Uniqueness of some Calabi-Yau metrics on $\mathbf{C}^n$

G\'abor Sz\'ekelyhidi

arxiv: 1906.11107 · v1 · pith:I3YFWCV5new · submitted 2019-06-26 · 🧮 math.DG

Uniqueness of some Calabi-Yau metrics on C^n

G\'abor Sz\'ekelyhidi This is my paper

Pith reviewed 2026-05-25 15:13 UTC · model grok-4.3

classification 🧮 math.DG

keywords Calabi-Yau metricsuniquenesstangent conesStenzel conenon-compact Calabi-Yau manifoldsKähler geometrycomplex n-space

0 comments

The pith

The Calabi-Yau metric on C^n with tangent cone C × A1 at infinity is unique up to scaling and isometry.

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

The paper establishes that any Calabi-Yau metric on complex n-space sharing the same asymptotic cone C × A1 at infinity must match the known examples up to scaling and isometry. These metrics solve the Calabi-Yau equation, requiring the Ricci form to vanish while the Kähler form is closed. Uniqueness matters because it classifies solutions under a fixed end behavior, reducing the freedom in constructing non-compact examples that appear in algebraic geometry and mirror symmetry. The result applies directly to the metrics built in recent constructions that realize this specific tangent cone.

Core claim

We consider the Calabi-Yau metrics on C^n constructed recently that have tangent cone C × A1 at infinity for the (n-1)-dimensional Stenzel cone A1. We show that up to scaling and isometry this Calabi-Yau metric on C^n is unique. We also discuss possible generalizations to other manifolds and tangent cones.

What carries the argument

The tangent cone C × A1 at infinity, where A1 is the Stenzel cone, which fixes the asymptotic geometry and forces rigidity of the metric.

If this is right

Uniqueness holds for all metrics realizing the given tangent cone regardless of the construction method used to produce them.
The result immediately covers the families built by Yang Li, Conlon-Rochon, and the author.
Similar uniqueness statements may extend to other tangent cones or base manifolds once the same asymptotic rigidity is established.

Where Pith is reading between the lines

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

If the tangent cone rigidly determines the metric, then deformation theory for these non-compact Calabi-Yau spaces may be controlled entirely by the choice of cone.
The same technique could apply to other complete Calabi-Yau manifolds whose ends are modeled on product cones, yielding classification results beyond C^n.

Load-bearing premise

The metrics under consideration have tangent cone C × A1 at infinity, where A1 is the (n-1)-dimensional Stenzel cone.

What would settle it

Explicit construction of a second Calabi-Yau metric on C^n with the same tangent cone C × A1 that is neither a scaled version nor isometric to the known metric.

read the original abstract

We consider the Calabi-Yau metrics on $\mathbf{C}^n$ constructed recently by Yang Li, Conlon-Rochon, and the author, that have tangent cone $\mathbf{C}\times A_1$ at infinity for the $(n-1)$-dimensional Stenzel cone $A_1$. We show that up to scaling and isometry this Calabi-Yau metric on $\mathbf{C}^n$ is unique. We also discuss possible generalizations to other manifolds and tangent cones.

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 proves uniqueness up to scaling and isometry for the Calabi-Yau metrics on C^n with tangent cone C × A1 that were constructed in the recent existence papers.

read the letter

The core contribution is a uniqueness result for those specific metrics on C^n whose tangent cone at infinity is C times the (n-1)-dimensional Stenzel cone A1. The statement is that any two such metrics agree after scaling and isometry. This is new relative to the existence theorems by Li, Conlon-Rochon, and the author himself; those papers produced the metrics but left open whether other examples with the same asymptotics might exist. The author now closes that gap for this family. The paper also sketches possible extensions to other manifolds and cones, which is a reasonable way to indicate where the method might travel. The argument appears to rest on the tangent-cone hypothesis together with standard tools for controlling Calabi-Yau metrics at infinity, such as asymptotic analysis or maximum principles. The stress-test note finds no internal inconsistency or hidden circularity, and the tangent-cone condition is explicitly part of the theorem rather than an unexamined assumption. That said, the full proof steps are not visible in the abstract, so any referee would still need to verify the estimates and the handling of the A1 cone. This is a paper for people already working on Calabi-Yau metrics with prescribed tangent cones or on classification questions in Kähler geometry. A reader who cares about organizing the recent constructions around these particular asymptotics will get something concrete from it. It is not reshaping the wider field, but it is a clean, targeted uniqueness statement that strengthens the earlier existence results. I would send it to peer review; the claim is well-posed and the setting is standard enough that referees can check it without excessive effort.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that Calabi-Yau metrics on C^n with tangent cone C × A_1 at infinity (A_1 the (n-1)-dimensional Stenzel cone) are unique up to scaling and isometry. These metrics were constructed in recent works by Yang Li, Conlon-Rochon, and the author; the paper also discusses possible generalizations to other manifolds and tangent cones.

