Analysis of the Geometric Heat Flow Equation: Computing Geodesics in Real-Time with Convergence Guarantees

Brett T. Lopez; Samuel G. Gessow

arxiv: 2510.11692 · v2 · submitted 2025-10-13 · 📡 eess.SY · cs.SY

Analysis of the Geometric Heat Flow Equation: Computing Geodesics in Real-Time with Convergence Guarantees

Samuel G. Gessow , Brett T. Lopez This is my paper

Pith reviewed 2026-05-18 07:21 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords geometric heat flowgeodesicsRiemannian manifoldsexponential stabilityL2 convergencepseudospectral methodChebyshev polynomialsmotion planning

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The pith

The geometric heat flow equation converges exponentially in L2 to geodesics on Riemannian manifolds whose sectional curvature is bounded above by a positive constant.

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

The paper analyzes the geometric heat flow equation, a parabolic PDE that evolves an initial curve into a geodesic on a Riemannian manifold. It proves exponential stability in the L2 norm when sectional curvature is bounded from above, while asymptotic L2 convergence to the geodesic holds without that restriction. A Chebyshev-polynomial pseudospectral discretization is introduced to solve the equation in milliseconds on non-Euclidean surfaces and to embed the result in a contraction-based nonlinear controller.

Core claim

The geometric heat flow equation is exponentially stable in L2 if the curvature of the Riemannian manifold does not exceed a positive bound and asymptotic convergence in L2 is always guaranteed. A pseudospectral method that leverages Chebyshev polynomials computes geodesics accurately in only a few milliseconds for non-contrived manifolds, with verification on common surfaces and inside a feedback controller using a non-flat metric.

What carries the argument

The geometric heat flow equation, the parabolic PDE whose solution evolves an arbitrary initial curve to a geodesic by curvature-adjusted diffusion on the manifold.

Load-bearing premise

The Riemannian manifold must have sectional curvature bounded above by some positive constant to secure the exponential L2 decay rate.

What would settle it

A numerical run on a manifold whose sectional curvature exceeds the stated positive bound in which the L2 error to the true geodesic fails to decay at an exponential rate.

Figures

Figures reproduced from arXiv: 2510.11692 by Brett T. Lopez, Samuel G. Gessow.

read the original abstract

We present an analysis on the convergence properties of the so-called geometric heat flow equation for computing geodesics (extremal curves) on Riemannian manifolds. Computing geodesics numerically in real time has become an important capability across several fields, including control and motion planning. The geometric heat flow equation involves solving a parabolic partial differential equation whose solution is a geodesic. In practice, solving this PDE numerically can be done efficiently, and tends to be more numerically stable and exhibit a better rate of convergence compared to numerical optimization. We prove that the geometric heat flow equation is exponentially stable in $L_2$ if the curvature of the Riemannian manifold does not exceed a positive bound and that asymptotic convergence in $L_2$ is always guaranteed. We also present a pseudospectral method that leverages Chebyshev polynomials to accurately compute geodesics in only a few milliseconds for non-contrived manifolds. Our analysis was verified with our custom pseudospectral method by computing geodesics on common non-Euclidean surfaces, and in feedback for a contraction-based controller with a non-flat metric for a nonlinear system.

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 proves L2 exponential stability for geometric heat flow under a curvature upper bound and pairs it with a fast Chebyshev pseudospectral solver, though the global rate may depend on unstated controls for curve length or parametrization.

read the letter

The main things to know are that this work establishes exponential L2 stability for the geometric heat flow equation when sectional curvature is bounded above, and it demonstrates a Chebyshev pseudospectral method that computes geodesics in milliseconds on non-flat manifolds. They also show the solver running inside a contraction-based controller for a nonlinear system with a non-Euclidean metric. That combination of analysis and real-time numerics is the practical hook for motion planning and geometric control work. The paper does well by grounding the numerics in concrete examples on standard surfaces and by keeping the implementation efficient enough for feedback loops. The pseudospectral choice avoids some of the stiffness issues that pure optimization methods can hit, and the verification steps look reproducible on the surfaces they tested. The soft spot sits in the stability claim. The stress-test note is on target: if the flow allows curve length to vary or the parametrization to drift, the integration-by-parts step that absorbs the curvature term into a uniform negative quadratic form can lose its constant. The abstract states both exponential decay under the bound and unconditional asymptotic convergence, so the proof likely separates those cases, but the exponential part needs an explicit a-priori length or arc-length control to hold globally. Without seeing that step closed, the bound feels sufficient only locally or on compact intervals. This paper is for people working at the overlap of Riemannian geometry, PDE methods, and nonlinear control. A reader who needs reliable, fast geodesic computation on manifolds with known curvature bounds will get usable material. It is not reshaping the field, but the specific stability result plus the solver is concrete enough to justify referee time. I would send it for peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript analyzes the geometric heat flow equation for computing geodesics on Riemannian manifolds. It proves exponential L2 stability when sectional curvature is bounded above by a positive constant and asymptotic L2 convergence in all cases. A pseudospectral method using Chebyshev polynomials is presented for real-time computation (milliseconds on non-contrived manifolds), with numerical verification on standard surfaces and integration into a contraction-based controller for a nonlinear system with non-flat metric.

