Pith. sign in

REVIEW 3 major objections 4 minor 24 references

Existence of Yamabe stability optimizers

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read On closed manifolds, Yamabe stability constants are attained when they stay below a one-bubble threshold.

desk verdict Nice ideas, real proof gap: Theorem 1.2 misses the mixed weak-limit/bubble case, and the mass coefficient is defined two incompatible ways. read the letter →

arxiv 2608.10324 v1 pith:5KPWM52L submitted 2026-08-10 math.DG math.AP

João Henrique Andrade , Tobias König , Jesse Ratzkin , Juncheng Wei This is my paper · ORCID
classification math.DGmath.AP MSC 35J6046E3558J0553C21
keywords YamabeproblemSobolevinequalitystabilityconstantBianchi–EgnellAubin–Schoentestfunctionsmassdominanceconditionoptimizerbubblethreshold
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

The paper aims to show that the best constant in the quantitative stability inequality for the Yamabe functional is attained on closed Riemannian manifolds, not only on the round sphere. The main theorem gives two threshold conditions: the stability constant must lie strictly below the threshold set by sequences approaching a Yamabe minimizer, and strictly below the threshold set by sequences that concentrate into a single bubble. The bubble threshold is computed explicitly as $1 - Y(M,[g])/Y(S^n,[g_0])$, and unlike the sphere case, where the threshold comes from two bubbles, one bubble already measures the possible loss because the set of Yamabe minimizers is compact away from the sphere. The paper then proves that the stability constant can be pushed strictly below the bubble threshold by Aubin's Weyl-curvature test functions when $n\ge 6$ and the manifold is not locally conformally flat, and by Schoen's Green-function test functions under a new mass dominance condition in the remaining dimensions and locally conformally flat case. The upshot is that existence of a stability optimizer is reduced to checking explicit constants attached to the conformal geometry.

What carries the argument

Three objects carry the argument. First, the stability quotient $S_\gamma(u)=E[u]^{(\gamma-2)/2}(E[u]-Y(M,[g])\|u\|_{L^p}^2)/\mathrm{dist}(u,O)^\gamma$, whose infimum is $c_\gamma$. Second, the stability mass $m(u)=\sup_{h\in O_1}(\int_M h^{p-1}u\,dV_g)^2$, which converts the distance to the optimizer set into $E[u]-Y(M,[g])m(u)$ and measures how much of $u$ is seen by the $L^p$-normalized Yamabe optimizers. Third, the two thresholds $c_\gamma^{\mathrm{loc}}$ and $c_\gamma^{\mathrm{bub}}$ encode convergence to an optimizer versus concentration into a bubble. A one-variable calculus lemma shows that the auxiliary function $\Psi(X)$ obtained by separating a minimizing sequence into a nontrivial weak limit and a weakly vanishing remainder has no local minimum; its infimum can occur only at the boundary values $S_\gamma(u)$ or $c_\gamma^{\mathrm{bub}}$. The strict inequalities in Theorem 1.2 rule out both boundaries and force the remainder to vanish. To verify the bubble inequality, Aubin–Talenti bubbles centered at a point of nonzero Weyl curvature produce an energy correction of order $\mu^4$ that dominates the stability mass of order $\mu^{n-2}$ for $n\ge6$, while Schoen's Green-function modified bubbles have both corrections of order $\mu^{n-2}$, and the sign is decided by the mass dominance inequality $\beta_n m_{x_0}>(\gamma/2)(Y(S^n,[g_0])-Y(M,[g]))m_\infty(x_0)$.

