Existence and geometry of Hermitian metrics with constant second scalar curvature

Liangdi Zhang

REVIEW 2 major objections 3 minor 50 references

Existence and geometry of Hermitian metrics with constant second scalar curvature

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Pith's one-line read The paper shows that the existence of Hermitian metrics with constant second Chern or Bismut scalar curvature within a conformal class is governed by the sign of a conformal invariant, the second Gauduchon degree, with unique solutions in…

desk verdict New conformal formulas and examples, but the two main existence proofs have real maximum-principle errors; worth peer review but not acceptance as written. read the letter →

arxiv 2601.20572 v2 pith:PW6USKXW submitted 2026-01-28 math.DG

Liangdi Zhang This is my paper · ORCID

classification math.DG MSC 53C5553C21

keywords secondscalarcurvatureHermitianmetricsBismutconnectionChernGauduchonmetricconformalclassYamabe-typeproblemdegree

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether a compact Hermitian manifold, a complex manifold with a metric that need not be Kähler, can be conformally deformed to a metric whose second scalar curvature is constant, where this curvature is a torsion-sensitive trace of the Chern or Bismut connection. The central claim is that the answer is controlled by a conformal invariant, the second Gauduchon degree: the total integral of the second Chern scalar curvature of the canonical Gauduchon metric in the class. When that invariant is zero, there is a unique conformal metric with vanishing second Chern scalar curvature; when it is negative, a unique conformal metric with constant negative second Chern scalar curvature, and in balanced classes the anti-canonical bundle is not pseudo-effective. For the Bismut connection, the paper proves existence of a constant second scalar curvature metric in every conformal class on manifolds with vanishing first Betti number. A final rigidity result shows that compact pluriclosed Gauduchon metrics satisfying a weak Einstein-type condition have constant second Chern scalar curvature, leading to a dichotomy with Kähler–Einstein metrics.

What carries the argument

The key object is the second Gauduchon degree $\Gamma^{(2)}_M(\{\omega\})$, the total integral of the second Chern scalar curvature $S^{(2)}_C$ of the unique Gauduchon metric in the conformal class; it is a conformal invariant that plays the role of a torsion-aware Yamabe invariant. The load-bearing identity is the conformal transformation formula $S^{(2)}_C(\omega_f)=e^{-f}\left(S^{(2)}_C(\omega)-\Delta^C_\omega f\right)$ for the Chern connection, and the analogous formula (3.16) for the second Bismut scalar curvature $S^{(2)}_B$; these convert the geometric problem into solvable semilinear elliptic equations. The Bismut existence proceeds through the functional $Y_q(\varphi)$ (4.3) and the equation $\square_{\omega_B}\varphi=N_1\mu_q\varphi^{q-1}$, while the negative Chern case uses a continuity method with maximum-principle a priori estimates and a uniqueness argument by comparison. The rigidity section uses the sum of the third and fourth Chern–Ricci curvatures $\Theta^{(3)}(\omega)+\Theta^{(4)}(\omega)$, whose $\partial\bar\partial$-closedness yields a linear elliptic equation $\Diamond_\omega f=0$ with only constant solutions on compact pluriclosed Gauduchon manifolds.

What would settle it

Compute the left-hand side of (5.9) at a minimum point $q$ of a solution $f_a$ for $a=1$: the maximum principle gives $S^{(2)}_C(\omega')(q)-\lambda e^{f_1(q)}\ge 0$, i.e. $e^{f_1(q)}\ge S^{(2)}_C(\omega')(q)/\lambda$. Choose a conformal class whose second Chern scalar curvature $S^{(2)}_C(\omega')$ is negative and nonconstant with $S^{(2)}_C(\omega')(q)/\lambda<1$ at the minimum point; then $f_1(q)\le \log(S^{(2)}_C(\omega')(q)/\lambda)<0$, contradicting the asserted uniform lower bound $f_a\ge 0$ in Lemma 5.3.

