Geometric uncertainty principles for Schr\"odinger evolutions on negatively curved manifolds

Changxing Miao; Ruihan Zhou; Yilin Song

arxiv: 2605.17233 · v1 · pith:ZD3E6YGYnew · submitted 2026-05-17 · 🧮 math.AP · math.DG

Geometric uncertainty principles for Schr\"odinger evolutions on negatively curved manifolds

Changxing Miao , Yilin Song , Ruihan Zhou This is my paper

Pith reviewed 2026-05-19 23:22 UTC · model grok-4.3

classification 🧮 math.AP math.DG

keywords uncertainty principleSchrödinger equationCartan-Hadamard manifoldhyperbolic geometryCarleman estimateslogarithmic convexityrigiditydispersive equations

0 comments

The pith

Schrödinger solutions with Gaussian decay at two times are identically zero on asymptotic hyperbolic manifolds.

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

The paper establishes that the Hardy uncertainty principle carries over from Euclidean space to Cartan-Hadamard manifolds equipped with an asymptotic hyperbolic metric. A solution to the Schrödinger equation that decays sufficiently fast like a Gaussian at two distinct times must be the zero solution. This matters because it shows that quantitative uniqueness for dispersive equations persists even when translation invariance and Fourier analysis are unavailable. The proof adapts the Euclidean strategy but introduces a new weight function for Carleman estimates and a mollifier built from the exponential map and Jacobi fields to recover logarithmic convexity in the presence of exponential volume growth.

Core claim

For the Schrödinger equation with L^∞ bounded potentials on Cartan-Hadamard manifolds with asymptotic hyperbolic metric in dimensions n at least 2, sufficiently strong Gaussian decay of the solution at two different times implies that the solution vanishes identically. The result follows from new Carleman estimates with a geometry-adapted weight function together with logarithmic convexity obtained via virial identities and an approximation argument that employs a mollifier constructed using the exponential map and Jacobi fields.

What carries the argument

A mollifier defined on the manifold via the exponential map and Jacobi fields, which replaces the absent convolution structure and enables the derivation of logarithmic convexity.

If this is right

The same two-time Gaussian decay condition implies uniqueness for solutions of the Schrödinger equation on these manifolds.
Carleman estimates and logarithmic convexity can be established for Schrödinger evolutions despite exponential volume growth.
Curvature modifies propagation mechanisms yet preserves the rigidity phenomenon for dispersive equations.
Virial identities combined with manifold-specific approximations yield convexity properties adapted to hyperbolic geometry.

Where Pith is reading between the lines

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

The methods may extend to other manifolds with negative sectional curvature bounds.
Similar rigidity results could apply to inverse problems or control questions on hyperbolic spaces.
Explicit verification on the standard hyperbolic space H^n would test the sharpness of the Gaussian decay threshold.
The framework might inform uncertainty principles for related equations such as the wave equation.

Load-bearing premise

The manifold must be Cartan-Hadamard with an asymptotic hyperbolic metric so that a weight function and mollifier can be constructed from the exponential map and Jacobi fields.

What would settle it

Exhibiting a non-zero solution to the Schrödinger equation on one of these manifolds that decays like a Gaussian at two distinct times but does not vanish everywhere would disprove the claim.

read the original abstract

In this paper, we study the Hardy type uncertainty principle for Schr\"odinger equations with $L^\infty$ bounded potentials on certain Cartan-Hadamard manifolds endowed with an asymptotic hyperbolic metric in dimensions $n\geq2$. The classical Hardy uncertainty principle in Euclidean space, as developed in the works of Escauriaza-Kenig-Ponce-Vega (JEMS, 2008; Duke Math. J., 2010), reveals a rigidity phenomenon for solution $u$ to Schr\"odinger equations: sufficiently strong Gaussian decay at two distinct times yields $u\equiv0$. In this work, we show that a similar rigidity persists in the setting of hyperbolic geometry, despite the absence of translation invariance and Fourier representation. Our approach follows a general strategy of Escauriaza-Kenig-Ponce-Vega, where the underlying geometry brings an essential change. This enables us to establish new Carleman estimates and logarithmic convexity. Unlike the Euclidean setting, the hyperbolic geometry exhibits exponential volume growth and nontrivial geodesic escape at infinity, which fundamentally alters the propagation mechanism of Schr\"odinger evolutions. Based on the newly-built virial identities and an approximation argument, we derive the logarithmic convexity. The main difficulty in proving the logarithmic convexity is the lack of convolution structure on general manifolds. By making use of the exponential map and Jacobi field, we define a new mollifier on curved geometry. Meanwhile, to establish the Carleman estimate adapted to hyperbolic space, we introduce a new weight function adapted to the curved manifold.Our results highlight the role of curvature in shaping quantitative uniqueness properties for dispersive equations.

