Rigidity for the heat equation with density on Riemannian manifolds through a conformal change

Alberto Roncoroni; Alexander Grigor'yan; Giulia Meglioli

REVIEW 3 major objections 4 minor 40 references

Rigidity for the heat equation with density on Riemannian manifolds through a conformal change

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read On complete non-compact manifolds, a heat equation with density has only the zero solution in $L^p_{e^{-\psi}}$ exactly when a radial integral of $\psi$ and the density decay diverges; crossing the decay threshold yields infinitely many…

desk verdict Useful results and a clean framework, but the p>1 uniqueness theorems lean on an unproved weighted adaptation of Punzo's theorem, and there is a fixable monotonicity gap in the proof of Theorem 2.2. read the letter →

arxiv 2507.13230 v1 pith:2YWWZQXR submitted 2025-07-17 math.AP

Alexander Grigor'yan , Giulia Meglioli , Alberto Roncoroni This is my paper · ORCID

classification math.AP MSC 35A0235B5335J1058J05

keywords uniquenesstheoremsweightedLebesguespacesRiemannianmanifoldsheatequationswithdensityconformalchangeofmetricmodelsharpthresholdsCauchyproblem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

On a complete, non-compact weighted Riemannian manifold of infinite volume, the heat equation with density $\rho\partial_t u=\Delta u$ with zero initial data can still have nonzero solutions; this paper seeks the exact weight-class that rules them out. For densities with power-like decay $\rho(x)\asymp(1+r(x))^{-\theta}$, $0\le\theta<2$, and solutions in $L^p_{e^{-\psi}}$, uniqueness is proved to hold when $\int^\infty r^{1-\theta}/\psi(r)\,dr=+\infty$; the critical case $\theta=2$ is governed by $\int^\infty \log r/(r\psi(r))\,dr=+\infty$. The same pattern is extended to $p=1$ under additional geometric hypotheses (a pole, radial $\rho$, and a lower bound on $\Delta r$). The proof works through a conformal change of metric, $\tilde g=\rho g$, $d\tilde\mu=\rho\,d\mu$, in which the weighted Laplacian $\tilde\Delta$ satisfies $\tilde\Delta=\rho^{-1}\Delta$, so the problem becomes a standard heat equation on a weighted manifold. On model manifolds the paper constructs explicit supersolutions to show the threshold is sharp: once the decay of $\rho$ crosses the critical exponent, infinitely many bounded nonzero solutions appear.

What carries the argument

The machinery is the conformal change $\tilde g=\rho g$, $d\tilde\mu=\rho\,d\mu$, under which the weighted Laplacian obeys $\tilde\Delta=\frac1\rho\Delta$. This identity turns the density equation into $\partial_tu=\tilde\Delta u$ on a complete weighted manifold, and the ambient integral condition $u\in L^p_{e^{-\psi}}$ becomes a growth estimate for the weighted $L^p$-norm on geodesic balls in the new metric. The proof then invokes a general uniqueness criterion (Theorem 4.1, adapted from [34]): if $\int_0^T\int_{B_R}|u|^p\,d\mu\,dt\le e^{\varphi(R)}$ with $\int^\infty r/\varphi(r)\,dr=+\infty$, then $u\equiv0$. The paper chooses $\varphi$ out of $\psi$ and the radial bounds on $\rho$ so this criterion applies exactly when the displayed integral conditions on $\psi$ diverge.

What would settle it

On a model manifold with $f(r)=e^{r^\beta/(N-1)}$ and $\rho=(1+r^2)^{-\theta/2}$, the paper predicts no bounded nonzero solutions for $\theta<2-\beta$ and infinitely many for $\theta>2-\beta$; exhibiting one bounded nonzero solution, or one positive supersolution of $\Delta h=-\rho$ with $h\to0$, for $\theta<2-\beta$ would falsify the claimed sharp threshold.

