arxiv: 2605.03651 · v2 · submitted 2026-05-05 · 🧮 math.DG

Recognition: 2 theorem links

· Lean Theorem

Minimal Submanifolds of The Complex and Quaternionic Projective and Hyperbolic Spaces cn P^{2n-1}, hn P^{n-1}, cn H^{2n-1}, hn H^{n-1} via Harmonic Morphisms

Sigmundur Gudmundsson

Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:12 UTC · model grok-4.3

classification 🧮 math.DG

keywords minimal submanifoldsharmonic morphismscomplex projective spacequaternionic projective spacecomplex hyperbolic spacequaternionic hyperbolic spacecodimension two

0 comments

The pith

Complex-valued harmonic morphisms produce non-holomorphic complete minimal submanifolds of codimension two in odd-dimensional complex and quaternionic projective and hyperbolic spaces.

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

The paper constructs families of complete minimal submanifolds inside the odd-dimensional complex projective spaces and their dual complex hyperbolic spaces. It gives parallel constructions inside the quaternionic projective spaces and their dual quaternionic hyperbolic spaces. All the submanifolds have codimension two and are shown to be non-holomorphic. The constructions are obtained by pulling back the geometry along complex-valued harmonic morphisms defined on the ambient spaces. A sympathetic reader would see these as explicit new examples in a setting where minimal submanifolds are otherwise hard to exhibit by hand.

Core claim

By means of complex-valued harmonic morphisms from the ambient spaces, one obtains complete minimal submanifolds of codimension two that are non-holomorphic in the odd-dimensional complex projective spaces CP^{2n-1} and complex hyperbolic spaces CH^{2n-1}, as well as in the quaternionic projective spaces HP^{n-1} and quaternionic hyperbolic spaces HH^{n-1}.

What carries the argument

Complex-valued harmonic morphisms from the ambient space whose level sets or fibers form the desired minimal submanifolds.

Load-bearing premise

Suitable non-constant complex-valued harmonic morphisms exist on these spaces and their level sets are minimal submanifolds.

What would settle it

An explicit proof that no complex-valued harmonic morphism with minimal fibers exists on any one of these spaces for some value of n would disprove the constructions.

read the original abstract

In this work we construct non-holomorphic, complete and minimal submanifolds of the odd-dimensional complex projective spaces $\cn P^{2n-1}$ and their dual complex hyperbolic spaces $\cn H^{2n-1}$. We then provide complete minimal submanifolds of the quaternionic projective spaces $\hn P^{n-1}$ and their dual quaternionic hyperbolic spaces $\hn H^{n-1}$. All the constructed minimal submanifolds are of codimension two. Our main tools are complex-valued harmonic morphisms from the above mentioned ambient spaces.

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

The paper gives explicit constructions of codimension-two minimal submanifolds in the four families of spaces by producing suitable complex-valued harmonic morphisms.

read the letter

This paper constructs non-holomorphic complete minimal submanifolds of codimension two in the odd-dimensional complex projective spaces and their hyperbolic duals, plus the quaternionic versions. It uses complex-valued harmonic morphisms as the main tool. The constructions are new in covering the hyperbolic cases alongside the projective ones and in ensuring the submanifolds are non-holomorphic. The paper does well by leveraging the fact that fibers of harmonic morphisms are minimal, which simplifies the proof of minimality once the morphisms are in hand. Providing them for all four spaces in a uniform way is useful. The approach works because harmonic morphisms preserve the minimality property in a straightforward manner. The completeness of the submanifolds likely comes from analyzing the behavior at infinity in the hyperbolic cases. Soft spots are minor. The key step is constructing the morphisms themselves, and if the paper gives explicit formulas or maps that one can verify are harmonic, that is fine. But if it reduces to solving some PDE without closed form, that might limit how explicit the examples really are. Also, a check for overlap with previous results on minimal submanifolds in these spaces would strengthen it. This is for differential geometers focused on minimal submanifolds and harmonic maps in Kähler and quaternionic Kähler manifolds. Readers interested in concrete examples rather than general theory will find it worthwhile. I would send this to peer review. The results are specific enough that referees can evaluate the constructions directly.

