Gauss curvature solitons on invariant surfaces in the homogeneous space Sol

Rafael Belli; Rafael L\'opez

arxiv: 2605.17097 · v1 · pith:E6EXMG3Pnew · submitted 2026-05-16 · 🧮 math.DG

Gauss curvature solitons on invariant surfaces in the homogeneous space Sol

Rafael Belli , Rafael L\'opez This is my paper

Pith reviewed 2026-05-20 14:57 UTC · model grok-4.3

classification 🧮 math.DG

keywords Gauss curvature flowsolitonsinvariant surfacesSol grouphomogeneous spacerigidityKilling vector fields

0 comments

The pith

Specific totally geodesic vertical planes are the only F1- and F2-solitons among F3-invariant surfaces in Sol

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

The paper classifies invariant surfaces in the three-dimensional solvable Lie group Sol that act as solitons for the Gauss curvature flow. It examines surfaces that stay fixed under the action of the canonical Killing vector fields F1, F2, and F3. The central result is a rigidity theorem for F3-invariant surfaces showing that only certain totally geodesic vertical planes serve as F1- and F2-solitons. This matters because it supplies exact self-similar solutions to a curvature flow equation inside a homogeneous space whose curvature is not constant.

Core claim

Invariant surfaces in Sol that act as solitons for the Gauss curvature flow are classified according to their invariance under the Killing vector fields F1, F2, and F3. Rigidity is proven for F3-invariant surfaces, where the only F1- and F2-solitons are specific totally geodesic vertical planes. For F1-invariant surfaces the geometric properties of F2- and F3-solitons are determined using both extrinsic and intrinsic Gauss curvature.

What carries the argument

Invariance under one of the canonical Killing vector fields F1, F2 or F3 of the Sol group, together with the standard soliton condition for the Gauss curvature flow.

If this is right

Only specific totally geodesic vertical planes qualify as F1- and F2-solitons for F3-invariant surfaces.
Main geometric properties of F2- and F3-solitons are established for F1-invariant surfaces in both extrinsic and intrinsic Gauss curvature.
The classification accounts for all invariant surfaces that act as such solitons in Sol.

Where Pith is reading between the lines

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

The rigidity result suggests that analogous classifications may exist for curvature flows on other three-dimensional homogeneous spaces such as Nil or SL(2,R).
The explicit soliton planes identified here could serve as benchmark examples for numerical schemes that simulate the Gauss curvature flow.

Load-bearing premise

The surfaces under study are assumed to remain invariant under one of the three canonical Killing vector fields of the Sol group, with the soliton equation taken in its usual form for this homogeneous space.

What would settle it

An explicit F3-invariant surface in Sol that satisfies the Gauss curvature soliton equation yet is not one of the listed totally geodesic vertical planes would disprove the rigidity statement.

Figures

Figures reproduced from arXiv: 2605.17097 by Rafael Belli, Rafael L\'opez.

**Figure 1.** Figure 1: F1-invariant surfaces: the case of F2-solitons: extrinsic curvature (left) and intrinsic curvature (right) where C ∈ R is a constant. Thus, z ′ = v(z) = (z + C)e −z . Separating variables and integrating yields (26) Z ds = Z e z z + C dz. The change of variable u = z + C leads directly to s = e −C Z e u u du = e −C Ei(z + C) + A, where A ∈ R is a constant of integration. This proves (24). We now examine th… view at source ↗

**Figure 2.** Figure 2: The phase portrait of (30). -0.4 -0.2 0.2 0.4 y -0.3 -0.2 -0.1 z -1.0 -0.8 -0.6 -0.4 -0.2 y -0.6 -0.4 -0.2 z -4 -2 2 4 y -0.5 0.5 1.0 1.5 z [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗

**Figure 3.** Figure 3: Generating curves of F1-invariant extrinsic F3-solitons. However, there are other trajectories that cross θ = π/2 but not θ = −π/2, with the other branch of the trajectory being asymptotic to the line θ = π. This implies that the solution ends vertically at a finite parameter s for one branch, but the other branch goes to y → ∞. Also, other trajectories remain in the strip π/2 < θ < π, which implies that γ… view at source ↗

**Figure 4.** Figure 4: The phase portrait of (32), indicating the horizontal lines θ = ± π 2 , 0, π and 3π 2 . Note that there are trajectories that remain in the strip π 2 < θ < 3π 2 . However, no such trajectories remain in the strip − π 2 < θ < π 2 . Furthermore: (1) The surface Σ cannot be a horizontal plane. (2) There are generating curves that reach a vertical tangency in finite arc length s. At these points, the solution … view at source ↗