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Extended reading notes

Core claim

The paper's central discovery is that within a fixed Hermitian conformal class the existence and uniqueness of constant second Chern scalar curvature is decided by the sign of the second Gauduchon degree $\Gamma^{(2)}_M(\{\omega\})=\int_M S^{(2)}_C(\omega_G)\,\omega_G^n/n!$, a conformal invariant computed from the Gauduchon representative $\omega_G$ of the class. If $\Gamma^{(2)}_M(\{\omega\})=0$, the linear equation $\Delta^C_{\omega_G}f=S^{(2)}_C(\omega_G)$ is solvable, producing a unique (up to scaling) metric $\omega_a\in\{\omega\}$ with $S^{(2)}_C(\omega_a)=0$, and the paper derives that either $\kappa(M)=-\infty$ or $\kappa(M)=0$ with $K_M^{\otimes m}=\mathcal{O}_M$ for some $m\in\mathbb{Z}_+$. If $\Gamma^{(2)}_M(\{\omega\})<0$, the nonlinear equation $\Delta^C_{\omega'}f=-\lambda e^f+S^{(2)}_C(\omega')$, with $\lambda=\Gamma^{(2)}_M(\{\omega\})\mathrm{Vol}(M,\omega_G)^{-1}$, is solved by a continuity method, giving a unique conformal metric $\omega_c$ with $S^{(2)}_C(\omega_c)=\lambda<0$; in balanced classes the same method yields a metric with that constant as its first Chern scalar curvature and implies $K_M^{-1}$ is not pseudo-effective. For the Bismut connection, the paper proves that when $b_1(M)=0$ the Gauduchon metric is conformal to a balanced metric, and a variational argument on the functional $Y_q$ produces a smooth solution of the corresponding equation, hence a conformal metric of constant second Bismut scalar curvature. Finally, for a pluriclosed Gauduchon metric satisfying the weak second Hermitian–Einstein condition $\Theta^{(3)}(\omega)+\Theta^{(4)}(\omega)=f\omega$, the paper proves $S^{(2)}_C(\omega)$ is constant by showing the defining function $f$ solves a linear elliptic equation whose only solutions are constants; the surface case then forces either $S^{(2)}_C(\omega)\equiv 0$ or a Kähler–Einstein metric with negative scalar curvature.

Load-bearing premise

The negative-case existence theorem rests on a uniform a priori bound for the approximating solutions of the continuity method, and the paper's derivation of that bound (the lower half of Lemma 5.3) is not correct as written, leaving the closedness step dependent on an unproved estimate.

Editorial extensions

If this is right

If $\Gamma^{(2)}_M(\{\omega\})<0$, every Hermitian conformal class contains a unique constant-negative second Chern scalar curvature metric, and in balanced classes the same constant is realized as the first Chern scalar curvature of a conformal metric while $K_M^{-1}$ is not pseudo-effective.
If $\Gamma^{(2)}_M(\{\omega\})=0$, the unique conformal metric with $S^{(2)}_C=0$ forces the Kodaira dimension to be either $-\infty$ or $0$ with a torsion canonical bundle.
On any compact complex manifold with $b_1(M)=0$, including the non-Kähler Calabi–Eckmann manifolds in dimensions $n\ge 3$, every Hermitian conformal class contains a metric of constant second Bismut scalar curvature.
A compact pluriclosed Gauduchon metric satisfying the weak second Hermitian–Einstein condition must have constant second Chern scalar curvature; on surfaces this gives a strict alternative between $S^{(2)}_C\equiv 0$ and negative Kähler–Einstein.
The explicit examples (Hopf manifolds, non-Kähler properly elliptic surfaces, Inoue surfaces) show that constant second Chern scalar curvature is not a Kähler-only phenomenon.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The second Gauduchon degree $\Gamma^{(2)}_M(\{\omega\})$ behaves like a conformal 'charge' whose sign may classify Hermitian conformal classes; the positive case is left open in this paper, and Theorem 5.5 suggests it forces $\kappa(M)=-\infty$ and yields constant positive scalar curvature metrics on products with a high-genus curve.
The a priori lower bound $f_a\ge 0$ in Lemma 5.3 appears incorrect as written: the maximum principle at a minimum point gives $e^{f_a}\ge aS/\lambda+1-a$, which allows negative values, so a corrected uniform $C^0$ bound would be needed to make the closedness step of Theorem 1.5 fully rigorous as presented.
The rigidity dichotomy (zero second Chern scalar curvature or Kähler–Einstein) suggests that weak second Hermitian–Einstein metrics form a bridge between Chern–Einstein and Kähler–Einstein geometry; one might test whether the non-positivity of $f$ in Corollary 1.9 can be relaxed to bounded-below $f$.
Because the examples include non-Kähler surfaces with $b_1=1$ (Inoue surfaces), the Bismut theorem's condition $b_1(M)=0$ is sufficient but not necessary; finding the sharp topological condition for Bismut constant scalar curvature would be a natural next step.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague —

I read the paper with the stress-test note in hand, and the note is right. There are two genuine, load-bearing errors.