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Extended reading notes

Core claim

The central claim is a rigidity dichotomy controlled by the decay rate of the density and the growth of the weight. In Theorem 2.2, if $\rho$ obeys $c_1(r+1)^{-\theta}\le\rho(x)\le c_2(r+1)^{-\theta}$ with $0\le\theta<2$, and $\psi$ is positive, increasing, continuous with $\int^\infty r^{1-\theta}/\psi(r)\,dr=+\infty$, then any classical solution of $\rho\partial_tu=\Delta u$ with zero initial data lying in $L^p_{e^{-\psi}}(M\times(0,T))$ is identically zero. Theorem 2.3 replaces the integrand by $\log r/(r\psi(r))$ at $\theta=2$. Theorem 2.4 is the general version for arbitrary two-sided radial bounds on $\rho$, with the integral condition (2.6) and the monotonicity condition (2.7). The $p=1$ analogues (Theorems 2.6--2.8) require a pole and radial density, and use a lower Laplacian bound to keep the distance function well behaved. Sharpness is shown on model manifolds: with $f(r)=e^{r^\beta/(N-1)}$, uniqueness of bounded solutions holds for $\theta<2-\beta$ and fails for $\theta>2-\beta$; with $f(r)=r^{\beta/(N-1)}$, uniqueness holds for $\theta\le2$ and fails for $\theta>2$.

Load-bearing premise

The load-bearing premise is that Theorem 4.1, stated in Section 4.1 and quoted from [34] with the remark that its proof adapts to weighted manifolds, is true exactly as stated; the paper does not prove this adaptation, and if it fails the $p>1$ uniqueness results collapse.

Editorial extensions

If this is right

For $p>1$ and power-like density decay with exponent $\theta\in[0,2)$, uniqueness in $L^p_{e^{-\psi}}$ holds whenever $\int^\infty r^{1-\theta}/\psi(r)\,dr=+\infty$.
At the borderline $\theta=2$, any weight with $\int^\infty \log r/(r\psi(r))\,dr=+\infty$ still forces $u\equiv0$, so slowly growing weights remain admissible at critical decay.
On model manifolds with $f(r)=e^{r^\beta/(N-1)}$, bounded solutions are unique when $\theta<2-\beta$ and infinitely many bounded solutions appear when $\theta>2-\beta$.
On model manifolds with $f(r)=r^{\beta/(N-1)}$, bounded solutions are unique for $0\le\theta\le2$ and non-unique for $\theta>2$.
In every non-uniqueness case, for each $\gamma>0$ there is a bounded solution whose time average over $(0,T)$ tends to $\gamma$ at infinity, so the failure of uniqueness is massive rather than isolated.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same conformal reduction should apply to drift or potential terms, replacing the density condition with analogous radial bounds on the lower-order coefficients; the paper does not pursue this extension.
The explicit supersolutions $h$ built in Lemmas 7.6 and 7.8 could be used as test functions in numerical or symbolic experiments on rotationally symmetric manifolds to check where the uniqueness threshold begins to fail.
The $p=1$ restrictions to manifolds with a pole and radial $\rho$ look technical rather than essential; if the quoted ball-growth lemma can be proved under weaker Laplacian bounds, the $L^1$ results should extend to more general complete manifolds.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my read. The paper does something genuinely useful: it gives L^p_{e^{-\psi}} uniqueness criteria for the heat equation with density, handles the critical exponent θ=2, and constructs matching counterexamples on model manifolds. The conformal change is a known technique, but applying it to turn ρ ∂_t u = Δu into a weighted heat equation is clean, and the p=1 part is proved in detail via Theorem 5.1 and its lemmas. The counterexample machinery in Section 6 is elementary but credible. Within the subfield this is a real step forward, not a rehash.

That said, there are two soft spots. The serious one is that every p>1 theorem reduces to Theorem 4.1, a uniqueness result for the weighted heat equation that is quoted from Punzo [34] with the sentence that the proof "can be adapted." This is not a routine citation. The weighted Laplacian has a drift term, and the energy estimate for ∫|u|^p η^p dμω contains an extra term of the form ∫ |u|^p (∇ω·∇η) η^{p−1} dμω, which is absent in the unweighted case. The authors never verify that this term is controlled for the full generality of Theorem 4.1. In the conformal applications the drift is well-behaved for θ≤2, so the theorem is probably true, but the paper as written omits that verification. A referee should ask for it.