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 adapts the EKP V rigidity argument to Cartan-Hadamard manifolds via a new mollifier and weight, but the error control under exponential volume growth remains the open question.

read the letter

The main takeaway is that this work extends the Euclidean Hardy uncertainty principle for Schrödinger solutions to Cartan-Hadamard manifolds with asymptotic hyperbolic metric. They replace the missing convolution with a mollifier built from the exponential map and Jacobi fields, and they introduce a curvature-adjusted weight to get Carleman estimates and logarithmic convexity from virial identities. That geometric adaptation is the concrete novelty. The strategy follows the Euclidean template but accounts for exponential volume growth and geodesic escape at infinity, which is a reasonable way to handle the lack of translation invariance. The constructions look like honest attempts to carry the approximation argument over rather than a direct reduction. The soft spot sits in the mollification errors. Jacobi fields grow like sinh(r) or cosh(r) in this geometry, and the volume element adds another exponential factor. Those terms appear in the mollified virial identity, and it is not obvious from the abstract whether the Gaussian decay plus L^infty potential can absorb them at large distances. If the remainders are not controlled, the convexity inequality breaks and the rigidity conclusion does not follow. The abstract sketches the plan but supplies no explicit error bounds or comparison lemmas, so the central step is hard to verify without the full estimates. This paper is for people working on dispersive equations in curved geometries who already know the Euclidean EKP V results. A reader looking for how curvature changes quantitative uniqueness would get value from the new tools, even if the final claim needs verification. It is coherent enough on its own terms to deserve a serious referee rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. The paper establishes a Hardy-type uncertainty principle for Schrödinger equations with L^∞-bounded potentials on Cartan-Hadamard manifolds equipped with an asymptotic hyperbolic metric, in dimensions n≥2. It proves that sufficiently strong Gaussian decay of a solution at two distinct times implies the solution is identically zero. The argument adapts the Escauriaza-Kenig-Ponce-Vega strategy, replacing Euclidean convolution with a new mollifier constructed via the exponential map and Jacobi fields, and introducing a geometry-adapted weight function to obtain Carleman estimates and logarithmic convexity from virial identities.

Significance. If the central result holds, the work provides a meaningful extension of Euclidean rigidity phenomena to negatively curved manifolds, showing that quantitative uniqueness persists despite exponential volume growth and the lack of translation invariance or Fourier representation. The construction of a manifold-adapted mollifier and weight function constitutes a genuine technical contribution that could serve as a template for other dispersive problems on non-flat geometries. The paper explicitly credits the adaptation of virial identities plus approximation arguments for overcoming the absence of convolution structure.

major comments (1)

[§4] §4 (Logarithmic convexity via mollification): The error terms arising in the mollified virial identity from the exponential growth of Jacobi fields (solutions to the Jacobi equation behave like sinh(r) or cosh(r)) and the associated volume element must be shown to be absorbable by the assumed Gaussian decay and the L^∞ bound on the potential. The manuscript should supply explicit bounds demonstrating that these curvature-induced errors remain o(1) or controllable at large distances; without such estimates the convexity inequality fails to close and the rigidity conclusion does not follow.

minor comments (2)

[Introduction] The introduction could include a short comparison table or bullet list highlighting the precise differences between the Euclidean mollifier and the new geometric mollifier defined via the exponential map.
[§3] Notation for the weight function in the Carleman estimate should be introduced with an explicit formula immediately after its definition rather than deferred to an appendix.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The positive assessment of the technical contributions is appreciated. We address the single major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [§4] §4 (Logarithmic convexity via mollification): The error terms arising in the mollified virial identity from the exponential growth of Jacobi fields (solutions to the Jacobi equation behave like sinh(r) or cosh(r)) and the associated volume element must be shown to be absorbable by the assumed Gaussian decay and the L^∞ bound on the potential. The manuscript should supply explicit bounds demonstrating that these curvature-induced errors remain o(1) or controllable at large distances; without such estimates the convexity inequality fails to close and the rigidity conclusion does not follow.