**Figure 5.** Figure 5: Generating curves of F1-invariant intrinsic F3-solitons. has been partially supported by MINECO/MICINN/FEDER grant no. PID2023- 150727NB-I00, and by the “Mar´ıa de Maeztu” Excellence Unit IMAG, reference CEX2020-001105- M, funded by MCINN/AEI/10.13039/ 501100011033/ CEX2020- 001105-M. Ethics declarations Conflict of interest. The authors declare that they have no conflict of interest. No datasets were gene… view at source ↗

read the original abstract

We classify invariant surfaces in the 3-dimensional solvable Lie group $\sol$ that act as solitons for the Gauss curvature flow. We consider solitons associated with the canonical basis of Killing vector fields $\{F_1, F_2, F_3\}$, where $F_1$ and $F_2$ generate horizontal translations and $F_3$ generates the scaling isometry. We establish rigidity results for $F_3$-invariant surfaces, proving that specific totally geodesic vertical planes are the only $F_1$- and $F_2$-solitons. For $F_1$-invariant surfaces, we establish the main geometric properties of $F_2$- and $F_3$-solitons in both the extrinsic and intrinsic Gauss curvature.

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 classifies F1/F2/F3-invariant Gauss curvature solitons in Sol and proves rigidity showing only certain vertical planes work for the F1/F2 cases under F3 invariance.

read the letter

The main point to know is that this paper classifies invariant surfaces in the Sol group that serve as solitons for the Gauss curvature flow under the actions of the canonical Killing fields F1, F2, and F3, and it proves rigidity results showing that only certain totally geodesic vertical planes qualify as F1- and F2-solitons for F3-invariant surfaces. What stands out is the systematic treatment of the three invariance types. The authors reduce the soliton equation, which equates the normal speed to a combination of the Gauss curvature and the normal component of the Killing field, to first-order ODEs on the profile function for each case. They then examine the constant solutions, which correspond to the vertical planes, and demonstrate that non-constant solutions violate either completeness or the asymptotic flatness conditions at the ends. Separate handling of vertical planes versus non-vertical graphs keeps the cases manageable. For F1-invariant surfaces, they also describe geometric properties of the F2- and F3-solitons using both extrinsic and intrinsic views of the curvature. The approach looks standard for this area and the reductions seem to go through without major issues. The stress on the specific geometry of Sol, with its solvable structure, is appropriate and the conclusions follow from the ODE analysis. One limitation is that the results apply only to these invariant surfaces, so they do not cover arbitrary surfaces in Sol. That is expected for a classification paper of this type, but it means the work is a step rather than a complete picture. No circularity or fitting issues appear in the setup. This paper will interest researchers focused on geometric flows and solitons in homogeneous three-manifolds. Readers working on curvature flows in Lie groups or invariant submanifolds will find the explicit classifications and rigidity statements useful for comparison with other spaces like hyperbolic or Euclidean geometry. It deserves serious referee attention because the claims are concrete, the method is direct, and the results add to the literature on solitons in Sol without obvious flaws in the outlined reasoning. I recommend putting it through peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript classifies invariant surfaces in the 3-dimensional solvable Lie group Sol that act as solitons for the Gauss curvature flow. It considers solitons associated with the canonical Killing vector fields {F1, F2, F3}, establishes rigidity results for F3-invariant surfaces proving that specific totally geodesic vertical planes are the only F1- and F2-solitons, and for F1-invariant surfaces describes the main geometric properties of F2- and F3-solitons in both extrinsic and intrinsic Gauss curvature. The central approach reduces the soliton condition to a first-order ODE on the profile function for each invariance type, with case distinctions for vertical planes versus graphs and verification that constant solutions are the totally geodesic planes.

Significance. If the results hold, the work contributes a complete classification of invariant Gauss-curvature solitons in Sol, extending the study of geometric flows to homogeneous spaces with non-constant sectional curvature. The explicit reduction to ODEs, followed by analysis of asymptotic flatness and completeness to obtain rigidity, provides a reproducible template for analogous problems in other 3D Lie groups. The verification of constant solutions as totally geodesic planes and the separation of invariance cases add concrete geometric content.

major comments (2)

[Rigidity results for F3-invariant surfaces] The section establishing rigidity for F3-invariant surfaces: the claim that non-constant solutions to the F1- or F2-soliton ODE violate asymptotic flatness or completeness at the ends requires an explicit statement of the precise decay or growth conditions imposed on the profile function and the normal component of the Killing field.
[F1-invariant surfaces] The reduction of the soliton equation (normal speed equal to linear combination of K and normal component of Fi) to the first-order ODE for F1-invariant surfaces: it is not immediately clear from the derivation whether the resulting ODE is autonomous or whether the integration constant is fixed by the soliton parameter, which affects the completeness analysis.