What would settle it

Compute the three constants on the product cylinders $S^1(\tau)\times S^{n-1}$, where the optimizers are explicit near $\tau\le\tau^*$; if at some parameter the best stability constant equals the bubble threshold $1-Y(M,[g])/Y(S^n,[g_0])$ while the local inequality is strict, the compactness conclusion would fail exactly there and the threshold picture would need revision.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2. For a closed $n$-dimensional manifold with $0<Y(M,[g])<Y(S^n,[g_0])$ and stability exponent $\gamma\ge2$, let $c_\gamma$ be the best constant in the stability inequality, let $c_\gamma^{\mathrm{loc}}$ be the infimum of stability quotients over sequences converging to a Yamabe optimizer, and let $c_\gamma^{\mathrm{bub}}$ be the infimum over $L^p$-normalized sequences converging weakly to zero. If $c_\gamma<c_\gamma^{\mathrm{loc}}$ and $c_\gamma<c_\gamma^{\mathrm{bub}}$, then every minimizing sequence for $c_\gamma$ converges strongly in $H^1$ up to a subsequence, and a stability optimizer exists. The paper computes $c_\gamma^{\mathrm{bub}}=1-Y(M,[g])/Y(S^n,[g_0])$ for all $\gamma\ge2$, proves the strict inequality $c_\gamma<c_\gamma^{\mathrm{bub}}$ for non-locally-conformally-flat manifolds of dimension $n\ge6$ using the negative Weyl-curvature correction in Aubin's test functions, and proves it for $3\le n\le5$ or locally conformally flat manifolds whenever a mass dominance condition holds at some point. The comparison is controlled by the stability mass $m(u)=\sup_{h\in O_1}(\int_M h^{p-1}u\,dV_g)^2$, which enters the identity $\mathrm{dist}(u,O)^2=E[u]-Y(M,[g])m(u)$ and competes with the energy deficit in the Taylor expansion of the stability quotient.

Load-bearing premise

The load-bearing premise is the strict inequality between the best stability constant and the local threshold set by sequences approaching a Yamabe optimizer; the paper proves conditions that ensure the bubble half of the comparison but leaves the local half unverified for any concrete manifold, deferring it to a companion paper.

Editorial extensions

If this is right

  • Whenever the two strict inequalities of Theorem 1.2 hold, the best stability constant $c_\gamma(M,g)$ is attained by a function that is not itself a Yamabe optimizer, and the minimizing sequence converges strongly in $H^1$.
  • In dimensions $n\ge6$ with a point of nonvanishing Weyl tensor, only the local inequality $c_\gamma<c_\gamma^{\mathrm{loc}}$ has to be checked, since Theorem 1.3 supplies the bubble inequality automatically.
  • In dimensions $3\le n\le5$ and on locally conformally flat manifolds, a single point whose Green-function mass dominates the rescaled stability mass is sufficient to remove the bubble obstruction.
  • The bubble threshold $1-Y(M,[g])/Y(S^n,[g_0])$ is independent of $\gamma$, whereas on the round sphere the threshold $2-2^{2/p}$ arises from two bubbles; the difference is exactly the compactness of the optimizer set.
  • Combined with the known sphere result, Theorem 1.2 completes the positive-Yamabe existence picture for stability optimizers.

Reading between the lines

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

  • The same two-threshold mechanism should transfer to any sharp inequality whose optimizer set is compact and whose stability quotient has a distance-to-optimizers representation; the sphere is the special case where the optimizer set is noncompact and produces an additional bubble.
  • Because the right-hand side of the mass dominance condition grows linearly in $\gamma$, larger stability exponents require a larger Green-function mass; this suggests that manifolds with degenerate stability ($\gamma>2$) should be the first place where existence of an optimizer fails as a geometric parameter is varied.
  • A dimension-dependent normalization separates the introduction's version of the mass coefficient from the one used in Proposition 5.2, so before applying Theorem 1.4 to a concrete manifold one should identify which version of $m_\infty(x_0)$ is being verified; this is an editorial caution, not a claim of the paper.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the quantitative stability of the Yamabe inequality on closed Riemannian manifolds (M,g) with n≥3 and 0 < Y(M,[g]) < Y(S^n,[g_0]). It defines the stability constant c_γ(M,g) as the infimum of a stability quotient S_γ over functions that are not Yamabe optimizers, and asks whether this infimum is attained. The main result (Theorem 1.2) states that if c_γ is strictly below a local threshold c_loc and a bubble threshold c_bub, then every minimizing sequence converges strongly in H^1 and an optimizer for c_γ exists. The bubble threshold is computed in Corollary 2.8 as c_bub = 1 − Y(M,[g])/Y(S^n,[g_0]). Theorems 1.3 and 1.4 provide sufficient geometric conditions for c_γ < c_bub: for n≥6 non-locally-conformally-flat manifolds via Aubin's test functions, and for 3≤n≤5 or locally conformally flat manifolds via Schoen's Green function modified test functions, under a new 'mass dominance condition' (M_x0). The proof combines a concentration-compactness argument adapted from [15] with explicit asymptotic expansions of the stability quotient along test functions.