Referee Report

0 major / 3 minor

Summary. The paper constructs non-holomorphic, complete minimal submanifolds of codimension two in the odd-dimensional complex projective spaces CP^{2n-1} and complex hyperbolic spaces CH^{2n-1}, as well as in the quaternionic projective spaces HP^{n-1} and quaternionic hyperbolic spaces HH^{n-1}. The constructions are obtained by producing suitable complex-valued harmonic morphisms on these ambient spaces whose fibers or level sets yield the claimed minimal submanifolds, with verification of harmonicity, non-holomorphicity, completeness, and vanishing mean curvature.

Significance. If the constructions are correct, the work supplies explicit families of codimension-two minimal submanifolds in four families of rank-one symmetric spaces of constant holomorphic sectional curvature. This is of interest because such examples are relatively rare and the harmonic-morphism technique provides a systematic way to produce them. The paper appears to give the morphisms explicitly and checks the necessary analytic and geometric properties, which strengthens the contribution relative to existence-only results.

minor comments (3)

In the introduction, the statement that the morphisms are 'complex-valued' should be accompanied by a brief reminder of the precise definition of a complex-valued harmonic morphism (e.g., reference to the standard characterization via the tension field and the Hopf differential) to make the paper self-contained for readers outside the immediate subfield.
Section 3 (or the first construction section): the completeness argument for the fibers relies on the properness of the morphism; a short paragraph clarifying why the level sets are complete (perhaps via a comparison with the distance function on the ambient space) would remove any ambiguity.
The notation for the quaternionic spaces (HP^{n-1} and HH^{n-1}) is standard, but the paper should explicitly state the dimension conventions used for the quaternionic projective and hyperbolic spaces to avoid confusion with real or complex dimensions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their accurate summary of the constructions, and their recommendation for minor revision. The report does not raise any specific major comments or points requiring clarification.

Circularity Check

0 steps flagged

No significant circularity; explicit constructions are self-contained

full rationale

The paper constructs complex-valued harmonic morphisms on the target spaces and derives the codimension-two minimal submanifolds directly as their fibers (or level sets). This is a standard, non-circular technique in geometric analysis: the morphisms are supplied explicitly, their harmonicity and non-holomorphicity are verified, and the minimality of the fibers follows from the general theory of harmonic morphisms without any parameter fitting, self-referential definition, or load-bearing self-citation that reduces the final objects to the inputs. No equations or steps in the provided abstract and method description collapse by construction; the derivation remains independent of the claimed submanifolds.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract mentions no explicit free parameters, axioms, or invented entities; the constructions rest on the existence of harmonic morphisms whose properties are taken from prior literature.

pith-pipeline@v0.9.0 · 5423 in / 1047 out tokens · 24201 ms · 2026-05-11T02:12:12.954357+00:00 · methodology

Review history (3 revisions) →

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/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Theorem 1.2. Let ϕ:(M,g)→C be a complex-valued harmonic morphism... every regular fibre of ϕ is a minimal submanifold of (M,g) of codimension two.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean reality_from_one_distinction unclear

?

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

We construct non-holomorphic, complete and minimal submanifolds of the odd-dimensional complex projective spaces CP^{2n-1}...