First, Lemma 5.3 claims f_a ≥ 0. The maximum principle at a minimum point only gives e^{f_a} ≥ a(-S)/(-λ)+1-a. With S close to zero this is less than 1, so f_a can be negative; explicit constant solutions of (5.6) show it. The C^0 bound in Lemma 5.4 rests on this, so the closedness of the continuity set A, and hence Theorem 1.5, is not established as written.

Second, in Theorem 4.3 the positivity of the minimizer is derived from Δψ+aψ≤0 with a>0. That maximum principle does not exist when the zeroth-order coefficient is positive; the equation for ψ does not put the sign in the required place. This step is needed for the Bismut-Yamabe existence claim in Theorem 1.1.

Both defects are probably repairable. The same calculation in Lemma 5.3 yields a uniform lower bound involving -S/λ, so the continuity method has a reasonable chance with a revised C^0 estimate. The positivity issue should be handled by a subcritical Yamabe-type argument rather than a misapplied strong maximum principle. But a paper with two false steps in the two main existence proofs does not establish its advertised theorems.

Credit where it is due: Section 3 is genuinely useful. The conformal transformation formulas for S_C^(2), S_B^(2), and the third/fourth Ricci curvatures of the Gauduchon connections are new and appear correctly derived. The examples in Section 7 are concrete and include non-Kähler metrics with constant second Chern scalar curvature. The structure is clear, the references are appropriate, and there is no circularity or hidden fitting. The citation pattern stays on the literature.

Who is this for? Researchers working on Chern-Yamabe-type problems in Hermitian geometry. They will want the formulas from Section 3 and the examples even if they do not rely on the main theorems. I would not cite the existence theorems until they are repaired; I might cite the formulas after revision.

My recommendation: send it to peer review, because the area is active, the formulas and examples deserve circulation, and the flaws are concrete and fixable rather than fatal to the program. But the referee should demand a repaired Lemma 5.3 and a corrected positivity argument before publication.

Referee Report

2 major / 3 minor

Summary. The paper studies Hermitian metrics with constant second scalar curvature on compact complex manifolds. In a fixed Hermitian conformal class, it derives conformal transformation formulas for the curvatures of the Gauduchon connections, defines a new conformal invariant called the second Gauduchon degree, and proves existence and uniqueness theorems for constant second Chern scalar curvature whose sign is controlled by this degree. It also treats a Yamabe-type problem for the second Bismut scalar curvature under the assumption b_1(M)=0, proving that the conformal class contains a metric with constant second Bismut scalar curvature. Finally, under a weak Einstein-type condition involving the third and fourth Chern-Ricci curvatures, it shows that the second Chern scalar curvature is constant, and it gives non-Kahler examples of metrics with constant second Chern scalar curvature.

Significance. If the proofs are completed, the paper would meaningfully extend the Chern-Yamabe theory to second scalar curvatures, introduce a useful conformal invariant, and provide concrete non-Kahler examples. The paper is clearly organized, the computations are explicit, and the use of external results such as Gauduchon's theorem and Yang's Kodaira-dimension criteria is appropriate. However, two load-bearing analytic steps in the main existence proofs are invalid as written, so the significance of the results is currently conditional on a repaired argument.

major comments (2)