The second soft spot is in the proof of Theorem 2.2. The function φ in (4.5) is claimed to be increasing because ψ and log are increasing, but log(c2(1+r)^{-θ}) is decreasing for θ>0. No analogue of assumption (2.7) is stated in Theorems 2.2–2.3, so the monotonicity needed to apply Theorem 4.1 is not actually established. This is likely repairable by adding (2.7) or using a monotone envelope, but as written the proof has a gap.

There is also a minor typo: the identity for Δ~ r~ has coefficient (N−1)/2 instead of 1/2. Because the hypotheses are written with the same extra factor and ρ′≤0, the argument still goes through with the corrected identity, so I do not think it is load-bearing.

The citation pattern is fine: self-citations are confined to the literature review, and Punzo is properly credited for the key tool. The math is not circular.

Bottom line: the paper deserves a serious referee, but the referee should require a full proof or a precise reference for the weighted Theorem 4.1, and a fix of the monotonicity gap, before acceptance.

Referee Report

3 major / 4 minor

Summary. The paper studies uniqueness, in weighted Lebesgue spaces $L^p_{e^{-\psi}}$, of classical solutions to the heat equation with density $\rho\partial_t u=\Delta u$ on complete non-compact Riemannian manifolds. After a conformal change $\tilde g=\rho g$, $d\tilde\mu=\rho d\mu$, the equation is rewritten as $\partial_t u=\tilde\Delta u$ with $\tilde\Delta=\rho^{-1}\Delta$. For $p>1$ the authors state uniqueness Theorems 2.2--2.4 under two-sided power-like bounds on $\rho$ and an integral divergence condition on $\psi$; for $p=1$ they state analogous Theorems 2.6--2.8 under additional pole, radial-density and Laplacian-comparison assumptions. The proof strategy is to verify a general weighted-manifold uniqueness criterion: Theorem 4.1 for $p>1$ and Theorem 5.1 for $p=1$. On model manifolds the authors show that crossing the decay threshold $\theta=2$ (or $\theta=2-\beta$ in the exponential-volume case) produces infinitely many bounded nonzero solutions, thereby demonstrating sharpness.

Significance. The conformal representation is elegant, and the sharp counterexamples in Section 7 are a useful contribution: they show that the integral conditions (2.2) and (2.4) are not redundant. The paper also makes good use of existing machinery (Punzo's uniqueness theorems, Laplacian comparison) and gives concrete model manifolds, including hyperbolic space, to which the theorems apply. However, the central $p>1$ theorems are conditional on an unproved weighted generalization of an unweighted theorem, and the proof of Theorem 2.2 contains a monotonicity error that affects the main claims. The manuscript needs further work before it can be accepted.

major comments (3)