Authors: We agree that making the control of curvature-induced error terms fully explicit will improve the clarity of the logarithmic convexity argument in §4. In the current draft the estimates are indicated via the Gaussian decay assumption and the construction of the mollifier through the exponential map, but we acknowledge they are not written out in sufficient detail. In the revised manuscript we will insert a new lemma (or expanded subsection) deriving explicit bounds. Specifically, we will show that the contributions from Jacobi field growth (∼ sinh(r), cosh(r)) and the volume distortion factor (∼ sinh^{n-1}(r)) produce error terms that are dominated by the assumed Gaussian decay e^{-α|x|^2} at both times. Because any exponential e^{C r} is absorbed by the quadratic exponential decay for large r, the integrated error is O(ε) where ε→0 as the mollification radius tends to zero, uniformly in the support of the solution. The L^∞ bound on the potential is used only to control the lower-order commutator terms, which remain bounded independently of curvature. These estimates close the convexity inequality exactly as in the Euclidean EKP V argument, yielding the desired rigidity. We will also add a short remark on the asymptotic hyperbolicity assumption ensuring the Jacobi fields behave at most exponentially at infinity. revision: yes

Circularity Check

0 steps flagged

No circularity: new geometric constructions yield independent Carleman estimates and convexity

full rationale

The derivation adapts the Escauriaza-Kenig-Ponce-Vega logarithmic-convexity strategy but replaces Euclidean convolution with a manifold-specific mollifier constructed explicitly from the exponential map and Jacobi fields, together with a curvature-adapted weight function. These objects are defined from the Cartan-Hadamard structure and asymptotic hyperbolic metric rather than presupposing the target rigidity statement; the resulting virial identities and error estimates are then shown to close under the assumed Gaussian decay and L^∞ potential. No step reduces by definition or by self-citation to the final uniqueness conclusion, and the cited Euclidean results supply only the overall proof template, not the geometric ingredients required on the manifold.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard properties of Cartan-Hadamard manifolds and hyperbolic geometry at infinity, with no free parameters or invented entities introduced beyond the new weight and mollifier constructions.

axioms (2)

domain assumption Cartan-Hadamard manifolds admit an exponential map with controlled Jacobi fields that can be used to define a mollifier.
Invoked to overcome the lack of convolution structure on general manifolds.
domain assumption The asymptotic hyperbolic metric allows construction of a weight function yielding Carleman estimates.
Central to adapting the Euclidean strategy to curved geometry.

pith-pipeline@v0.9.0 · 5826 in / 1306 out tokens · 25922 ms · 2026-05-19T23:22:15.188357+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.

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

By making use of the exponential map and Jacobi field, we define a new mollifier on curved geometry... ∂²ρ/∂t² = cothρ(|Ṗ|² − ρ_t²) + ⟨P̈,∇ρ⟩
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_strictMono_of_one_lt unclear

?

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

We obtain the lower bound for the commutator St + [S,A] ... Δ²g(ρ²) ≤ Cn + O(ρ^{-m-1})

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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

24 extracted references · 24 canonical work pages

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Escauriaza, C

L. Escauriaza, C. Kenig, G. Ponce and L. Vega, The sharp Hardy uncertainty principle for Schr¨ odinger evolutions, Duke Math. J.155(2010), no. 1, 163–187. 2 UNCERTAINTY PRINCIPLE AND UNIQUENESS 53

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B. Wilson and X. Yu, Global well-posedness and scattering for the defocusing mass-critical Schr¨ odinger equation in the three-dimensional hyperbolic space, arxiv: 2310.12277, to appear in Transaction of AMS. 3 Changxing Miao Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China. National Key Laboratory of Computational Physic...

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[1] [1]

Almgren, Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents, inMinimal submanifolds and geodesics (Proc

F. Almgren, Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents, inMinimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977), pp. 1-6, North-Holland, Amsterdam-New York. 5

work page 1977

[2] [2]

Anderson, Hardy’s uncertainty principle on hyperbolic spaces, Bull

N. Anderson, Hardy’s uncertainty principle on hyperbolic spaces, Bull. Aust. Math. Soc. 66 (2002), 163-

work page 2002

[3] [3]

Anderson,L p versions of Hardy type uncertainty principle on hyperbolic space, Proc

N. Anderson,L p versions of Hardy type uncertainty principle on hyperbolic space, Proc. Amer. Math. Soc. 131 (2003), 2797-2807. 3

work page 2003

[4] [4]

Anker and V

J.-P. Anker and V. Pierfelice, Nonlinear Schr¨ odinger equation on real hyperbolic spaces, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire26(2009), no. 5, 1853-1869. 3

work page 2009

[5] [5]

Banica, The nonlinear Schr¨ odinger equation on hyperbolic space, Comm

V. Banica, The nonlinear Schr¨ odinger equation on hyperbolic space, Comm. Partial Differential Equations 32(2007), no. 10-12, 1643-1677. 3

work page 2007

[6] [6]

Barcel´ o, L

J. Barcel´ o, L. Fanelli, S. Guiti´ errez, A. Ruiz and M. Vilela, Hardy uncertainty principle and unique continuation properties of covariant Schr¨ odinger flows, J. Funct. Anal.264(2013), no. 10, 2386-2415. 3

work page 2013

[7] [7]

