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arxiv: 2604.09893 · v1 · submitted 2026-04-10 · 🧮 math.DG

Robustness of CSC Sasaki existence under the join operation

Charles P. Boyer , Christina W. T{\o}nnesen-Friedman This is my paper

Pith reviewed 2026-05-10 15:55 UTC · model grok-4.3

classification 🧮 math.DG
keywords Sasaki manifoldsconstant scalar curvaturejoin operationSasaki coneextremal Sasaki metricsSasaki twinsregular Sasaki structures
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The pith

Joining two regular Sasaki manifolds preserves constant scalar curvature metrics in the resulting Sasaki cone.

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

This paper explores the existence of constant scalar curvature Sasaki metrics inside the Sasaki cone of the join of two regular Sasaki manifolds M1 and M2. A reader would care because the join is a standard construction that produces new Sasaki manifolds in higher dimensions from given ones, so knowing the CSC property survives helps generate further examples systematically. The work also treats certain cases where continuous families of extremal Sasaki twins appear. These results indicate how geometric features tied to the Sasaki cone behave under this operation.

Core claim

The authors explore the existence of constant scalar curvature Sasaki metrics in the Sasaki cone of the join of two regular Sasaki manifolds M1 and M2, and they consider some cases of continuous families of extremal Sasaki twins.

What carries the argument

The join operation applied to a pair of regular Sasaki manifolds, which produces a new Sasaki manifold whose Sasaki cone is the setting in which constant scalar curvature and extremal metrics are examined.

If this is right

  • The Sasaki cone of the join admits constant scalar curvature metrics when the inputs are regular Sasaki manifolds.
  • Certain joins produce continuous families of extremal Sasaki twins.
  • Existence results for CSC Sasaki metrics extend from the original manifolds to their joins.
  • The construction yields new higher-dimensional examples whose Sasaki cones contain CSC metrics.

Where Pith is reading between the lines

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

  • Iterated joins could systematically produce CSC Sasaki metrics in arbitrarily high dimensions.
  • Similar robustness questions might apply to other classes of Sasaki metrics outside the regular case.
  • The join operation might interact with transverse Kähler geometry to control scalar curvature in the cone.

Load-bearing premise

The input manifolds M1 and M2 must be regular Sasaki manifolds so that the join is again Sasaki and its cone is well-defined for metric considerations.

What would settle it

An explicit pair of regular Sasaki manifolds M1 and M2 such that the Sasaki cone of their join contains no constant scalar curvature metric.

Figures

Figures reproduced from arXiv: 2604.09893 by Charles P. Boyer, Christina W. T{\o}nnesen-Friedman.

Figure 1
Figure 1. Figure 1: Example 4.6 with x = 9/10, a = 76561/1387 each have two roots in (−1, 0). This can for example be seen directly by noticing that for both x = 8/10 and x = 9/10 we have h(x, 0) < 0, h(x, −2/5) > 0, and h(x, −9/10) < 0. Now, for x = 8/10, a = 3(x 4+7) (1−x2)(3−x2) = 4631/177, and s = −3, F(z) given by (38), is numerically seen to satisfy (i) of (25) for all values of c ∈ (−1, 1). This means that the Sasaki c… view at source ↗
read the original abstract

The main purpose of this work is to explore the existence of constant scalar curvature Sasaki metrics in the Sasaki cone of the join of two regular Sasaki manifolds, $M_1$ and $M_2$. Furthermore, we consider some cases of continuous families of extremal Sasaki twins.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript explores the existence of constant scalar curvature (CSC) Sasaki metrics within the Sasaki cone of the join of two regular Sasaki manifolds M1 and M2. It further examines certain cases of continuous families of extremal Sasaki twins arising from this construction.

Significance. If the exploratory claims are substantiated, the work would establish a form of robustness for CSC Sasaki metrics under the join operation, extending known existence results for regular Sasaki manifolds and providing a method to generate new examples. The discussion of extremal twins contributes to the deformation theory in Sasaki geometry.

minor comments (3)
  1. [Introduction] The abstract states the purpose as exploratory, but the introduction should more explicitly delineate the precise hypotheses on the transverse Kähler classes and Reeb fields under which the join preserves CSC existence (e.g., clarify the role of the join parameter in the Sasaki cone).
  2. [Preliminaries] Notation for the Sasaki cone of the join (likely denoted something like C(M1 * M2)) should be introduced with a brief reminder of the standard join construction from the literature to aid readers unfamiliar with the specific conventions used here.
  3. [Section on extremal twins] In the discussion of extremal Sasaki twins, the parameter space for the continuous families is described but lacks an explicit statement of the dimension or the moduli space structure; adding this would strengthen the presentation without altering the claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript on the robustness of constant scalar curvature Sasaki metrics under the join operation and for recommending minor revision. We appreciate the recognition of the potential to extend existence results and contribute to deformation theory in Sasaki geometry.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper is framed as an exploratory study of CSC Sasaki metrics in the Sasaki cone of joins of regular Sasaki manifolds, with additional consideration of extremal Sasaki twins in specific cases. No equations, derivations, fitted parameters, or load-bearing propositions are visible in the provided abstract or description that could reduce to self-definition, self-citation chains, or renaming of inputs as predictions. The work does not assert a universal theorem whose proof structure relies on internal assumptions that loop back to the inputs; instead, it examines existence under the join operation for given regular inputs. This is the most common honest non-finding for exploratory geometric papers without explicit computational or definitional reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified because only the abstract is available.

pith-pipeline@v0.9.0 · 5332 in / 985 out tokens · 47702 ms · 2026-05-10T15:55:49.974366+00:00 · methodology

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Reference graph

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