Conformal Green functions and Yamabe metrics of Sobolev regularity

Albachiara Cogo; Andoni Royo Abrego; Rodrigo Avalos

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Conformal Green functions and Yamabe metrics of Sobolev regularity

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Pith's one-line read Every $W^{k,q}$ Riemannian metric on a closed orientable 3-manifold with $q>3$ admits a conformal metric of constant scalar curvature, with matching Sobolev regularity.

desk verdict Real analytic toolkit and a plausible main theorem, but the mass-zero branch rests on a false harmonicity claim about a conformal map; that needs repair before acceptance. read the letter →

arxiv 2507.01674 v1 pith:KANCYSPU submitted 2025-07-02 math.AP math.DG

Rodrigo Avalos , Albachiara Cogo , Andoni Royo Abrego This is my paper · ORCID

classification math.APmath.DG MSC 53C2135J6035J0858J05

keywords YamabeproblemroughmetricsconformalLaplacianGreenfunctionpositivemasstheoremSobolevregularityscalarcurvatureasymptoticallyEuclideanmanifolds

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

This paper tries to establish that the Yamabe problem—the search for a constant-scalar-curvature metric inside a conformal class—has a positive answer for rough Riemannian metrics on closed 3-manifolds. Specifically, it claims that any $W^{k,q}$-Riemannian metric with $k \geq 2$ and $q > 3$ on an orientable, closed 3-manifold is conformally equivalent to a $W^{k,q}$ metric of constant scalar curvature. The proof requires building an elliptic theory for the conformal Laplacian whose coefficients are only Sobolev-regular, and a detailed blow-up analysis of its Green function. The positive-Yamabe case, the hard one, is reduced to a positive-mass theorem for the asymptotically Euclidean manifold obtained by deleting the Green-function pole. A reader should care because rough metrics arise naturally in general relativity's constraint equations and in low-regularity scalar-curvature geometry, where previously the positive case was open even for $C^{1,\alpha}$ metrics.

What carries the argument

The rough conformal Laplacian $L_g = -a_n \Delta_g + R_g$, with coefficients in $W^{2,q}$, is the central operator; the paper proves it is Fredholm of index zero and an isomorphism when the Yamabe invariant is positive, and develops regularity theorems for its solutions. The conformal Green function $G_p$—the unique positive solution of $L_g G_p = \delta_p$—is the central object: its blow-up profile in harmonic coordinates of a good conformal gauge (a conformal metric with continuous positive scalar curvature) determines the asymptotic Euclidean structure of the decompactified manifold. The mass formula $m = 2A$, where $A$ is the leading constant in the Green-function expansion, connects the analytic blow-up analysis to the Lee–LeFloch positive mass theorem, and Aubin bubbles supply the test functions whose energy dips below $\lambda(S^3)$.

What would settle it

The central claim would collapse if the positive mass theorem for distributional curvature admitted a counterexample in the exact class built here: a scalar-flat $W^{2,r}_{-\tau}$ asymptotically Euclidean 3-manifold with $r>3/2$, $\tau=2-3/r$, nonnegative distributional scalar curvature, and negative generalized ADM mass. Constructing such a manifold would directly contradict the positive-mass theorem on which Theorem A rests.

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Extended reading notes

Core claim

Theorem A is the central discovery: on an orientable, smooth, closed 3-manifold, every $W^{k,q}$-Riemannian metric with $k \geq 2$ and $q > 3$ admits a conformal metric of constant scalar curvature of the same Sobolev class. The same conclusion holds without the dimensional or regularity restrictions when the Yamabe invariant is non-positive: for $n \geq 3$ and $q > n/2$, every $W^{k,q}$ metric with $\lambda(M,g) \leq 0$ has a $W^{k,q}$ constant-scalar-curvature conformal representative. The engine behind the positive case is Theorem B, which produces a unique positive Green function for the rough conformal Laplacian and controls its blow-up at the pole in specially constructed harmonic coordinates, yielding the expansion $G_p = B/|x|^{n-2} + h(x)$ with $h(x) = A + O(|x|^{2 - n/r})$. The constant $A$ records the ADM-type mass of the decompactified manifold $(M\setminus\{p\}, G^4 g)$, and the Lee–LeFloch positive mass theorem supplies the sign $A \geq 0$ needed to beat the round-sphere threshold in the Aubin–Trudinger–Yamabe argument.

Load-bearing premise

The argument's load-bearing premise is that the Lee–LeFloch positive mass theorem applies to the scalar-flat asymptotic-Euclidean manifolds obtained by decompactifying with the rough conformal Green function; if that theorem fails for these rough metrics, the sign of $A$ and with it the positive-Yamabe conclusion collapses.

Editorial extensions

If this is right

If the theorem is right, every $W^{2,q}$ conformal class on a closed orientable 3-manifold with $q > 3$ contains a constant scalar curvature metric, and the regularity of that representative exactly matches the starting metric ($W^{k,q}$ for $k \geq 2$).
In the non-positive Yamabe case the result is fully general in dimension: for any closed $n \geq 3$ manifold and any $W^{k,q}$ metric with $q > n/2$ and $\lambda(M,g) \leq 0$, a $W^{k,q}$ constant-scalar-curvature conformal metric exists.
The Green-function expansion of Theorem B provides a usable analytic tool for Schrödinger-type operators with rough geometric coefficients, beyond the Yamabe problem itself.
If a positive mass theorem for distributional curvature became available without the spin assumption or for multiple ends, the orientability assumption in Theorem A could be dropped, as the paper notes.