Bertolin and E

F. Bertolin and E. Malinnikova, Dynamical versions of Hardy’s uncertainty principle: A survey, Bull. Amer. Math. Soc.58 (2021), no. 3, 357-375. 2

work page 2021

[8] [8]

Bourgain, On the compactness of the support of solutions of dispersive equations, Internat

J. Bourgain, On the compactness of the support of solutions of dispersive equations, Internat. Math. Res. Notices1997, no. 9, 437-447. 2

work page

[9] [9]

Cassano and L

B. Cassano and L. Fanelli, Sharp Hardy uncertainty principle and Gaussian profiles of covariant Schr¨ odinger evolutions, Trans. Amer. Math. Soc.367(2015), no. 3, 2213-2233. 3

work page 2015

[10] [10]

B. Chow, P. Lu and L. Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, 77, Amer. Math. Soc., Providence, RI, 2006 Sci. Press Beijing, New York, 2006. 45

work page 2006

[11] [11]

Cowling, L

M. Cowling, L. Escauriaza, C. Kenig, G. Ponce and L. Vega, The Hardy uncertainty principle revisited, Indiana Univ. Math. J.59(2010), no. 6, 2007–2025; MR2919746. 2

work page 2010

[12] [12]

Escauriaza, L

L. Escauriaza, L. Fanelli and L. Vega, Carleman estimates and necessary conditions for the existence of waveguides, Indiana Univ. Math. J.61(2012), no. 1, 15-30. 3

work page 2012

[13] [13]

Escauriaza, C

L. Escauriaza, C. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of Schr¨ odinger equations, Comm. Partial Differential Equations31(2006), no. 10-12, 1811-1823. 2, 4, 5, 7

work page 2006

[14] [14]

Escauriaza, C

L. Escauriaza, C. Kenig, G. Ponce and L. Vega, Hardy’s uncertainty principle, convexity and Schr¨ odinger evolutions, J. Eur. Math. Soc. (JEMS)10(2008), no. 4, 883-907. 2, 3, 5, 6, 18, 22, 27

work page 2008

[15] [15]

Escauriaza, C

L. Escauriaza, C. Kenig, G. Ponce and L. Vega, The sharp Hardy uncertainty principle for Schr¨ odinger evolutions, Duke Math. J.155(2010), no. 1, 163–187. 2 UNCERTAINTY PRINCIPLE AND UNIQUENESS 53

work page 2010

[16] [16]

Federico, Z

S. Federico, Z. Li and X. Yu, On the uniqueness of variable coefficient Schr¨ odinger equations, Commun. Contemp. Math.27(2025), no. 3, Paper No. 2450016, 45 pp. 3

work page 2025

[17] [17]

Hardy, A theorem concerning Fourier transform, J

G. Hardy, A theorem concerning Fourier transform, J. London Math. Soc.8(1933), no. 3, 227-231. 2

work page 1933

[18] [18]

Krist´ aly, Sharp uncertainty principles on Riemannian manifolds: the influence of curvature, J

C. Krist´ aly, Sharp uncertainty principles on Riemannian manifolds: the influence of curvature, J. Math. Pure Anal. 119 (2018), 326–346. 3

work page 2018

[19] [19]

Logunov, E

A. Logunov, E. Malinnikova, N. Nadirashvili and F. Nazarov, The Landis conjecture on exponential decay, Invent. Math. 241 (2025), no. 2, 465-508. 4

work page 2025

[20] [20]

Miao and R

C. Miao and R. Shen,Regularity and scattering of dispersive wave equations—multiplier method and Morawetz estimate, De Gruyter Studies in Mathematics, 100, De Gruyter, Berlin, (2025). 3

work page 2025

[21] [21]

C. Miao, B. Zhang and J. Zheng, Harmonic analysis methods in partial differential equations, De Gruyter Studies in Mathematics, 102, De Gruyter, Berlin, (2025). 3

work page 2025

[22] [22]

Petersen, Riemannian geometry

P. Petersen, Riemannian geometry. Second edition. Graduate Texts in Mathematics, 171. Springer, New York, 2006. xvi+401 pp. ISBN: 978-0387-29246-5. 45

work page 2006

[23] [23]

Shen and G

R. Shen and G. Staffilani, A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation onR 2, Trans. Amer. Math. Soc. 368 (2016), no. 4, 2809-2864. 3

work page 2016

[24] [24]

Wilson and X

B. Wilson and X. Yu, Global well-posedness and scattering for the defocusing mass-critical Schr¨ odinger equation in the three-dimensional hyperbolic space, arxiv: 2310.12277, to appear in Transaction of AMS. 3 Changxing Miao Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China. National Key Laboratory of Computational Physic...

work page arXiv