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A one-parameter Hermitian deformation of the Yamabe problem adds natural torsion terms to the scalar curvature on compact complex manifolds.

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load-bearing objection The paper defines a one-parameter deformation of the Yamabe equation on Hermitian manifolds by adding torsion terms and studies existence for the resulting family.

arxiv 2605.31028 v1 pith:7ADHIAXU submitted 2026-05-29 math.DG

Yamabe-type problems on compact Hermitian manifolds

Daniele Angella , Francesco Pediconi , Carlo Scarpa , Cristiano Spotti , Oluwagbenga Joshua Windare This is my paper
classification math.DG
keywords Yamabe problemHermitian manifoldstorsion termslocally conformally Kählerscalar curvaturedeformationexistence criteriacomplex manifolds
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper introduces a deformed Yamabe problem on compact complex manifolds that varies continuously with one parameter. This is done by adding natural torsion terms to the Riemannian scalar curvature to form a family of elliptic equations. The family recovers the classical Yamabe equation and also includes a scalar-curvature equation from locally conformally Kähler geometry viewed as a momentum map. The authors study criteria for the existence of solutions and give several examples. A sympathetic reader would care because this supplies a single elliptic framework connecting two separate curvature problems in the Hermitian setting.

Core claim

We introduce and study a one-parameter Hermitian deformation of the Yamabe problem on compact complex manifolds. The deformation is defined by adding natural torsion terms to the Riemannian scalar curvature, and includes both the classical Yamabe equation and a scalar-curvature equation arising in locally conformally Kähler geometry as a momentum map. We analyze criteria for the existence of solutions, and discuss several examples.

What carries the argument

The one-parameter family of elliptic equations obtained by adding natural torsion terms to the Riemannian scalar curvature on compact Hermitian manifolds.

Load-bearing premise

The torsion terms added to the Riemannian scalar curvature are natural and yield a well-defined one-parameter family of elliptic equations on any compact Hermitian manifold.

What would settle it

A compact Hermitian manifold on which the deformed equation fails to be elliptic for some parameter value, or on which the stated existence criteria do not correctly predict solvability.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The classical Yamabe equation is recovered as a special case of the deformed equation.
  • A scalar-curvature equation from locally conformally Kähler geometry appears as another special case.
  • Existence of solutions can be analyzed using criteria developed for the one-parameter family.
  • Several examples on compact Hermitian manifolds illustrate the solutions and the framework.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Varying the parameter continuously might allow interpolation between geometric properties on the same manifold.
  • The momentum map interpretation could connect the problem to variational methods in symplectic or Kähler geometry.
  • The framework might suggest new approaches to prescribing curvature on non-Kähler complex manifolds.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 1 minor

Summary. The paper introduces and studies a one-parameter Hermitian deformation of the Yamabe problem on compact complex manifolds. The deformation is defined by adding natural torsion terms to the Riemannian scalar curvature, and includes both the classical Yamabe equation and a scalar-curvature equation arising in locally conformally Kähler geometry as a momentum map. The authors analyze criteria for the existence of solutions and discuss several examples.

Significance. If the existence criteria and ellipticity claims hold, the construction would provide a unified one-parameter family interpolating between the classical Yamabe problem and the LCK momentum-map equation on Hermitian manifolds, offering a new tool for studying conformal geometry beyond the Kähler setting.

minor comments (1)
  1. The title refers to Hermitian manifolds while the abstract refers to complex manifolds; clarify the precise setting in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript. The recommendation is listed as uncertain, but the report contains no major comments or specific points for us to address.

Circularity Check

0 steps flagged

No significant circularity; new equation family introduced by definition

full rationale

The paper defines a one-parameter deformation of the Yamabe equation by adding natural torsion terms to the Riemannian scalar curvature on compact Hermitian manifolds. This construction is presented as a direct definition that recovers the classical Yamabe equation and the LCK momentum-map equation at parameter endpoints. No load-bearing step reduces a claimed prediction or uniqueness result to a fitted input, self-citation chain, or prior ansatz from the same authors. Existence criteria are analyzed via standard elliptic PDE methods on the newly defined family, without circular reduction to the inputs. The work is self-contained as an introduction of a new problem rather than a derivation from pre-existing fitted quantities.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Assessment based solely on the abstract; the full paper may introduce additional parameters or assumptions.

free parameters (1)
  • deformation parameter
    The one-parameter family is introduced but its status (free or fitted) is not specified in the abstract.
axioms (1)
  • domain assumption The manifold is compact and Hermitian.
    The problem is posed on compact complex manifolds equipped with Hermitian metrics.

pith-pipeline@v0.9.1-grok · 5591 in / 1186 out tokens · 27508 ms · 2026-06-28T21:13:33.681732+00:00 · methodology

0 comments
read the original abstract

We introduce and study a one-parameter Hermitian deformation of the Yamabe problem on compact complex manifolds. The deformation is defined by adding natural torsion terms to the Riemannian scalar curvature, and includes both the classical Yamabe equation and a scalar-curvature equation arising in locally conformally K\"ahler geometry as a momentum map. We analyze criteria for the existence of solutions, and discuss several examples.

discussion (0)

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Reference graph

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