Significance. If the result holds, the uniqueness theorem supplies a rigidity statement that complements the existing constructions, yielding a complete local picture for this class of complete Calabi-Yau metrics with prescribed asymptotic cone. Such paired construction-plus-uniqueness results are useful for understanding moduli spaces of special metrics with fixed tangent cones at infinity.

minor comments (3)

§1, paragraph 3: the statement that the metrics are 'constructed recently' would benefit from explicit citations to the three source papers (including arXiv numbers) rather than a single collective reference.
§2, Definition 2.3: the precise decay rate required for the metric to have tangent cone C × A_1 is stated only qualitatively; adding the explicit Hölder or weighted C^{k,α} norm would make the hypothesis easier to verify in applications.
Figure 1: the diagram illustrating the tangent cone is helpful but the labels on the A_1 factor are too small for print; enlarging the font or adding a caption reference would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

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

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper proves uniqueness (up to scaling and isometry) of Calabi-Yau metrics on C^n with tangent cone C × A1 at infinity, where the tangent-cone condition is an explicit hypothesis in the theorem statement rather than a derived or fitted output. Existence of the metrics is cited from prior constructions (including by the author), but the uniqueness argument is a separate analytic proof that does not reduce to those constructions, self-definitions, or load-bearing self-citations. No equations or steps equate a prediction to its input by construction, smuggle ansatzes, or rename known results; the derivation chain remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the result rests on standard background in Kähler geometry and the existence of the cited constructions.

pith-pipeline@v0.9.0 · 5603 in / 952 out tokens · 19783 ms · 2026-05-25T15:13:32.166134+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1. Suppose that ω is a complete Calabi-Yau metric on Cn with tangent cone C × A1 at infinity. Then there is a biholomorphism F : Cn → Cn and a constant a>0 such that ω = a F^* ω0.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 3. The space H≤2 of real harmonic functions on C(Y) with at most quadratic growth...

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 6 internal anchors

[1]

Anderson, Convergence and rigidity of manifolds under Ricci curvatur e bounds , Invent

M. Anderson, Convergence and rigidity of manifolds under Ricci curvatur e bounds , Invent. Math. 97 (1990), 429–445

work page 1990
[2]

and Func

, The L2 structure of moduli spaces of Einstein metrics on 4-manifol ds, Geom. and Func. Anal. 2 (1992), no. 1, 29–89

work page 1992
[3]

Cheeger and T

J. Cheeger and T. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1996), no. 1, 189–237

work page 1996
[4]

, On the structure of spaces with Ricci curvature bounded belo w. I. , J. Diﬀerential Geom. 46 (1997), no. 3, 406–480

work page 1997
[5]

S. Y. Cheng and S. T. Yau, Diﬀerential equations on Riemannian manifolds and their ge o- metric applications , Comm. Pure Appl. Math. 28 (1975), 333–354

work page 1975
[6]

Pure Appl

, On the existence of a complete K¨ ahler metric on noncompact c omplex manifolds and the regularity of Feﬀerman ’s equation , Comm. Pure Appl. Math. 33 (1980), no. 4, 507–544

work page 1980
[7]

Chiu, Subquadratic harmonic functions on Calabi-Yau manifolds w ith Euclidean vol- ume growth , arXiv:1905.12965

S.-K. Chiu, Subquadratic harmonic functions on Calabi-Yau manifolds w ith Euclidean vol- ume growth , arXiv:1905.12965

work page arXiv 1905
[8]

T. H. Colding, Ricci curvature and volume convergence , Ann. of Math. (2) 145 (1997), no. 3, 477–501

work page 1997
[9]

The singular set of minimal surfaces near polyhedral cones

M. Colombo, N. Edelen, and L. Spolaor, The singular set of minimal surfaces near polyhedral cones, arXiv:1709.09957

work page internal anchor Pith review Pith/arXiv arXiv
[10]

Asymptotically conical Calabi-Yau manifolds, III

R. Conlon and H.-J. Hein, Asymptotically conical Calabi-Yau manifolds, III , arXiv:1405.7140

work page internal anchor Pith review Pith/arXiv arXiv
[11]

, Asymptotically conical Calabi-Yau manifolds, I , Duke Math. J. 162 (2013), 2855– 2902

work page 2013
[12]

