are defined on CP 2 by blowing up one or two points.

Proof of Theorem B.
We begin by characterizing the Gauduchon condition as follows. Proof. Let (ẽ α ) be a Q p -orthonormal basis for p. Then, a straightforward computation based on Equation (4.1) yields

r) and so g is Gauduchon if and only if
, h(r) are positive for any r ∈ I and

follows that g is Gauduchon if and only if Equation (4.2) is satisfied.
Let us assume now that (M (i,n) (G, K), J) has a singular orbit, that is i = 2 or i = 4. Then, the smoothness conditions at r = 0 imply that k = 0 in Equation (4.2). Therefore, in this case, g is Gauduchon if and only if it is Kähler.
Since the balanced condition is equivalent to ϑ = 0, from Proposition 3.17 and Equation (3.11) we immediately get   [26,58], namely, they do not satisfy the same symmetries as in the Kähler case.
Concerning the pluriclosed condition, a tedious but straightforward computation (see Appendix A) shows that where ρ(X, Y ) = Q p (JX, Y ) is again the G-invariant Kähler-Einstein form on P . Hence, together with Equation (3.11), this proves the following which completes the proof of Theorem B.

Constant Chern-scalar curvature and second-Chern-Einstein metrics
In this section, we investigate the existence of second-Chern-Einstein metrics and of metrics with constant Chern-scalar curvature on the Bérard-Bergery standard cohomogeneity one Hermitian manifolds. In particular, we first prove a local existence and uniqueness result for second-Chern-Einstein metrics with prescribed Chern-scalar curvature. Then, we prove Theorem C and Theorem D.

The second-Chern-Einstein equations.
Let (M (i,n) (G, K), J, g) be a Bérard-Bergery standard cohomogeneity one Hermitian manifold and fix a unit speed geodesic γ : I → M which intersects orthogonally any G-orbit. Then, by means of Proposition 3.15, the second-Chern-Einstein equation where λ : I → R is an admissible function (see Remark 3.10). Notice that, in this case, it holds that λ = scal Ch (g). Our first result in this section concerns the local existence and uniqueness of second-Chern-Einstein metrics, with prescribed Chern-scalar curvature, in a neighborhood of a singular orbit. More precisely Then, a straightforward computation shows that the Equations with v(r) := (x(r), y(r), z(r), w(r)) t , A := Moreover, the smoothness conditions for the functions f and h, together with the equation h(0) = a, imply that v o = (1, 0, a, 0) t . (5.4) We stress now that the following conditions are satisfied: Then, by the Malgrange Theorem [39, Theorem 7.1], see also [14,Theorem 2.2], there exists a unique solution v(r), defined on an interval (−ε, ε), to the Equation (5.3) with initial condition (5.4), which depends continuously on the data a, λ.
By Equation (5.2), we obtain a pair (f, h) of smooth functions f, h : (−ε, ε) → R which satisfy Equations (5.1) such that Since the pair (f ,ĥ) of functions defined bŷ satisfy Equations (5.1) with the initial conditions (5.5), by uniqueness we conclude that f is odd and h is even. Therefore, these functions give rise to a smooth Hermitian metric g = g(f, h) on the open set U reg ε ⊂ M (i,n) (G, K) reg , which admits a unique smooth extension over the singular orbit G · γ(0).
Concerning complete solutions to the Equations 5.1, we point out the following . Therefore, in the following, we will focus on Bérard-Bergery manifolds (M (i,n) (G, K), J) with singular orbits, namely, the cases i = 2 and i = 4.

Complete second-Chern-Einstein metrics in case of singular orbits.
In this section, we will construct complete second-Chern-Einstein metrics on the manifolds (M (2,n) (G, K), J) by using the same technique as [9, Section 11].
Then, from Equations (5.10) and (5.11), we get the following Cauchy problem for u(t): The unique solution to Equation (5.12) is the function u : [1, +∞) → R defined by is smooth, positive, increasing and, by construction, its inverse φ := ϕ −1 solves the Cauchy problem (5.10). Therefore, by means of Equation (5.9), the proof is completed.
Remark 5.4. The Chern-scalar curvature of the metric g φ = g(f φ , h φ ) constructed from Equation (5.9) by solving the Cauchy problem (5.10) is given by Notice that, if mn − p = 0, which correspond to the manifolds O CP m−1 (−1), the function (5.13) is u(t) = 4(t − 1). Hence, we recover the family of examples introduced in Formula (5.8).
Fix a complex manifold (M (i,n) (G, K), J), with i ∈ {2, 4}, and set for some smooth, increasing, positive, function φ : I → R. Notice that, in this case, and so this metric is necessarily non-Kähler by Equation (3.11). Let c ∈ R to be fixed later. Then, the constant Chern-scalar curvature equation We look for a solution of the form φ ′ (r) = u(φ(r)) (5. 16) for some smooth real function u = u(t). The, we get the following ODE which can be explicitly integrated. Indeed, the following cases occur.