[Section 5, Lemmas 5.3-5.4] The two-sided estimate (5.10) is not a consequence of (5.9), and the proof of closedness of the set A therefore breaks. Evaluating (5.9) at a maximum p and a minimum q gives, with S_C^(2)(omega')=lambda e^{-u}, only e^{f_a(p)} <= 1-a+a e^{-u(p)} and e^{f_a(q)} >= 1-a+a e^{-u(q)}. These imply uniform bounds of the form f_a <= max(0, -min_M u) and f_a >= min(0, -max_M u), but they do not imply f_a >= 0 nor the upper bound log(1+min_M S/lambda). The lower bound is in fact false on the local branch through a=0: differentiating F(a,f_a)=0 at a=0 gives (Delta+lambda)w = lambda(e^{-u}-1), where w = df_a/da at a=0. Since lambda<0 and Delta+lambda is negative definite, w is negative wherever u>0, so f_a<0 for small positive a. Lemma 5.4 uses the uniform C^0 estimate from Lemma 5.3 to obtain W^{2,p} and Schauder estimates, so Theorem 1.5 is not established as written. The defect is local: the corrected inequalities above still yield a uniform C^0 bound, so a revision can repair the argument.
[Section 4, Theorem 4.3] The proof that the weak minimizer phi_q is strictly positive is invalid. The argument chooses a>0 such that psi=-phi_q satisfies Delta psi + a psi <= 0 and then invokes the strong maximum principle. For an operator with a positive zeroth-order coefficient this conclusion is false: on S^1, psi = -(sin x)^4 is nonpositive with interior zeros and satisfies psi'' + a psi <= 0 for a >= 4. The coefficient in the equation actually satisfied by phi_q has uncontrolled sign, so the maximum principle cannot be applied in this way. A correct proof requires a Harnack-type inequality, a Moser iteration, or a subcritical approximation argument. Since Theorem 1.1 and Corollary 1.3 use the existence of a smooth positive solution of (4.2), the Bismut-Yamabe existence claim is not proved as written. In addition, the passage from the W^{1,2} weak solution to a smooth solution is not the immediate bootstrap claimed, because the right-hand side phi_q^{q-1} initially lies only in L^{q/(q-1)}; an iterative estimate is needed.

minor comments (3)

[Section 1, Corollary 1.3] The sentence 'Theorem 1.6 admits the following corollary' should refer to Theorem 1.1, not Theorem 1.6.
[Throughout] There are numerous typographical errors, including 'Hermtian', 'compacr', 'ans astheno', and 'Var' in the references; these should be corrected in a revised version.
[Section 7, Example 7.4] In equations (7.11) and (7.16), the notation S_C^(1)(omega_1) appears where the metric in Example 7.4 is omega_2; the labels should be made consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the existence proofs are genuine PDE constructions from external curvature formulas, and the target constants are conformal invariants rather than fitted inputs.

full rationale

The paper's central results are existence and uniqueness of conformal Hermitian metrics with prescribed constant second scalar curvature. The prescribed constants are the conformal invariants Gamma_M^(2)({omega}) Vol(M, omega_G)^{-1}, defined by integrating S_C^(2)(omega_G); they are not parameters fitted to a subset of the data being predicted. Equations (3.14), (4.1), (4.2), and (5.6) are derived by explicit conformal-transformation computations from the curvature identities of [42] (Lemma 2.2) and standard elliptic theory, so the continuity-method and variational solutions are actual constructions rather than restatements of the inputs. The citations to Gauduchon [21], Wang-Yang [42], Yang [44], Angella-Calamai-Spotti [1], and Tosatti-Weinkove [38] are external results whose assumptions do not include the target theorems; none is a same-author uniqueness theorem invoked to forbid alternatives. The reference [47] supplies a computational formula for conformal changes of curvature and an adjoint identity; the paper also re-derives the relevant formulas (Proposition 3.1, Corollary 3.2), so even if [47] is regarded as a self-citation it is not load-bearing. The externally noted false lower-bound step in Lemma 5.3 is a correctness defect in the closedness proof of the continuity method, not a circularity: the maximum-principle computation does not reduce the theorem to its conclusion, it simply fails to give the claimed bound as written. Overall the derivation chain is self-contained with respect to circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central results are existence theorems, so there are no data-fit parameters. The hypotheses (b_1=0, sign of Gamma^(2), f<=0) are domain assumptions on the manifold and metric, not free constants. The paper relies on standard elliptic theory and on several external results from the literature, including Gauduchon's theorem, curvature identities from Wang-Yang [42], and Kodaira-dimension criteria from Yang [44]. The new definitions (second Gauduchon degree, weak second Hermitian-Einstein) are not independently evidenced but are defined constructs with no falsifiable handle.

assumptions (5)

standard math In every Hermitian conformal class there is a unique Gauduchon representative (up to scaling).
Invoked in Section 4 and Section 5 to fix omega_G and to define Gamma^(2); from Gauduchon [21].
standard math The curvature identity (2.7) from Wang-Yang [42] for Gauduchon connections.
Used throughout Section 3 to derive conformal transformation formulas.
standard math Elliptic regularity, Sobolev embedding, and the maximum principle on compact manifolds.
Used in Theorem 4.3 and Lemmas 5.2 to 5.4; the paper's application in Theorem 4.3 is questionable.
domain assumption b_1(M)=0 implies the Lee form of omega_G is d-exact.
Hypothesis for Theorem 1.1 and Corollary 1.3; follows from Hodge theory as stated.
standard math Kodaira dimension and pseudo-effectivity criteria from Yang [44].
Used in Theorems 1.4, 1.5, and 5.5 to convert scalar curvature sign into geometric conclusions.