Significance. If the stability analysis holds, the combination of rigorous L2 convergence guarantees and an efficient, reproducible pseudospectral implementation would be valuable for real-time geodesic computation in control and motion planning. The explicit numerical verification and control application are strengths that enhance practical relevance.

major comments (1)

[§3 (L2 stability analysis)] §3 (L2 stability analysis): The exponential decay claim invokes an upper bound on sectional curvature to absorb a term into a negative quadratic form in the energy estimate. However, the derivation does not appear to include an a-priori bound on the evolving curve length or a fixed arc-length reparametrization. Without such control, the integration-by-parts step may lose its uniform constant, restricting the result to local-in-time decay rather than global exponential L2 stability. Please supply the missing estimate or clarify the manifold compactness assumption used to close the argument.

minor comments (1)

[Numerical Results] The numerical examples would benefit from a table reporting L2 error norms and CPU times across multiple initial curves and manifolds to quantify the claimed millisecond convergence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the stability analysis. We address the major comment below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

Referee: §3 (L2 stability analysis): The exponential decay claim invokes an upper bound on sectional curvature to absorb a term into a negative quadratic form in the energy estimate. However, the derivation does not appear to include an a-priori bound on the evolving curve length or a fixed arc-length reparametrization. Without such control, the integration-by-parts step may lose its uniform constant, restricting the result to local-in-time decay rather than global exponential L2 stability. Please supply the missing estimate or clarify the manifold compactness assumption used to close the argument.

Authors: We agree that the current presentation of the energy estimate in §3 would benefit from an explicit a-priori bound on curve length to ensure the integration-by-parts constant remains uniform. The proof relies on the compactness of the manifold (implicit in the bounded-curvature hypothesis and the global existence of the flow), which guarantees that the evolving curve stays within a compact subset where length can be controlled by the initial energy via the dissipation identity. We will revise the manuscript to state the compactness assumption explicitly in the theorem, add the length bound derivation immediately before the integration-by-parts step, and confirm that this yields global (rather than local-in-time) exponential L2 stability under the stated curvature bound. revision: yes

Circularity Check

0 steps flagged

No circularity: direct PDE stability analysis

full rationale

The paper derives exponential L2 stability and asymptotic convergence for the geometric heat flow equation via standard energy estimates on the parabolic PDE, invoking a sectional curvature upper bound only to obtain the decay rate. No step reduces a claimed prediction or rate to a fitted parameter, self-definition, or load-bearing self-citation by construction; the abstract and proof outline treat the curvature condition as an external hypothesis on the manifold rather than an output of the derivation itself. The pseudospectral method is presented as a numerical implementation, not as the source of the analytic claims. This is a self-contained mathematical argument without the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only: relies on standard definitions of Riemannian manifolds and the geometric heat flow PDE; no explicit free parameters, invented entities, or ad-hoc axioms are stated.

axioms (1)

standard math Riemannian manifold with well-defined sectional curvature
Invoked throughout the stability statements and numerical examples.

pith-pipeline@v0.9.0 · 5727 in / 1198 out tokens · 26608 ms · 2026-05-18T07:21:27.023342+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

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

Theorem 1. The Jacobi heat flow equation (3) is exponentially stable in L2 if ... ⟨J, R(J, ∂sc)∂sc⟩g <4⟨J, J⟩.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

dE(c(τ))/dτ = −α ∫ ⟨D/ds ∂sc, D/ds ∂sc⟩g ds ... LaSalle’s invariance principle

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.
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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

15 extracted references · 15 canonical work pages

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M. P. Do Carmo and J. Flaherty Francis,Riemannian geometry. Springer, 1992, vol. 2

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K. Leung and I. R. Manchester, “Nonlinear stabilization via con- trol contraction metrics: A pseudospectral approach for computing geodesics,” in2017 American Control Conference (ACC). IEEE, 2017, pp. 1284–1289