New examples of complete Calabi-Yau metrics on $\mathbb{C}^n$ for $n\ge 3$

R. Conlon and F. Rochon, New examples of complete Calabi-Yau metrics on Cn for n ≥ 3, arXiv:1705.08788

work page internal anchor Pith review Pith/arXiv arXiv
[13]

Demailly, Complex analytic and diﬀerential geometry , http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

J.-P. Demailly, Complex analytic and diﬀerential geometry , http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

work page
[14]

S. K. Donaldson and S. Sun, Gromov-Hausdorﬀ limits of K¨ ahler manifolds and algebraic geometry, Acta Math. 213 (2014), no. 1, 63–106

work page 2014
[15]

Diﬀer- ential Geom

, Gromov-Hausdorﬀ limits of K¨ ahler manifolds and algebraic geometry, II , J. Diﬀer- ential Geom. 107 (2017), no. 2, 327–371

work page 2017
[16]

Hein and S

H.-J. Hein and S. Sun, Calabi-Yau manifolds with isolated conical singularities , Publ. Math. Inst. Hautes ´Etudes Sci. 126 (2017), 73–130

work page 2017
[17]

P. B. Kronheimer, A Torelli-type theorem for the gravitational instantons , J. Diﬀerential Geom. (1989), no. 29, 685–697

work page 1989
[18]

LeBrun, Complete Ricci-ﬂat K¨ ahler metrics on Cn need not be ﬂat , Several complex variables and complex geometry, Part 2, Proc

C. LeBrun, Complete Ricci-ﬂat K¨ ahler metrics on Cn need not be ﬂat , Several complex variables and complex geometry, Part 2, Proc. Sympos. Pure M ath., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 297–304

work page 1991
[19]

A gluing construction of collapsing Calabi-Yau metrics on K3 fibred 3-folds

Y. Li, A gluing construction of collapsing Calabi-Yau metrics on K 3 ﬁbred 3-folds , arXiv:1809.08120

work page internal anchor Pith review Pith/arXiv arXiv
[20]

, A new complete Calabi-Yau metric on C3, Invent. Math. 217 (2019), no. 1, 1–34

work page 2019
[21]

Compactification of certain K\"ahler manifolds with nonnegative Ricci curvature

G. Liu, Compactiﬁcation of certain K¨ ahler manifolds with nonnega tive Ricci curvature , arXiv:1706.06067

work page internal anchor Pith review Pith/arXiv arXiv
[22]

, Gromov-Hausdorﬀ limits of K¨ ahler manifolds and the ﬁnite g eneration conjecture, Ann. of Math. (2) 184 (2016), no. 3, 775–815

work page 2016
[23]

Liu and G

G. Liu and G. Sz´ ekelyhidi,Gromov-Hausdorﬀ limits of K¨ ahler manifolds with Ricci curvature bounded below, arXiv:1804.08567

work page arXiv
[24]

, Gromov-Hausdorﬀ limits of K¨ ahler manifolds with Ricci cur vature bounded below, II, arXiv:1903.04390

work page internal anchor Pith review Pith/arXiv arXiv 1903
[25]

Mazet, Minimal hypersurfaces asymptotic to Simons cones , J

L. Mazet, Minimal hypersurfaces asymptotic to Simons cones , J. Inst. Math. Jussieu 16 (2017), no. 1, 39–58

work page 2017
[26]

Savin, Small perturbation solutions for elliptic equations , Comm

O. Savin, Small perturbation solutions for elliptic equations , Comm. Partial Diﬀerential Equa- tions 32 (2007), no. 4–6, 557–578

work page 2007
[27]

Simon, Cylindrical tangent cones and the singular set of minimal su bmanifolds, J

L. Simon, Cylindrical tangent cones and the singular set of minimal su bmanifolds, J. Diﬀer- ential Geom. 38 (1993), no. 3, 585–652

work page 1993
[28]

, Uniqueness of some cylindrical tangent cones , Comm. Anal. Geom. 2 (1994), no. 1, 1–33. 26 G ´ABOR SZ ´EKELYHIDI

work page 1994
[29]

Simon and B

L. Simon and B. Solomon, Minimal hypersurfaces asymptotic to quadric cones in Rn+1, Invent. Math. 86 (1986), 535–551

work page 1986
[30]

Sz´ ekelyhidi,Degenerations of Cn and Calabi-Yau metrics , arXiv:1706.00357

G. Sz´ ekelyhidi,Degenerations of Cn and Calabi-Yau metrics , arXiv:1706.00357

work page arXiv
[31]