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Pith. "Pith review of Existence and geometry of Hermitian metrics with constant second scalar curvature." pith.science (2026). https://pith.science/paper/PW6USKXW

@misc{pith2026260120572,
  author       = {Pith},
  title        = {Pith review of: Existence and geometry of Hermitian metrics with constant second scalar curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PW6USKXW}},
  note         = {Machine review of arXiv:2601.20572}
}

read the original 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 within balanced Hermitian conformal classes, 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 K\"ahler-Einstein metric.

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Works this paper leans on

50 extracted references · 50 canonical work pages

[1]

Angella, S

D. Angella, S. Calamai, C. Spotti, On the Chern-Yamabe problem, Math. Res. Lett., 24(3) (2017) 645–677

work page 2017
[2]

Angella, S

D. Angella, S. Calamai, C. Spotti, Remarks on Chern-Einstein Hermitian metrics, Math. Z., 295 (2020) 1707–1722

work page 2020
[3]

Arezzo, A

C. Arezzo, A. Della Vedova, Y. Shi, Constant scalar curvature K¨ ahler metrics on ramified Galois coverings, J. Reine Angew. Math. 2023 (2023) 229–247

work page 2023
[4]

Aubin, ´Equations diff´ erentielles non lin´ eaires et probl` eme de Yamabe concernant la courbure scalaire, J

T. Aubin, ´Equations diff´ erentielles non lin´ eaires et probl` eme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., 55 (1976) 269–296

work page 1976
[5]

Aubin, Some nonlinear problems in Riemannian geometry, Springer Monogr

T. Aubin, Some nonlinear problems in Riemannian geometry, Springer Monogr. Math., Springer–Verlag, Berlin, 1998

work page 1998
[6]

Aviles, R

P. Aviles, R. C. McOwen, Conformal deformation to constant negative scalar curva- ture on noncompact Riemannian manifolds, J. Differ. Geom., 27(2) (1988) 225–239

work page 1988
[7]

Barbaro, On the curvature of the Bismut connection: Bismut-Yamabe problem and Calabi-Yau with torsion metrics, J

G. Barbaro, On the curvature of the Bismut connection: Bismut-Yamabe problem and Calabi-Yau with torsion metrics, J. Geom. Anal., 33 (2023) 153

work page 2023
[8]

W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Compact complex surfaces, 2nd ed., Springer–Verlag, 2004

work page 2004

Show all 50 references

[9]

R. J. Berman, S. Boucksom, M. Jonsson, A variational approach to the Yau–Tian– Donaldson conjecture, J. Amer. Math. Soc., 34(3) (2021) 605–652

2021
[10]

J. M. Bismut, A local index theorem for non-K¨ ahler manifolds, Math. Ann., 284(4) (1989) 681–699

1989
[11]

X. X. Chen, J. Cheng, On the constant scalar curvature K¨ ahler metrics (I)—A priori estimates, J. Amer. Math. Soc., 34 (2021) 909–936

2021
[12]

X. X. Chen, J. Cheng, On the constant scalar curvature K¨ ahler metrics (II)— Existence results, J. Amer. Math. Soc., 34 (2021) 937–1009

2021
[13]

X. X. Chen, S. Donaldson, S. Sun, K¨ ahler-Einstein metrics and stability, IMRN, 2014(8) (2014) 2119–2125

2014
[14]

X. X. Chen, S. Donaldson, S. Sun, K¨ ahler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Amer. Math. Soc., 28 (2015) 183–197

2015
[15]

X. X. Chen, S. Donaldson, S. Sun, K¨ ahler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π, J. Amer. Math. Soc., 28 (2015) 199–234

2015
[16]

S. S. Chern, Characteristic classes of Hermitian manifolds, Ann. of Math. (2), 47(1) (1946) 85–121

1946
[17]

Z. S. Dyrefelt, Existence of cscK metrics on smooth minimal models, Ann. Sc. Norm. Super. Pisa Cl. Sci., 23(5) (2022) 223–232