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Control contraction metrics: Convex and intrinsic criteria for nonlinear feedback design,

I. R. Manchester and J.-J. E. Slotine, “Control contraction metrics: Convex and intrinsic criteria for nonlinear feedback design,”IEEE Trans. on Automatic Control, vol. 62, no. 6, pp. 3046–3053, 2017

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M. A. Belabbas and S. Liu, “New method for motion planning for non-holonomic systems using partial differential equations,” in2017 American Control Conference (ACC). IEEE, 2017, pp. 4189–4194

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Affine geometric heat flow and motion planning for dynamic systems,

S. Liu, Y . Fan, and M.-A. Belabbas, “Affine geometric heat flow and motion planning for dynamic systems,”IFAC-PapersOnLine, vol. 52, no. 16, pp. 168–173, 2019

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H. Tsukamoto, S.-J. Chung, and J.-J. E. Slotine, “Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview,”Annual Reviews in Control, vol. 52, pp. 135–169, 2021

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Krstic and A

M. Krstic and A. Smyshlyaev,Boundary control of PDEs: A course on backstepping designs. SIAM, 2008

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M. P. Do Carmo and J. Flaherty Francis,Riemannian geometry. Springer, 1992, vol. 2

work page 1992

[2] [2]

Nonlinear stabilization via con- trol contraction metrics: A pseudospectral approach for computing geodesics,

K. Leung and I. R. Manchester, “Nonlinear stabilization via con- trol contraction metrics: A pseudospectral approach for computing geodesics,” in2017 American Control Conference (ACC). IEEE, 2017, pp. 1284–1289

work page 2017

[3] [3]

Control contraction metrics: Convex and intrinsic criteria for nonlinear feedback design,

I. R. Manchester and J.-J. E. Slotine, “Control contraction metrics: Convex and intrinsic criteria for nonlinear feedback design,”IEEE Trans. on Automatic Control, vol. 62, no. 6, pp. 3046–3053, 2017

work page 2017

[4] [4]

Jost,Riemannian geometry and geometric analysis

J. Jost,Riemannian geometry and geometric analysis. Springer, 2005

work page 2005

[5] [5]

New method for motion planning for non-holonomic systems using partial differential equations,

M. A. Belabbas and S. Liu, “New method for motion planning for non-holonomic systems using partial differential equations,” in2017 American Control Conference (ACC). IEEE, 2017, pp. 4189–4194

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Affine geometric heat flow and motion planning for dynamic systems,

S. Liu, Y . Fan, and M.-A. Belabbas, “Affine geometric heat flow and motion planning for dynamic systems,”IFAC-PapersOnLine, vol. 52, no. 16, pp. 168–173, 2019

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Adaptive nonlinear control with contraction metrics,

B. T. Lopez and J.-J. E. Slotine, “Adaptive nonlinear control with contraction metrics,”IEEE Control Systems Letters, vol. 5, no. 1, pp. 205–210, 2020

work page 2020

[8] [8]

Robust feedback motion planning via contraction theory,

S. Singhet al., “Robust feedback motion planning via contraction theory,”The Int. Journal of Robotics Research, vol. 42, no. 9, pp. 655–688, 2023

work page 2023

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Continuous-time dynamic realization for nonlinear stabilization via control contraction metrics,

R. Wang and I. R. Manchester, “Continuous-time dynamic realization for nonlinear stabilization via control contraction metrics,” in2020 American Control Conference (ACC). IEEE, 2020, pp. 1619–1624

work page 2020

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Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview,

H. Tsukamoto, S.-J. Chung, and J.-J. E. Slotine, “Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview,”Annual Reviews in Control, vol. 52, pp. 135–169, 2021

work page 2021

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W. E. Schiesser,The numerical method of lines: integration of partial differential equations. Elsevier, 2012

work page 2012

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Bring the heat: Rapid trajectory optimization with pseudospectral techniques and the affine geometric heat flow equation,

C. E. Aduet al., “Bring the heat: Rapid trajectory optimization with pseudospectral techniques and the affine geometric heat flow equation,”IEEE Robotics and Automation Letters, 2025

work page 2025

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Krstic and A

M. Krstic and A. Smyshlyaev,Boundary control of PDEs: A course on backstepping designs. SIAM, 2008

work page 2008

[14] [14]

J. P. Boyd,Chebyshev and Fourier spectral methods. Courier Corporation, 2001

work page 2001

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Hardy,Inequalities

G. Hardy,Inequalities. Cambridge University Press, 1952

work page 1952