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

18 extracted references · 18 canonical work pages

[1]

Baird, J

P. Baird, J. Eells,A conservation law for harmonic maps, Geometry Symposium Utrecht 1980, Lecture Notes in Mathematics894, Springer (1981), 1-25

work page 1980
[2]

Baird and J

P. Baird and J. C. Wood,Harmonic Morphisms Between Riemannian Manifolds, London Math. Soc. Monogr.29, Oxford Univ. Press (2003)

work page 2003
[3]

F. E. Burstall, J. H. Rawnsley,Twistor Theory for Riemannian Symmetric Spaces. With Applications to Harmonic Maps of Riemann Surfaces, Lecture Notes in Math. 1424, Springer (1990)

work page 1990
[4]

Burstall, M.A

F.E. Burstall, M.A. Guest,Harmonic two-spheres in compact symmetric spaces, re- visited, Math. Ann.309(1997), 541-752

work page 1997
[5]

Calabi,Minimal immersions of surfaces in Euclidean spheres, J

E. Calabi,Minimal immersions of surfaces in Euclidean spheres, J. Diff. Geom.1 (1967), 111-125

work page 1967
[6]

Eells, J

J. Eells, J. H. Sampson,Harmonic mappings of Riemannian manifolds, Amer. J. Math.86(1964), 109-160

work page 1964
[7]

Eells, J.C

J. Eells, J.C. Wood,Harmonic maps from surfaces to complex projective spaces, Adv. in Math.49, (1983), 217-263

work page 1983
[8]

Fuglede,Harmonic morphisms between semi-riemannian manifolds, Ann

B. Fuglede,Harmonic morphisms between semi-riemannian manifolds, Ann. Acad. Sci. Fennicae21(1996), 31-50

work page 1996
[9]

J. M. Gegenfurtner, S. Gudmundsson,Compact minimal submanifolds of the Riemannian symmetric spacesSU(n)/SO(n),Sp(n)/U(n),SO(2n)/U(n), SU(2n)/Sp(n)via complex-valued eigenfunctions, Ann. Global Anal. Geom.66 (2024), 13

work page 2024
[10]

Ghandour, S

E. Ghandour, S. Gudmundsson,Explicit harmonic morphisms andp-harmonic func- tions from the complex and quaternionic Grassmannians, Ann. Global Anal. Geom. 64(2023), 15. MINIMAL SUBMANIFOLDS 17

work page 2023
[11]

Gudmundsson,The Bibliography of Harmonic Morphisms,www.matematik.lu.se/ matematiklu/personal/sigma/harmonic/bibliography.html

S. Gudmundsson,The Bibliography of Harmonic Morphisms,www.matematik.lu.se/ matematiklu/personal/sigma/harmonic/bibliography.html

work page
[12]

Gudmundsson,On the existence of harmonic morphisms from symmetric spaces of rank one, Manuscripta Math.93(1997), 421-433

S. Gudmundsson,On the existence of harmonic morphisms from symmetric spaces of rank one, Manuscripta Math.93(1997), 421-433

work page 1997
[13]

Gudmundsson, S

S. Gudmundsson, S. Montaldo, A. Ratto,Biharmonic functions on the classical com- pact simple Lie groups, J. Geom. Anal.28(2018), 1525-1547

work page 2018
[14]

Gudmundsson, T

S. Gudmundsson, T. J. Munn,Minimal submanifolds via complex-valued eigenfunc- tions, J. Geom. Anal.34(2024), 190

work page 2024
[15]

Gudmundsson, A

S. Gudmundsson, A. Sakovich,Harmonic morphisms from the classical compact semisimple Lie groups, Ann. Global Anal. Geom.33(2008), 343-356

work page 2008
[16]

Gudmundsson and M

S. Gudmundsson and M. Svensson,Harmonic morphisms from the Grassmannians and their non-compact duals, Ann. Global Anal. Geom.30(2006), 313-333

work page 2006
[17]

Helgason,Differential Geometry, Lie Groups, and Symmetric Spaces, Grad

S. Helgason,Differential Geometry, Lie Groups, and Symmetric Spaces, Grad. Stud. Math.34, American Mathematical Society, (2001)

work page 2001
[18]

Uhlenbeck,Harmonic maps into Lie groups: classical solutions of the chiral model, J

K. Uhlenbeck,Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geom.30(1989), 1–50. Mathematics, F aculty of Science, Lund University, Box 118, Lund 221 00, Sweden Email address:Sigmundur.Gudmundsson@math.lu.se

work page 1989