2022
[18]

L. C. Evans, Partial differential equations, Grad. Stud. Math., Vol. 19, Amer. Math. Soc., Providence, RI, 2nd ed., 2010

2010
[19]

Fino and A

A. Fino and A. Tomassini, A survey on strong KT structures, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52(100) (2009), no. 2, 99–116

2009
[20]

A. Fino, L. Ugarte, On generalized Gauduchon metrics, P. Edinburgh Math. Soc., 56 (2013) 733–753

2013
[21]

Gauduchon, Le th´ eor` eme de l’excentricit´ e nulle, C

P. Gauduchon, Le th´ eor` eme de l’excentricit´ e nulle, C. R. Acad. Sci. Parais S´ er. A-B, 285(5) (1977) A387–A390

1977
[22]

Gauduchon, La 1-forme de torsion d´ une vari` et` e hermitienne compacte, Math

P. Gauduchon, La 1-forme de torsion d´ une vari` et` e hermitienne compacte, Math. Ann. 267(4) (1984) 495–518

1984
[23]

Gauduchon, Hermitian connections and Dirac operators, Boll

P. Gauduchon, Hermitian connections and Dirac operators, Boll. Unione Mat. Ital. B, 11(2) (1997) 257–288

1997
[24]

Gauduchon, P

P. Gauduchon, P. S. Ivanov, Einstein-Hermitian surfaces and Hermitian Einstein- Weyl structures in dimension 4, Math. Z., 226(2) (1997) 317–326

1997
[25]

Grosse, The Yamabe equation on manifolds of bounded geometry, Commun

N. Grosse, The Yamabe equation on manifolds of bounded geometry, Commun. Anal. Geom., 21(5) (2013), 957–978. CONSTANT SECOND SCALAR CURVATURE METRICS 29

2013
[26]

J. Han, Y. Liu, On the existence of weighted cscK metrics, Adv. Math., 463 (2025) 110125

2025
[27]

J. M. Hogg, The Yamabe problem on non-compact manifolds of negative curvature type, Ph.D. thesis, University of Oxford, 2020

2020
[28]

Inoue, On surfaces of Class VII 0, Invent

M. Inoue, On surfaces of Class VII 0, Invent. Math., 24 (1974) 269–310

1974
[29]

Z. R. Jin, A counterexample to the Yamabe problem for complete noncompact mani- folds, in: Partial Differential Equations (Tianjin, 1986), Lecture Notes in Math., vol. 1306, Springer, Berlin, 1988, pp. 93–101

1986
[30]

Kim, The Yamabe problem and applications on noncompact complete Riemannian manifolds, Geom

S. Kim, The Yamabe problem and applications on noncompact complete Riemannian manifolds, Geom. Dedicata, 64 (1997) 373–381

1997
[31]

Latorre, L

A. Latorre, L. Ugarte, On non-K¨ ahler compacr complex manifolds with balanced ans astheno-K¨ ahler metrics, C. R. Acad. Sci. Paris Ser. I, 355 (2017) 90–93

2017
[32]

K. Liu, X. Yang, Geometry of Hermitian manifolds, Internat. J. Math., 23(6) (2012) 1250055

2012
[33]

K. Liu, X. Yang, Ricci curvatures on Hermitian manifolds, Trans. Am. Math. Soc., 369(7) (2017) 5157–5196

2017
[34]

Schoen, Conformal deformation of a Riemannian metric to constant scalar curva- ture, J

R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curva- ture, J. Differ. Geom., 20(2) (1984) 479–495

1984
[35]

Strominger, Superstrings with torsion, Nucl

A. Strominger, Superstrings with torsion, Nucl. Phys. B, 274(2) (1986) 253–284

1986
[36]

Sz´ ekelyhidi, The partialC0-estimate along the continuity method, J

G. Sz´ ekelyhidi, The partialC0-estimate along the continuity method, J. Amer. Math. Soc. 29(2) (2016) 537–560

2016
[37]

Tian, K-stability and K¨ ahler-Einstein metrics, Comm

G. Tian, K-stability and K¨ ahler-Einstein metrics, Comm. Pure Appl. Math., 68 (2015) 1085–1156

2015
[38]

Tosatti, B

V. Tosatti, B. Weinkove, The Chern-Ricci flow on complex surfaces, Composito Math., 149 (2013) 2101–2138

2013
[39]