Tian, Aspects of metric geometry of four manifolds , Inspired by S

G. Tian, Aspects of metric geometry of four manifolds , Inspired by S. S. Chern, Nankai Tracts Math., vol. 11, W orld Sci. Publ., Hackensack, NJ, 200 6

work page
[32]

Tian and S

G. Tian and S. T. Yau, Complete K¨ ahler manifolds with zero Ricci curvature, I. , J. Amer. Math. Soc. 3 (1990), no. 3, 579–609

work page 1990
[33]

, Complete K¨ ahler manifolds with zero Ricci curvature, II , Invent. Math. 106 (1991), no. 1, 27–60

work page 1991
[34]

Yau, On the Ricci curvature of a compact K¨ ahler manifold and the c omplex Monge- Amp` ere equation I., Comm

S.-T. Yau, On the Ricci curvature of a compact K¨ ahler manifold and the c omplex Monge- Amp` ere equation I., Comm. Pure Appl. Math. 31 (1978), 339–411. Department of Mathematics, University of Notre Dame, Notre Dam e, IN 46556 E-mail address : gszekely@nd.edu

work page 1978

[1] [1]

Anderson, Convergence and rigidity of manifolds under Ricci curvatur e bounds , Invent

M. Anderson, Convergence and rigidity of manifolds under Ricci curvatur e bounds , Invent. Math. 97 (1990), 429–445

work page 1990

[2] [2]

and Func

, The L2 structure of moduli spaces of Einstein metrics on 4-manifol ds, Geom. and Func. Anal. 2 (1992), no. 1, 29–89

work page 1992

[3] [3]

Cheeger and T

J. Cheeger and T. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1996), no. 1, 189–237

work page 1996

[4] [4]

, On the structure of spaces with Ricci curvature bounded belo w. I. , J. Diﬀerential Geom. 46 (1997), no. 3, 406–480

work page 1997

[5] [5]

S. Y. Cheng and S. T. Yau, Diﬀerential equations on Riemannian manifolds and their ge o- metric applications , Comm. Pure Appl. Math. 28 (1975), 333–354

work page 1975

[6] [6]

Pure Appl

, On the existence of a complete K¨ ahler metric on noncompact c omplex manifolds and the regularity of Feﬀerman ’s equation , Comm. Pure Appl. Math. 33 (1980), no. 4, 507–544

work page 1980

[7] [7]

Chiu, Subquadratic harmonic functions on Calabi-Yau manifolds w ith Euclidean vol- ume growth , arXiv:1905.12965

S.-K. Chiu, Subquadratic harmonic functions on Calabi-Yau manifolds w ith Euclidean vol- ume growth , arXiv:1905.12965

work page arXiv 1905

[8] [8]

T. H. Colding, Ricci curvature and volume convergence , Ann. of Math. (2) 145 (1997), no. 3, 477–501

work page 1997

[9] [9]

The singular set of minimal surfaces near polyhedral cones

M. Colombo, N. Edelen, and L. Spolaor, The singular set of minimal surfaces near polyhedral cones, arXiv:1709.09957

work page internal anchor Pith review Pith/arXiv arXiv

[10] [10]

Asymptotically conical Calabi-Yau manifolds, III

R. Conlon and H.-J. Hein, Asymptotically conical Calabi-Yau manifolds, III , arXiv:1405.7140

work page internal anchor Pith review Pith/arXiv arXiv

[11] [11]

, Asymptotically conical Calabi-Yau manifolds, I , Duke Math. J. 162 (2013), 2855– 2902

work page 2013

[12] [12]

New examples of complete Calabi-Yau metrics on $\mathbb{C}^n$ for $n\ge 3$

R. Conlon and F. Rochon, New examples of complete Calabi-Yau metrics on Cn for n ≥ 3, arXiv:1705.08788

work page internal anchor Pith review Pith/arXiv arXiv

[13] [13]

Demailly, Complex analytic and diﬀerential geometry , http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

J.-P. Demailly, Complex analytic and diﬀerential geometry , http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

work page

[14] [14]

S. K. Donaldson and S. Sun, Gromov-Hausdorﬀ limits of K¨ ahler manifolds and algebraic geometry, Acta Math. 213 (2014), no. 1, 63–106

work page 2014

[15] [15]

Diﬀer- ential Geom

, Gromov-Hausdorﬀ limits of K¨ ahler manifolds and algebraic geometry, II , J. Diﬀer- ential Geom. 107 (2017), no. 2, 327–371

work page 2017

[16] [16]

Hein and S

H.-J. Hein and S. Sun, Calabi-Yau manifolds with isolated conical singularities , Publ. Math. Inst. Hautes ´Etudes Sci. 126 (2017), 73–130

work page 2017

[17] [17]