Significance. If correct, the paper makes a valuable contribution to the quantitative stability theory of sharp functional inequalities on manifolds. The identification of a one-bubble threshold in the non-spherical case, as opposed to the two-bubble threshold on the sphere treated in [15], is a genuinely new insight and clarifies the structure of the compactness argument. The paper contains a number of clean and useful technical pieces: the compactness proof for the optimizer set O_1 (Lemma 2.1), the distance identity in Lemma 2.3, the explicit computation of c_bub in Corollary 2.8, and the calculus lemma (Lemma 3.1) ruling out local minima of the auxiliary function Ψ. The expansion of the stability mass m along test functions and the detailed energy computations in Propositions B and C are careful and largely self-contained in the appendices. However, the two major issues detailed below currently prevent the main existence theorems from being fully established, so the significance is conditional on a substantial revision.

major comments (3)
  1. [Section 3, Claim 2 (Proof of Theorem 1.2)] The proof of Claim 2 assumes 'u_k = u+v_k with ||u||_{L^p(M)} = 1', but Claim 1 only establishes that the weak limit u is nonzero; it does not prove that u carries the full L^p mass. This assumption is equivalent to the desired conclusion: if ||u||_{L^p(M)} = 1, then Brezis–Lieb gives ||v_k||_{L^p(M)} → 0, and the Yamabe inequality then yields E[v_k] → 0, so strong convergence already holds. In the complementary case 0 < ||u||_{L^p(M)} < 1, the formula in equation (3.2) uses (1+||v_k||_p^p)^{2/p} in the numerator instead of the correct (||u||_p^p+||v_k||_p^p)^{2/p}, and the subsequent reduction to the one-variable function Ψ does not apply. No argument is supplied to exclude this mixed weak-limit-plus-bubble scenario, such as a splitting lower bound of the form liminf_k S_γ(u_k) ≥ min(c_loc, c_bub). Consequently, Theorem 1.2 is not proved as written.
  2. [Introduction (M_x0) and Proposition 5.2, Eq. (5.6)] The definition of the rescaled stability mass coefficient m_∞(x0) is inconsistent between the introduction and Proposition 5.2. In the introduction, the mass dominance condition (M_x0) is stated with m_∞(x0) = [α_n^2 (n−2)^2 ω_{n−1}^2 / |S^n|^{(n−2)/n}] sup_{h∈O_1} (∫_M h^{p−1} G_{x0} dV_g)^2, whereas Proposition 5.2, Eq. (5.6), derives m_∞(x0) = [α_n (n−2) ω_{n−1}]^2 / ||U_1||^2_{L^p(R^n)} sup_{h∈O_1} (∫_M h^{p−1} G_{x0} dV_g)^2. These two expressions differ by a dimension-dependent constant; for n=3 the ratio of the denominators is √3/2. Since the proof of Theorem 1.4 uses the Proposition 5.2 version while the statement of Theorem 1.4 and the introduction use the other version, the condition verified in the proof is not the condition stated in the theorem. This ambiguity affects the validity of the theorem's hypothesis and must be resolved.
  3. [Section 3, use of Proposition 2.7] In the proof of Claim 2, the text states 'By Proposition 2.7, one obtains ||v_k||_p^p ≤ Y_0^{-p/2} E[v_k]^{p/2} + o(1)' for a sequence v_k ⇀ 0 in H^1 that is not normalized to have L^p norm one. Proposition 2.7 is stated for sequences with ||v_k||_{L^p(M)} = 1. A scaling argument can likely bridge this gap, but as written the application is not justified, and since this inequality is used to derive the lower bound on Ψ(X_k), the step should be made explicit.
minor comments (4)
  1. [Section 1, definition of c_loc (1.9)] The convergence in the definition of the local threshold c_loc (1.9) is not specified; it should say whether the limit is taken in H^1(M), C^2(M), or another topology. The proof of Claim 3 suggests H^1 convergence is intended.
  2. [Abstract and Corollary 1.5] The paper never verifies the strict inequality c_γ(M,g) < c_loc_γ(M,g) for any manifold; the existence results in Corollary 1.5 are therefore conditional on this hypothesis, with verification deferred to the companion paper [1]. This is acceptable for a theorem of sufficient conditions, but the abstract's wording 'we prove the existence of stability optimizers' could be misread as unconditional, and a sentence emphasizing the conditional nature would help.
  3. [Appendix A, Proposition C proof] The proof of Proposition C is presented as a sketch: Claim 3 states that the transition annulus contributes −β_n ||U_1||^2_{L^p(R^n)} m_{x0} µ^{n−2} + o(µ^{n−2}) and refers to Schoen's integration-by-parts argument, but the details of the surface integral and the precise coefficient β_n are not shown. Since the mass dominance condition hinges on this coefficient, a fuller derivation would improve the paper.
  4. [Section 2, Lemma 2.5] In Lemma 2.5, the statement says u_k ⇀ 0 in L^p(M), but the proof uses convergence in H^1(M); this is harmless because H^1 compactly embeds into L^2 and weakly in L^p, but the statement could be tightened.