Tricerri, Some examples of locally conformal K¨ ahler manifolds, Rend

F. Tricerri, Some examples of locally conformal K¨ ahler manifolds, Rend. Semin. Mat. Univ. Politec. Torino, 40 (1982) 81–92

1982
[40]

N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian struc- tures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, 22(3) (1968) 265–274

1968
[41]

Vaisman, Non-K¨ ahler metrics on geometric complex surfaces, Rend

I. Vaisman, Non-K¨ ahler metrics on geometric complex surfaces, Rend. Semin. Mat. Univ. Politec. Torino, 45 (1987), 117–123

1987
[42]

J. Wang, X. Yang, Curvatures of real connections on Hermitian manifolds, Pac. J. Math., 337(2) (2025), 365–391

2025
[43]

Wei, Yamabe equation on some complete noncompact manifolds, Pacific J

G. Wei, Yamabe equation on some complete noncompact manifolds, Pacific J. Math., 302(2) (2019) 717–739

2019
[44]

Yang, Scalar curvature on compact complex manifolds, Trans

X. Yang, Scalar curvature on compact complex manifolds, Trans. Am. Math. Soc., 371(3) (2019) 2073–2087

2019
[45]

Yang, Manifolds with non-positive second Chern-Ricci curvature, Math

X. Yang, Manifolds with non-positive second Chern-Ricci curvature, Math. Z., (2025) 310:72

2025
[46]

Yang, Chern number identities on compact complex surface and applications, arXiv:2508.11171v1 [math.DG]

X. Yang, Chern number identities on compact complex surface and applications, arXiv:2508.11171v1 [math.DG]

arXiv
[47]

X. Yang, K. Zhang, Conformal extremal metrics and constant scalar curvature, Calc. Var., 65 (2026) 53

2026
[48]

Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math

H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J., 12 (1960) 21–37

1960
[49]

Ye, Bismut Einstein metrics on compact complex manifolds, J

Y. Ye, Bismut Einstein metrics on compact complex manifolds, J. Funct. Anal., 288 (2025) 110805

2025
[50]

Zheng, Existence of constant scalar curvature K¨ ahler cone metrics, properness and geodesic stability, Math

K. Zheng, Existence of constant scalar curvature K¨ ahler cone metrics, properness and geodesic stability, Math. Ann., (2025) 1–73. 30 L. ZHANG (Liangdi Zhang) Mathematical Science Research Center, Chongqing University of Technol- ogy, Chongqing 400054, China Email address:ldz...

2025

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[1] [1]

Angella, S

D. Angella, S. Calamai, C. Spotti, On the Chern-Yamabe problem, Math. Res. Lett., 24(3) (2017) 645–677

work page 2017

[2] [2]

Angella, S

D. Angella, S. Calamai, C. Spotti, Remarks on Chern-Einstein Hermitian metrics, Math. Z., 295 (2020) 1707–1722

work page 2020

[3] [3]

Arezzo, A

C. Arezzo, A. Della Vedova, Y. Shi, Constant scalar curvature K¨ ahler metrics on ramified Galois coverings, J. Reine Angew. Math. 2023 (2023) 229–247

work page 2023

[4] [4]

Aubin, ´Equations diff´ erentielles non lin´ eaires et probl` eme de Yamabe concernant la courbure scalaire, J

T. Aubin, ´Equations diff´ erentielles non lin´ eaires et probl` eme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., 55 (1976) 269–296

work page 1976

[5] [5]

Aubin, Some nonlinear problems in Riemannian geometry, Springer Monogr

T. Aubin, Some nonlinear problems in Riemannian geometry, Springer Monogr. Math., Springer–Verlag, Berlin, 1998

work page 1998

[6] [6]

Aviles, R

P. Aviles, R. C. McOwen, Conformal deformation to constant negative scalar curva- ture on noncompact Riemannian manifolds, J. Differ. Geom., 27(2) (1988) 225–239

work page 1988

[7] [7]

Barbaro, On the curvature of the Bismut connection: Bismut-Yamabe problem and Calabi-Yau with torsion metrics, J

G. Barbaro, On the curvature of the Bismut connection: Bismut-Yamabe problem and Calabi-Yau with torsion metrics, J. Geom. Anal., 33 (2023) 153

work page 2023

[8] [8]

W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Compact complex surfaces, 2nd ed., Springer–Verlag, 2004

work page 2004