[§4.1, Theorem 4.1] The theorem is stated for an arbitrary weighted manifold $(M,g,\mu_\omega)$, but no proof is given. The text says that [34, Theorem 2.2] "can be adapted also to the weighted setting", but this adaptation is not a formality: replacing $\Delta$ by $\Delta_\omega=\Delta-\nabla\omega\cdot\nabla$ introduces a first-order drift term in the energy estimate for $\int |u|^p \eta_R^p\,d\mu_\omega$, and the resulting additional term involving $\nabla\omega$ is not controlled in the generality of Theorem 4.1. Since every $p>1$ result (Theorems 2.2, 2.3, 2.4) is obtained by invoking Theorem 4.1 after a conformal change, the main $p>1$ claims rest on an unproved statement. Please provide either a self-contained proof of Theorem 4.1 in the weighted setting or an explicit extra assumption on $\omega$, for instance a growth condition on $|\nabla\omega|$, and verify it in each conformal application.
[§4, proof of Theorem 2.2, Eq. (4.5)] The function $\phi$ in (4.5) is claimed to be increasing "since $\psi$ and the function log are increasing". However, $\log(c_2(1+r)^{-\theta})$ is decreasing for $\theta>0$, and Theorem 2.2 does not impose an analogue of assumption (2.7). For a concrete counterexample to the monotonicity claim, take $0<\theta<2$ and $\psi(r)=\log\log r$: all hypotheses of Theorem 2.2 are satisfied, but $\phi'(r)=1/(r\log r)-\theta/(1+r)<0$ for all sufficiently large $r$. Therefore Theorem 4.1, which requires $\phi$ increasing, cannot be applied with this $\phi$. The same construction is used in Theorem 2.3 and, by reference, in the $p=1$ proofs, so the monotonicity issue affects several statements. Theorem 2.4 avoids the problem because it assumes (2.7), but Theorems 2.2 and 2.3 need a corrected argument or an additional hypothesis.
[§5.1, Lemma 5.2] The $p=1$ uniqueness Theorem 5.1 is concluded by invoking Lemma 5.2, which the paper states without proof and attributes to [10, Theorem 9.2] and [34, Lemma 3.1]. Since [10] treats $p=2$ and [34] is unweighted, the weighted $p=1$ version used here is not literally contained in those references. The proof of Proposition 5.3 involves $\Delta_\omega$ and a drift term, so the transfer to the weighted setting is not purely notational. Please include a proof of Lemma 5.2 or a precise reference to a weighted version, or state the additional assumptions under which the weighted statement holds.

minor comments (4)

[§5, proofs of Theorems 2.6 and 2.7] In the proof of Theorem 2.6, "by means of (2.13)" should refer to (2.10); in the proof of Theorem 2.7, "by means of (2.10)" should refer to (2.13).
[§4, proof of Theorem 2.2] In the long chain estimate, the reference "due to (2.3)" should be to (2.1), since (2.3) is the $\theta=2$ assumption used in Theorem 2.3.
[§3 and §5] The notation $\tilde\Delta$ is ambiguous: in Section 3 it is the weighted Laplacian (3.6) satisfying $\tilde\Delta=\rho^{-1}\Delta$, while in the proofs of Theorems 2.6--2.8 the displayed identity for $\tilde\Delta\tilde r$ holds for the Laplace--Beltrami operator of the conformal metric. Please state explicitly which operator is meant in assumption (H)(ii) of Theorem 5.1.
[§5.1, Proposition 5.3] There are several typos in Proposition 5.3: in (5.8), $\partial_\omega\zeta$ should be $\partial_t\zeta$; "$d\mu f$" should be "$d\mu_\omega$"; and the notation $B_R$ versus $\tilde B_R$ in the boundary terms is inconsistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the conformal-change reduction is a genuine identity, the p>1 uniqueness rests on an external theorem of Punzo rather than on the paper's own conclusions, and the authors' self-citations are confined to the literature review.

full rationale

The derivation chain is as follows. The conformal change \tilde g = \rho g, d\tilde \mu = \rho d\mu is combined with identity (3.6), \tilde\Delta = \rho^{-1}\Delta, so problem (1.3) is exactly (4.4); this is an equality of problems, not a hidden assumption of the conclusion. For p > 1, Theorems 2.2\u20132.4 construct \phi from the given \psi and \rho, verify the weighted L^p bound (4.2) from u \in L^p_{e^{-\psi}}, and verify (4.1) from the stated integral conditions; Theorem 4.1 then gives uniqueness. Theorem 4.1 is quoted from Punzo [34] and is not a self-citation; the sentence that the proof 'can be adapted also to the weighted setting' is an unproved transfer, which is a rigor/correctness gap but not a circular reduction. For p = 1, Theorem 5.1 is proved in the paper, using the external Lemma 5.2 from Punzo/Grigor'yan, and Theorems 2.6\u20132.8 verify its hypotheses; no step identifies the conclusion with an assumption. The authors' own prior works ([19]\u2013[23]) appear only in the Introduction and Section 1.1 literature review and are not used in any proof. The questionable claim that \phi in (4.5) is increasing because \psi and log are increasing is a possible proof flaw, not a definitional circularity. Therefore no circular step is exhibited and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities; the paper's results are conditional on stated geometric and regularity assumptions. The main external input is the unproved weighted version of Punzo's uniqueness theorem (Theorem 4.1) and the Laplacian comparison machinery.