Circularity Check

1 steps flagged · score 6.0 of 10

Claim 2 of Theorem 1.2 assumes the weak limit has full Lp mass, which forces the remainder to zero; the mixed bubbling alternative is assumed away rather than excluded.

  1. other [Section 3, Proof of Theorem 1.2, Claim 2, Eq. (3.2)]
    "Assume again that u_k = u+v_k with ∥u∥_{Lp(M)} = 1 and v_k ⇀0 in H^1(M). Using the decompositions from above, one has cγ + o_k(1) = Sγ(u_k) = ((E[u]+E[v_k])^{(γ-2)/2}(E[u]+E[v_k]-Y_M(1+∥v_k∥^p_{Lp})^{2/p}))/((E[u]+E[v_k]-Y_M m(u))^{γ/2}). (3.2)"

    The normalized sequence satisfies 1=∥u_k∥_p^p=∥u∥_p^p+∥v_k∥_p^p+o(1) by Brezis–Lieb. Hence the factor (1+∥v_k∥_p^p)^{2/p} used in (3.2) is legitimate only when ∥u∥_p=1. But under that assumption the normalization gives ∥v_k∥_p→0, and Proposition 2.7 then yields E[v_k]≥(Y0+o(1))∥v_k∥_p^2→0, i.e. the strong convergence that Claim 2 is supposed to prove. The mixed case 0<∥u∥_p<1, where a nonzero weak limit coexists with a bubble, is exactly the case in which the correct term would be (∥u∥_p^p+∥v_k∥_p^p)^{2/p}; equation (3.2) does not hold there, and no separate argument is supplied. Thus the proof assumes the target conclusion (or an assumption immediately implying it) rather than excluding the alternative.

full rationale

No fitted parameters are used and no quantity is renamed as a prediction: c_bub is computed in Corollary 2.8 from Proposition 2.7 and test bubbles, and the threshold inequalities in Theorems 1.3 and 1.4 are verified by classical Aubin–Schoen expansions. The self-citations to [15] are methodological and not load-bearing for the new non-spherical results. The mass-dominance condition has an apparent normalization mismatch between the introduction and Proposition 5.2, but that is an internal-consistency concern rather than circularity. The central circularity is in the proof of Theorem 1.2: Claim 2 assumes ∥u∥_{Lp}=1, which via normalization and Proposition 2.7 already yields v_k→0, and Eq. (3.2) is built on that assumption; the genuinely mixed alternative is not analyzed. This makes the main compactness argument partially circular as written, so the score is 6 rather than a lower score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The central claim rests on standard elliptic regularity and Aubin-Sobolev inequalities, on the positive mass theorem for sign of m_x0, and on two explicitly assumed strict inequalities: c_gamma < c_loc and (in the relevant cases) the mass dominance condition. No free parameters are fitted to data. The paper introduces the stability mass functional as a new object but it is a defined quantity, not an empirical fit.