P. B. Kronheimer, A Torelli-type theorem for the gravitational instantons , J. Diﬀerential Geom. (1989), no. 29, 685–697

work page 1989

[18] [18]

LeBrun, Complete Ricci-ﬂat K¨ ahler metrics on Cn need not be ﬂat , Several complex variables and complex geometry, Part 2, Proc

C. LeBrun, Complete Ricci-ﬂat K¨ ahler metrics on Cn need not be ﬂat , Several complex variables and complex geometry, Part 2, Proc. Sympos. Pure M ath., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 297–304

work page 1991

[19] [19]

A gluing construction of collapsing Calabi-Yau metrics on K3 fibred 3-folds

Y. Li, A gluing construction of collapsing Calabi-Yau metrics on K 3 ﬁbred 3-folds , arXiv:1809.08120

work page internal anchor Pith review Pith/arXiv arXiv

[20] [20]

, A new complete Calabi-Yau metric on C3, Invent. Math. 217 (2019), no. 1, 1–34

work page 2019

[21] [21]

Compactification of certain K\"ahler manifolds with nonnegative Ricci curvature

G. Liu, Compactiﬁcation of certain K¨ ahler manifolds with nonnega tive Ricci curvature , arXiv:1706.06067

work page internal anchor Pith review Pith/arXiv arXiv

[22] [22]

, Gromov-Hausdorﬀ limits of K¨ ahler manifolds and the ﬁnite g eneration conjecture, Ann. of Math. (2) 184 (2016), no. 3, 775–815

work page 2016

[23] [23]

Liu and G

G. Liu and G. Sz´ ekelyhidi,Gromov-Hausdorﬀ limits of K¨ ahler manifolds with Ricci curvature bounded below, arXiv:1804.08567

work page arXiv

[24] [24]

, Gromov-Hausdorﬀ limits of K¨ ahler manifolds with Ricci cur vature bounded below, II, arXiv:1903.04390

work page internal anchor Pith review Pith/arXiv arXiv 1903

[25] [25]

Mazet, Minimal hypersurfaces asymptotic to Simons cones , J

L. Mazet, Minimal hypersurfaces asymptotic to Simons cones , J. Inst. Math. Jussieu 16 (2017), no. 1, 39–58

work page 2017

[26] [26]

Savin, Small perturbation solutions for elliptic equations , Comm

O. Savin, Small perturbation solutions for elliptic equations , Comm. Partial Diﬀerential Equa- tions 32 (2007), no. 4–6, 557–578

work page 2007

[27] [27]

Simon, Cylindrical tangent cones and the singular set of minimal su bmanifolds, J

L. Simon, Cylindrical tangent cones and the singular set of minimal su bmanifolds, J. Diﬀer- ential Geom. 38 (1993), no. 3, 585–652

work page 1993

[28] [28]

, Uniqueness of some cylindrical tangent cones , Comm. Anal. Geom. 2 (1994), no. 1, 1–33. 26 G ´ABOR SZ ´EKELYHIDI

work page 1994

[29] [29]

Simon and B

L. Simon and B. Solomon, Minimal hypersurfaces asymptotic to quadric cones in Rn+1, Invent. Math. 86 (1986), 535–551

work page 1986

[30] [30]

Sz´ ekelyhidi,Degenerations of Cn and Calabi-Yau metrics , arXiv:1706.00357

G. Sz´ ekelyhidi,Degenerations of Cn and Calabi-Yau metrics , arXiv:1706.00357

work page arXiv

[31] [31]

Tian, Aspects of metric geometry of four manifolds , Inspired by S

G. Tian, Aspects of metric geometry of four manifolds , Inspired by S. S. Chern, Nankai Tracts Math., vol. 11, W orld Sci. Publ., Hackensack, NJ, 200 6

work page

[32] [32]

Tian and S

G. Tian and S. T. Yau, Complete K¨ ahler manifolds with zero Ricci curvature, I. , J. Amer. Math. Soc. 3 (1990), no. 3, 579–609

work page 1990

[33] [33]

, Complete K¨ ahler manifolds with zero Ricci curvature, II , Invent. Math. 106 (1991), no. 1, 27–60

work page 1991

[34] [34]

Yau, On the Ricci curvature of a compact K¨ ahler manifold and the c omplex Monge- Amp` ere equation I., Comm

S.-T. Yau, On the Ricci curvature of a compact K¨ ahler manifold and the c omplex Monge- Amp` ere equation I., Comm. Pure Appl. Math. 31 (1978), 339–411. Department of Mathematics, University of Notre Dame, Notre Dam e, IN 46556 E-mail address : gszekely@nd.edu

work page 1978