The Fefferman-Szeg\H{o} Sphericity Criterion in Complex Dimension Three

Venkata Siddharth Pendyala

arxiv: 2606.24907 · v1 · pith:GQENN64Unew · submitted 2026-06-16 · 🧮 math.CV

The Fefferman-SzegH{o} Sphericity Criterion in Complex Dimension Three

Venkata Siddharth Pendyala This is my paper

Pith reviewed 2026-06-26 21:30 UTC · model grok-4.3

classification 🧮 math.CV

keywords CR sphericityFefferman-Szegő metricChern-Moser curvaturestrongly pseudoconvex domainscomplex dimension threeMonge-Ampère determinantboundary expansion

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The pith

Vanishing of the second-order deviation from the ball model is equivalent to local sphericity in complex dimension three.

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

The paper establishes that for smoothly bounded strongly pseudoconvex domains in complex three-space, local CR sphericity is equivalent to the vanishing of the second-order coefficient in the boundary expansion of the normalized determinant of the Fefferman-Szegő metric. This coefficient is a universal multiple of the squared Chern-Moser curvature. A sympathetic reader would care because this supplies an analytic test for when a domain boundary is locally equivalent to the sphere, a basic structural question in CR geometry. The argument relies on deriving the expansion explicitly and invoking a logarithmic stability theorem for the associated Monge-Ampère determinant to handle the remainder term, thereby finishing the three-dimensional case.

Core claim

We establish a Fefferman-Szegő characterization of local CR sphericity for smoothly bounded strongly pseudoconvex domains in complex dimension three. We derive the boundary expansion of the normalized determinant of the Fefferman-Szegő metric and prove that its second-order coefficient is a universal multiple of the squared Chern-Moser curvature. Hence, vanishing of the second-order deviation from the ball model is equivalent to local sphericity. A logarithmic stability theorem for the associated Monge-Ampère determinant controls the remainder and completes the dimension-three case.

What carries the argument

The normalized determinant of the Fefferman-Szegő metric, whose second-order boundary-expansion coefficient is a universal multiple of the squared Chern-Moser curvature.

If this is right

If the second-order coefficient vanishes, the domain is locally CR spherical.
The result supplies a complete characterization in complex dimension three.
The logarithmic stability theorem for the Monge-Ampère determinant ensures the higher-order remainder does not interfere with the second-order test.

Where Pith is reading between the lines

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

If the expansion formula holds, one could numerically evaluate the determinant on sample boundaries to test for sphericity without computing the full curvature tensor.
The dimension-three restriction suggests that analogous expansions might exist in higher dimensions but would require separate stability arguments.
The criterion isolates a single scalar invariant whose vanishing forces the entire Chern-Moser tensor to vanish locally.

Load-bearing premise

The boundary expansion of the normalized determinant of the Fefferman-Szegő metric exists up to second order with the stated coefficient relation, together with the logarithmic stability theorem for the Monge-Ampère determinant that controls the remainder term.

What would settle it

Explicit computation of the second-order coefficient for a concrete non-spherical strongly pseudoconvex domain in C^3 (such as a small perturbation of the ball) and verification that the coefficient is nonzero and matches the squared Chern-Moser curvature.

read the original abstract

We establish a Fefferman-Szeg\H{o} characterization of local CR sphericity for smoothly bounded strongly pseudoconvex domains in complex dimension three. We derive the boundary expansion of the normalized determinant of the Fefferman-Szeg\H{o} metric and prove that its second-order coefficient is a universal multiple of the squared Chern-Moser curvature. Hence, vanishing of the second-order deviation from the ball model is equivalent to local sphericity. A logarithmic stability theorem for the associated Monge-Amp\`ere determinant controls the remainder and completes the dimension-three case.

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.

Desk Editor's Note private letter to a colleague

This paper completes the Fefferman-Szegő sphericity criterion in complex dimension three by tying the second-order term in the normalized determinant expansion directly to the squared Chern-Moser curvature.

read the letter

The paper derives the boundary expansion of the normalized Fefferman-Szegő determinant up to second order for smoothly bounded strongly pseudoconvex domains in dimension three and shows that its coefficient is a universal multiple of the squared Chern-Moser curvature. Vanishing of that term is then equivalent to local sphericity, with a logarithmic stability result for the Monge-Ampère determinant controlling the remainder. This finishes the three-dimensional case once the lower-dimensional results are in place.

The explicit coefficient relation is the concrete new piece. The argument follows the usual pattern in this area—expand the metric quantity near the boundary and match it to an existing invariant—so the logic is straightforward if the calculation holds. The stability theorem is presented as the step that turns the expansion into a full equivalence.

The soft spot is that the abstract gives no sample terms from the expansion or the numerical value of the multiple, so the central computation cannot be spot-checked from the summary alone. In CR geometry these expansions are often coordinate-heavy, and a reader would want to see at least the leading non-trivial term to gauge the work. The scope is also narrow by design: only smooth strongly pseudoconvex boundaries in dimension three.

This is for people already working on Fefferman-Szegő metrics, Chern-Moser invariants, or low-dimensional CR characterizations. A specialist who needs the dimension-three case filled in would get direct value. The abstract is internally consistent and the claimed steps match standard techniques in the field, so the paper deserves a serious referee even if the proofs turn out to need tightening.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes a Fefferman-Szegő characterization of local CR sphericity for smoothly bounded strongly pseudoconvex domains in complex dimension three. It derives the boundary expansion of the normalized determinant of the Fefferman-Szegő metric and proves that its second-order coefficient is a universal multiple of the squared Chern-Moser curvature. Hence vanishing of the second-order deviation from the ball model is equivalent to local sphericity, with a logarithmic stability theorem for the associated Monge-Ampère determinant controlling the remainder term and completing the dimension-three case.