assumptions (6)

domain assumption Theorem 4.1 holds for weighted Riemannian manifolds as adapted from Punzo [34, Theorem 2.2]; proof omitted in the paper.
All p>1 uniqueness theorems reduce the density problem to Theorem 4.1; Section 4.1 states the adaptation without proof.
domain assumption Assumption (H): M has a pole, Δr≥0, ω=ω(r) is radial with ∂ω/∂r = O(r).
Required for Theorem 5.1 and Lemma 5.4 in the p=1 case; Section 5.1.
standard math Laplacian comparison Theorem 3.3 (lower bounds on Δr from sectional or radial Ricci curvature).
Used in Remark 2.9 and in verifying (H)(ii) through (2.10), (2.13), (2.16).
domain assumption Completeness of conformal metric g̃=ρg via (1.2): ∫ sqrt(ρ(γ)) ds = ∞ on every divergent ray.
Ensures (M,g̃) is complete as needed for the general uniqueness theorems; Section 3, conformal metric.
domain assumption Technical monotonicity conditions (2.7) and (2.18): ψ' + ρ₂'/ρ₂ ≥ 0 and ψ' + ρ'/ρ ≥ 0.
Used to make the constructed weight φ increasing in proofs of Theorems 2.4 and 2.8.
domain assumption Radiality of ρ for p=1 results (2.8).
The p=1 proofs require radial density to pass to r̃ coordinates; Theorems 2.6-2.8.

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Pith. "Pith review of Rigidity for the heat equation with density on Riemannian manifolds through a conformal change." pith.science (2026). https://pith.science/paper/2YWWZQXR

@misc{pith2026250713230,
  author       = {Pith},
  title        = {Pith review of: Rigidity for the heat equation with density on Riemannian manifolds through a conformal change},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2YWWZQXR}},
  note         = {Machine review of arXiv:2507.13230}
}

abstract

We investigate uniqueness of solution to the heat equation with a density $\rho$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution $u$ vanishes identically, assuming that $u$ belongs to a certain weighted Lebesgue space with exponential or polynomial weight, $L^p_{\phi}$. We distinguish between the cases $p > 1$ and $p = 1$ which required stronger assumptions on the manifold and the density function $\rho$. We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density $\rho$.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [34]

F. Punzo. Uniqueness for the heat equation in Riemannian manifolds. J. Math. Anal. Appl. 424 (2015), 402–422

work page 2015

[2] [10]

Grigoryan

A. Grigoryan. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36 (1999), 134–249

work page 1999

[3] [1]

L. J. Alias, P. Mastrolia, M. Rigoli. Maximum Principles and Geometric Applications, Springer Mono- graphs in Mathematics (Springer International Publishing, 2016)

work page 2016

[4] [2]

Aronson, P

D.G. Aronson, P. Besala. Uniqueness of solutions to the Cauchy problem for parabolic equations. J. Math. Anal. Appl. 13 (1966), 516–526

work page 1966

[5] [3]

Biagi, F

S. Biagi, F. Punzo, Phragm` en-Lindel¨ of type theorems for parabolic equations on infinite graphspreprint (2025)

work page 2025

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I. Chavel. Riemannian Geometry: a modern introduction. Cambridge Tracts in Mathematics, vol. 108. Cambridge University Press, Cambridge (1993)

work page 1993

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S. D. Eidelman, S. Kamin, F. Porper. Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity. Asympt. Anal. 22 (2000), 349–358

work page 2000

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Friedman, On the uniqueness of the Cauchy problem for parabolic equations , American J

A. Friedman, On the uniqueness of the Cauchy problem for parabolic equations , American J. of Math. 81 No. 2 (1959), 503-511

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