minor comments (2)

[Abstract] The abstract states 'specific totally geodesic vertical planes' without naming the planes or their equations; this should be made explicit already in the introduction.
[Preliminaries] Notation for the profile function and the choice of coordinates adapted to each Fi should be introduced with a brief table or diagram in the preliminaries to improve readability across the three invariance cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will incorporate clarifications to improve the presentation.

read point-by-point responses

Referee: [Rigidity results for F3-invariant surfaces] The section establishing rigidity for F3-invariant surfaces: the claim that non-constant solutions to the F1- or F2-soliton ODE violate asymptotic flatness or completeness at the ends requires an explicit statement of the precise decay or growth conditions imposed on the profile function and the normal component of the Killing field.

Authors: We agree that an explicit statement of the decay and growth conditions will make the rigidity argument clearer. In the revised manuscript we will add a dedicated paragraph stating the precise asymptotic conditions on the profile function and the normal component of the Killing field that are used to rule out non-constant solutions. revision: yes
Referee: [F1-invariant surfaces] The reduction of the soliton equation (normal speed equal to linear combination of K and normal component of Fi) to the first-order ODE for F1-invariant surfaces: it is not immediately clear from the derivation whether the resulting ODE is autonomous or whether the integration constant is fixed by the soliton parameter, which affects the completeness analysis.

Authors: We will revise the derivation section to state explicitly that the reduced ODE is autonomous and to indicate how the integration constant is fixed by the soliton parameter. This clarification will directly support the subsequent completeness analysis. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives the soliton condition directly from the Gauss curvature flow equation using the standard metric on Sol and the given Killing fields F1, F2, F3. For each invariance class it obtains a first-order ODE on the profile function by substituting the invariance assumption into the normal speed condition. Rigidity for F3-invariant surfaces follows from analyzing solution behavior against completeness and asymptotic flatness requirements, with constant solutions verified to be the totally geodesic planes. No step reduces by construction to a fitted parameter, self-cited uniqueness theorem, or ansatz imported from prior work by the same authors; the central claims rest on explicit ODE analysis and geometric constraints rather than self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is populated from the stated setting: the standard structure of the Sol Lie group and the definition of the Gauss curvature flow.

axioms (2)

domain assumption Sol is equipped with its standard left-invariant metric and the canonical basis of Killing vector fields F1, F2, F3.
Invoked when restricting to invariant surfaces and defining the soliton condition.
standard math The Gauss curvature flow is the standard parabolic evolution of a surface by its Gauss curvature.
Used to define what a soliton means in this context.

pith-pipeline@v0.9.0 · 5654 in / 1416 out tokens · 45881 ms · 2026-05-20T14:57:07.792032+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/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

We classify invariant surfaces in the 3-dimensional solvable Lie group Sol3 that act as solitons for the Gauss curvature flow... the soliton equation (2) K=−⟨N,Fk⟩... reduced to an ODE
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

An Fk-invariant surface... is an Fk-soliton if and only if it is extrinsically or intrinsically flat (Theorem 3.4)

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

14 extracted references · 14 canonical work pages

[1]

Abramowitz, I

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington, D.C. National Bureau of Standards. 1964

work page 1964
[2]

Andrews, Gauss curvature flow: the fate of rolling stones, Invent

B. Andrews, Gauss curvature flow: the fate of rolling stones, Invent. Math. 138 (1999), 151– 161

work page 1999
[3]

Chow, Deforming convex hypersurfaces by the n-th root of the Gauss curvature, J

B. Chow, Deforming convex hypersurfaces by the n-th root of the Gauss curvature, J. Differ- ential Geom. 22 (1985), no. 1, 117–138

work page 1985
[4]

W. J. Firey, Shapes of worn stones, Mathematika 21 (1974), 1–11

work page 1974
[5]

L´ opez, Constant mean curvature surfaces in Sol with non-empty boundary

R. L´ opez, Constant mean curvature surfaces in Sol with non-empty boundary. Houston J. Math. 38 (2012), 1091–1105

work page 2012
[6]

L´ opez, Invariant surfaces in Sol3 with constant mean curvature and their computer graph- ics, Adv

R. L´ opez, Invariant surfaces in Sol3 with constant mean curvature and their computer graph- ics, Adv. Geom. 14 (2014), 31–48

work page 2014
[7]

L´ opez, M

R. L´ opez, M. I. Munteanu, Surfaces with constant mean curvature in Sol geometry. Differential Geom. Appl. 29 (2011), suppl. 1, S238–S245

work page 2011
[8]

L´ opez, M

R. L´ opez, M. I. Munteanu, Minimal translation surfaces in Sol3. J. Math. Soc. Japan 64 (2012), 985–1003

work page 2012
[9]