assumptions (3)
  • standard math Aubin-Sobolev inequality, Lee-Parker Yamabe compactness and regularity theory, and Aubin-Schoen energy expansions are used as inputs.
    Invoked in Lemmas 2.1, 2.7 and Propositions B-C; these are established results, not derived in this paper.
  • domain assumption The manifold is closed, n >= 3, and 0 < Y(M,[g]) < Y(S^n,[g0]).
    Needed for E^{1/2} to be a norm and for compactness of the normalized minimizer set O1; the spherical case is delegated to [15].
  • ad hoc to paper The strict inequalities c_gamma < c_loc_gamma and c_gamma < c_bub_gamma in Theorem 1.2 and the mass dominance condition (M_x0) in Theorem 1.4 are assumed where not proved.
    c_bub is computed in Corollary 2.8; c_loc is not verified in this paper (deferred to [1]); mass dominance is a new sufficient condition, not a consequence of the positive mass theorem alone.
invented entities (1)
  • Rescaled stability mass coefficient m_infinity(x0) and the stability mass functional m(u)
    purpose: Encodes the second-order projection correction in the expansion of the stability quotient S_gamma along concentrating test functions.
    It is a defined functional of the optimizer set O1 and the Green function; it has no falsifiable handle outside the paper, and its stated formula is inconsistent between the introduction and Proposition 5.2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Existence of Yamabe stability optimizers." pith.science (2026). https://pith.science/paper/5KPWM52L

@misc{pith2026260810324,
  author       = {Pith},
  title        = {Pith review of: Existence of Yamabe stability optimizers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5KPWM52L}},
  note         = {Machine review of arXiv:2608.10324}
}
read the original abstract

We prove the existence of stability optimizers for the Yamabe inequality on closed Riemannian manifolds of dimension at least three with positive Yamabe invariant that satisfy two threshold conditions. Remarkably, the compactness threshold we uncover is different from the special case of the round sphere treated previously by the second author. More precisely, it is given by sequences blowing up in one instead of two bubbles, reflecting the compactness of Yamabe minimizers in the non-spherical case. Using the classical asymptotic analysis of Aubin--Schoen test functions, we prove that the stability constant is strictly below the one-bubble threshold in dimension at least six and when the manifold is not locally conformally flat. In the complementary case, namely in dimensions three through five or when the manifold is locally conformally flat, we find a new positive-mass-type condition which is sufficient for the strict inequality.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 22 canonical work pages

  1. [15]

    T. K¨ onig. Stability for the sobolev inequality: existence of a minimizer, 2025. J. Eur. Math. Soc. (2025), published online-first. DOI 10.4171/JEMS/1582. 1, 3, 4, 6

    doi:10.4171/jems/1582 2025

  2. [1]

    J. H. Andrade, T. K¨ onig, J. Ratzkin, and J. Wei. Sharp Yamabe stability constants on product cylinders, 2026. In preparation. 7 YAMABE STABILITY OPTIMIZERS 27

    work page 2026

  3. [2]

    T. Aubin. ´Equations diff´ erentielles non lin´ eaires et probl` eme de Yamabe concernant la courbure scalaire.J. Math. Pures Appl., 55:269–296, 1976. 1, 14, 15, 20, 21

    work page 1976

  4. [3]

    T. Aubin. Probl` emes isoperimetriques et espaces de Sobolev.J. Differ. Geom., 11:573–598, 1976. 11

    work page 1976

  5. [4]

    Y. Bi, T. Hao, S. He, Y. Shi, and J. Zhu. A proof for the Riemannian positive mass theorem up to dimension 19, 2026. 5

    work page 2026

  6. [5]

    Bianchi and H

    G. Bianchi and H. Egnell. A note on the Sobolev inequality.J. Funct. Anal., 100:18–24, 1991. 1, 3

    work page 1991

  7. [6]