Significance. If the derivations of the second-order expansion and the stability theorem hold, the result supplies a new analytic criterion for local sphericity in CR geometry that directly ties the Fefferman-Szegő metric to the Chern-Moser curvature tensor. This strengthens the toolkit for studying invariants of strongly pseudoconvex domains in dimension three and may inform related questions on stability and rigidity.

minor comments (2)

[Abstract] Abstract: the phrase 'universal multiple' is used without stating the explicit constant; adding the numerical factor (or its derivation reference) would improve immediate readability.
The manuscript would benefit from a brief comparison paragraph situating the new criterion against existing Fefferman-Szegő or Chern-Moser characterizations in the literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No circularity: derivation is a direct expansion and coefficient identification

full rationale

The paper states it derives the second-order boundary expansion of the normalized Fefferman-Szegő determinant and identifies its coefficient as a universal multiple of the squared Chern-Moser curvature, then invokes a logarithmic stability result for the Monge-Ampère determinant to control remainders. These steps are presented as explicit computations and a stability theorem application for dimension three; no equation reduces to a prior fitted parameter, self-defined quantity, or load-bearing self-citation by construction. The central equivalence follows from the expansion identity rather than renaming or ansatz smuggling. The derivation chain is therefore self-contained against external CR geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no information on free parameters, background axioms, or new entities introduced in the proof.

pith-pipeline@v0.9.1-grok · 5622 in / 1015 out tokens · 29666 ms · 2026-06-26T21:30:52.745730+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 5 canonical work pages

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Pith/arXiv arXiv 2026
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Barrett, D.E., Lee, L.: On the Szegő metric. J. Geom. Anal.24, 104–117 (2014). https://doi.org/10.1007/s12220-012-9329-x

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[3]

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Acta Math

Chern, S.-S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974). https://doi.org/10.1007/BF02392146

work page doi:10.1007/bf02392146 1974
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Fefferman, C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math.26, 1–65 (1974). https://doi.org/10.1007/BF01406845

work page doi:10.1007/bf01406845 1974
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Fefferman, C.: Monge–Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. Math.103, 395–416 (1976)

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Fefferman, C.: Parabolic invariant theory in complex analysis. Adv. Math.31, 131–262 (1979). https://doi.org/10.1016/0001-8708(79)90025-2

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Hirachi, K.: Transformation law for the Szegő projectors on CR manifolds. Osaka J. Math.27, 301–308 (1990)

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Hirachi, K., Komatsu, G., Nakazawa, N.: Two methods of determining local invari- ants in the Szegő kernel. In: Komatsu, H. (ed.) Complex Geometry, Lecture Notes in Pure and Applied Mathematics, vol. 143, pp. 77–96. Dekker, New York (1993)

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Acta Math.148, 159–192 (1982)

Lee, J.M., Melrose, R.B.: Boundary behavior of the complex Monge–Ampère equation. Acta Math.148, 159–192 (1982). https://doi.org/10.1007/BF02392727 14

work page doi:10.1007/bf02392727 1982

[1] [1]

arXiv:2605.25455 (2026)

Bhatnagar, A., Fan, J.: The invariant Szegő metric on strongly pseudoconvex domains. arXiv:2605.25455 (2026)

Pith/arXiv arXiv 2026

[2] [2]

Barrett, D.E., Lee, L.: On the Szegő metric. J. Geom. Anal.24, 104–117 (2014). https://doi.org/10.1007/s12220-012-9329-x

work page doi:10.1007/s12220-012-9329-x 2014

[3] [3]

Astérisque34–35, 123–164 (1976)

Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegő. Astérisque34–35, 123–164 (1976)

1976

[4] [4]

Acta Math

Chern, S.-S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974). https://doi.org/10.1007/BF02392146

work page doi:10.1007/bf02392146 1974

[5] [5]

Fefferman, C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math.26, 1–65 (1974). https://doi.org/10.1007/BF01406845

work page doi:10.1007/bf01406845 1974

[6] [6]

Fefferman, C.: Monge–Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. Math.103, 395–416 (1976)

1976

[7] [7]

Fefferman, C.: Parabolic invariant theory in complex analysis. Adv. Math.31, 131–262 (1979). https://doi.org/10.1016/0001-8708(79)90025-2

work page doi:10.1016/0001-8708(79)90025-2 1979

[8] [8]

Math.64, 133–155 (1987)

Graham, C.R.: HigherasymptoticsofthecomplexMonge–Ampèreequation.Compos. Math.64, 133–155 (1987)

1987

[9] [9]

In: Berenstein, C.A

Graham, C.R.: Scalar boundary invariants and the Bergman kernel. In: Berenstein, C.A. (ed.) Complex Analysis II, Lecture Notes in Mathematics, vol. 1276, pp. 108–135. Springer, Berlin (1987)

1987

[10] [10]

Hirachi, K.: Transformation law for the Szegő projectors on CR manifolds. Osaka J. Math.27, 301–308 (1990)

1990

[11] [11]

In: Komatsu, H

Hirachi, K., Komatsu, G., Nakazawa, N.: Two methods of determining local invari- ants in the Szegő kernel. In: Komatsu, H. (ed.) Complex Geometry, Lecture Notes in Pure and Applied Mathematics, vol. 143, pp. 77–96. Dekker, New York (1993)

1993

[12] [12]

Acta Math.148, 159–192 (1982)

Lee, J.M., Melrose, R.B.: Boundary behavior of the complex Monge–Ampère equation. Acta Math.148, 159–192 (1982). https://doi.org/10.1007/BF02392727 14

work page doi:10.1007/bf02392727 1982