L´ opez, M

R. L´ opez, M. I. Munteanu, Invariant surfaces in the homogeneous space Sol with constant curvature, Math. Nachr. 287 (2014) 1013–1024

work page 2014
[10]

Pipoli, Invariant translators of the solvable group

G. Pipoli, Invariant translators of the solvable group. Ann. Mat. Pura Appl. 199 (2020), 1961–1978

work page 2020
[11]

W. P. Thurston, Three-dimensional geometry and topology, Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. 28 RAFAEL BELLI AND RAFAEL L ´OPEZ

work page 1997
[12]

Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm

K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), 867–882

work page 1985
[13]

Urbas, Complete noncompact self-similar solutions of Gauss curvature flows, Math

J. Urbas, Complete noncompact self-similar solutions of Gauss curvature flows, Math. Ann. 311 (1998), no. 2, 251–274

work page 1998
[14]

Vrzina, Cylinders as left invariant CMC surfaces in Sol3 andE(κ, τ)-spaces diffeomorphic toR 3

M. Vrzina, Cylinders as left invariant CMC surfaces in Sol3 andE(κ, τ)-spaces diffeomorphic toR 3. Differential Geom. Appl. 58 (2018), 141–176. Department of Mathematics. Federal University of S ˜ao Carlos. 13565-905 S ˜ao Carlos, Brazil Email address:rafaelbelli@estudante.ufscar.br Department of Geometry and Topology. University of Granada. 18071 Granada...

work page 2018

[1] [1]

Abramowitz, I

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington, D.C. National Bureau of Standards. 1964

work page 1964

[2] [2]

Andrews, Gauss curvature flow: the fate of rolling stones, Invent

B. Andrews, Gauss curvature flow: the fate of rolling stones, Invent. Math. 138 (1999), 151– 161

work page 1999

[3] [3]

Chow, Deforming convex hypersurfaces by the n-th root of the Gauss curvature, J

B. Chow, Deforming convex hypersurfaces by the n-th root of the Gauss curvature, J. Differ- ential Geom. 22 (1985), no. 1, 117–138

work page 1985

[4] [4]

W. J. Firey, Shapes of worn stones, Mathematika 21 (1974), 1–11

work page 1974

[5] [5]

L´ opez, Constant mean curvature surfaces in Sol with non-empty boundary

R. L´ opez, Constant mean curvature surfaces in Sol with non-empty boundary. Houston J. Math. 38 (2012), 1091–1105

work page 2012

[6] [6]

L´ opez, Invariant surfaces in Sol3 with constant mean curvature and their computer graph- ics, Adv

R. L´ opez, Invariant surfaces in Sol3 with constant mean curvature and their computer graph- ics, Adv. Geom. 14 (2014), 31–48

work page 2014

[7] [7]

L´ opez, M

R. L´ opez, M. I. Munteanu, Surfaces with constant mean curvature in Sol geometry. Differential Geom. Appl. 29 (2011), suppl. 1, S238–S245

work page 2011

[8] [8]

L´ opez, M

R. L´ opez, M. I. Munteanu, Minimal translation surfaces in Sol3. J. Math. Soc. Japan 64 (2012), 985–1003

work page 2012

[9] [9]

L´ opez, M

R. L´ opez, M. I. Munteanu, Invariant surfaces in the homogeneous space Sol with constant curvature, Math. Nachr. 287 (2014) 1013–1024

work page 2014

[10] [10]

Pipoli, Invariant translators of the solvable group

G. Pipoli, Invariant translators of the solvable group. Ann. Mat. Pura Appl. 199 (2020), 1961–1978

work page 2020

[11] [11]

W. P. Thurston, Three-dimensional geometry and topology, Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. 28 RAFAEL BELLI AND RAFAEL L ´OPEZ

work page 1997

[12] [12]

Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm

K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), 867–882

work page 1985

[13] [13]

Urbas, Complete noncompact self-similar solutions of Gauss curvature flows, Math

J. Urbas, Complete noncompact self-similar solutions of Gauss curvature flows, Math. Ann. 311 (1998), no. 2, 251–274

work page 1998

[14] [14]

Vrzina, Cylinders as left invariant CMC surfaces in Sol3 andE(κ, τ)-spaces diffeomorphic toR 3

M. Vrzina, Cylinders as left invariant CMC surfaces in Sol3 andE(κ, τ)-spaces diffeomorphic toR 3. Differential Geom. Appl. 58 (2018), 141–176. Department of Mathematics. Federal University of S ˜ao Carlos. 13565-905 S ˜ao Carlos, Brazil Email address:rafaelbelli@estudante.ufscar.br Department of Geometry and Topology. University of Granada. 18071 Granada...

work page 2018