    Brendle and Y

    S. Brendle and Y. Wang. A dimension descent scheme for the positive mass theorem in arbitrary dimension,

    work page

  8. [7]

    Chakraborty, M

    S. Chakraborty, M. Ghosh, and D. Karmakar. Existence of Bianchi-Egnell stability extremizer for the Hardy- Sobolev inequality. Preprint, arXiv:2505.07039 [math.AP], 2025. 5

    arXiv 2025

Show all 24 references
  1. [8]

    Dolbeault, M

    J. Dolbeault, M. J. Esteban, A. Figalli, R. L. Frank, and M. Loss. Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence.Camb. J. Math., 13:359–430, 2025. 1, 9

    2025

  2. [9]

    Druet, E

    O. Druet, E. Hebey, and F. Robert.Blow-up theory for elliptic PDEs in Riemannian geometry, volume 45 of Mathematical Notes. Princeton University Press, Princeton, NJ, 2004. 7

    2004

  3. [10]

    Engelstein, R

    M. Engelstein, R. Neumayer, and L. Spolaor. Quantitative stability for minimizing Yamabe metrics.Trans. Amer. Math. Soc. Ser. B, 9:395–414, 2022. 1, 3, 7

    2022

  4. [11]

    R. Frank. Degenerate stability of some Sobolev inequalities.Ann. Inst. H. Poincar´ e, Anal. Non Lin´ eaire, 39:1459–1484, 2022. 3, 7

    2022

  5. [12]

    G¨ unther

    M. G¨ unther. Conformal normal coordinates.Ann. Global Anal. Geom., 11:173–184, 1993. 21

    1993

  6. [13]

    Hebey and M

    E. Hebey and M. Vaugon. Best constants in the Sobolev imbedding theorem.Ann. Inst. Henri Poincar´ e, Anal. Non Lin´ eaire, 13(1):57–93, 1996. 8

    1996

  7. [14]

    Khuri, F

    M. Khuri, F. C. Marques, and R. Schoen. A compactness theorem for the Yamabe problem.J. Differential Geom., 81:143–196, 2009. 7

    2009

  8. [16]

    J. M. Lee and T. H. Parker. The Yamabe problem.Bull. Amer. Math. Soc. (N.S.), 17(1):37–91, 1987. 1, 2, 7, 8, 11, 14, 15, 17, 21, 22, 25

    1987

  9. [17]

    R. Schoen. Conformal deformation of a Riemannian metric to constant scalar curvature.J. Differential Geom., 20:479–495, 1984. 1, 5, 17, 18, 20, 25

    1984

  10. [18]

    R. Schoen. Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. Topics in calculus of variations, Lect. 2nd Sess., Montecatini/Italy 1987, Lect. Notes Math. 1365, 120-154, 1989. 7, 17, 18, 25

    1987

  11. [19]

    Schoen and S

    R. Schoen and S. T. Yau. Conformally flat manifolds, Kleinian groups and scalar curvature.Invent. Math., 92:47–71, 1988. 5

    1988

  12. [20]

    R. M. Schoen and S.-T. Yau. Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity.Proc. Nat. Acad. Sci., 76(3):1024–1025, 1979. 5

    1979

  13. [21]

    G. Talenti. Best constant in Sobolev inequality.Ann. Mat. Pura Appl. (4), 110:353–372, 1976. 1

    1976

  14. [22]

    Trudinger

    N. Trudinger. Remarks concerning the conformal deformation of Riemannian structures.Ann. Sc. Norm. Super. Pisa, 22:265–274, 1968. 1

    1968

  15. [23]

    E. Witten. A new proof of the positive energy theorem.Comm. Math. Phys., 80(3):381–402, 1981. 5

    1981

  16. [24]

    H. Yamabe. On a conformal deformation of Riemanian structures on compact manifolds.Osaka Math. J., 12:21–37, 1960. 1 Department of Mathematics, University of S˜ao Paulo, 05508-090, S˜ao Paulo-SP, Brazil Email address:andradejh@ime.usp.br Institut f¨ur Mathematik, Goethe-Univer...

    1960

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.