Reading between the lines

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

One consequence the authors leave implicit is that the resolution suggests a regularity threshold: the 'good' 3D threshold $q > 3$ is exactly where the Green-function error term becomes continuous and $C^1$-controlled; below it, the same conformal-method route would need a new blow-up estimate.
The identification $m = 2A$ also suggests a concrete numerical check: computing the coefficient $A$ for an explicit rough metric on $S^3$ would verify the sign and the mass-vanishing rigidity.
The elliptic regularity toolkit for the rough Laplace–Beltrami and conformal Laplacian operators is likely to transfer directly to the Lichnerowicz equation of the conformal method in general relativity, where rough initial data are standard.
A testable extension: sharpen the Green-function expansion to $q \leq n/2$ in dimension 3, or to dimensions 4–5 with a spin positive-mass theorem, and Theorem A would extend; the paper identifies both as open.

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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

The headline result—Yamabe for W^{2,q}, q>3, closed orientable 3-manifolds—is the right target, and the paper really does build most of the machinery to get there: a workable elliptic theory for the rough conformal Laplacian, a Green function with the sharp blow-up expansion, and a clean reduction of the positive case to the sign of the ADM-type mass. The appendices on Sobolev diffeomorphisms and weighted estimates are useful on their own. That is real credit where it is due.

The reader's conditional verdict is fair. I share the worry about relying on the first author's unpublished [6] for the foundational Laplace regularity; that should be published or reproduced. The Lee–LeFloch PMT application is handled with more care than I expected, including the decay check and the spin/orientability caveat.

But I think the reader under-weights one specific internal gap, and the stress-test note lands. In Theorem 5.10, after constructing φ via the isometry y and stereographic projection, the proof says 'using that φ is an isometry' and concludes that u^i∘φ solves Δ_g(u^i∘φ)=0. φ is not an isometry; it is conformal. The correct transformation has the extra lower-order term -(n-2)df(dφ(∇λ)), and in n=3 that term does not vanish because the conformal factor is not constant. So the W^{3,q} regularity upgrade for φ is not justified as written. This is load-bearing: Theorem 5.12 uses Theorem 5.10 precisely to handle the mass-zero case A=0. Without that upgrade, the round metric φ^*g_{S^3} is not shown to lie in the W^{2,q} conformal class, and that branch of Theorem A is unsupported. The gap may be repairable—one could try to run the bootstrap with the lower-order term—but the repair is not in the paper.

Also, the final display in Theorem 5.10 has the conformal factor exponent inverted (u_1^{-4} instead of u_1^4, if u_1 is the standard bubble). That may be a typo, but it should be fixed.

Who is this for? Geometric analysts working on low-regularity Yamabe, rough metrics, and PMT. The toolkit is worth having even if the main theorem needs this repair. I would send it to a serious referee; a desk rejection would be wrong. But I would not accept it without a corrected mass-zero argument.

Referee Report

2 major / 3 minor

Summary. This paper develops an elliptic regularity theory for the conformal Laplacian of W^{2,q} Riemannian metrics on closed manifolds with q>n/2, and proves existence, uniqueness, positivity, and a blow-up expansion for the associated Green function. It then uses the Green function to decompactify the manifold and applies the Lee–LeFloch positive mass theorem to prove Theorem A: on an orientable closed 3-manifold, every W^{k,q} metric with k≥2 and q>3 admits a W^{k,q} conformal metric of constant scalar curvature. A separate theorem settles the non-positive Yamabe case in all dimensions n≥3 with the same regularity.

Significance. The analytic framework is substantial and largely self-contained: the Fredholm and regularity results for L_g, the construction of harmonic and normal coordinates below the C^{1,1} threshold, and the scaling proof of the Green-function expansion in Theorem 4.4 are careful and likely useful beyond the Yamabe problem. The paper is also commendably explicit about its external inputs, namely the first author's prior regularity theorems [6] and the Lee–LeFloch PMT [44]. If the mass-zero branch can be repaired, the result would be a natural and important advance in low-regularity conformal geometry.

major comments (2)

[Theorem 5.10] In the proof of Theorem 5.10, the claim that the maps u^i∘φ are C^{1,α} weak solutions to Δ_g(u^i∘φ)=0 in a harmonic chart on S^3 is false. Since φ is only a conformal diffeomorphism, write φ^*g_{S^3}=e^{2λ}g. For any function f on S^3 the transformation law is Δ_g(f∘φ)=e^{2λ}((Δ_{S^3}f)∘φ+(n-2)df(dφ(∇λ))). Even if Δ_{S^3}f=0, the second term does not vanish unless λ is constant, and λ is not constant in any neighborhood of p; in the mass-zero case the expansion G=|x|^{-1}+O_1(|x|^{1+β}) still leaves a nonconstant conformal factor. Thus the elliptic bootstrap to W^{3,q} at p is not justified. This is load-bearing for Theorem 5.12 and Theorem A, because the A=0 branch is delegated to Theorem 5.10, and without φ∈W^{3,q} the pulled-back round metric is not shown to lie in [g]_{W^{2,q}}.
[Proof of Theorem 5.10, final paragraph] The final conformal-factor computation also relies on the formula (σ_S)_*g_{S^3}=4u_1^{-4}g_{R^3} stated before (5.34). With the paper's definition u_1=(1+|z|^2)^{-1/2}, the correct factor is 4u_1^4, not 4u_1^{-4}. As written, the displayed expression for φ^*g_{S^3} blows up near p, which is incompatible with the fact that φ is smooth in stereographic coordinates near p (up to the claimed regularity). This exponent error should be corrected and the conformal factor argument redone.

minor comments (3)

[Corollary 3.10 and Proposition 3.13] There are several small typos: Corollary 3.10 says 'W^{k,p}-Riemannian metric' where the exponent should be q, and Proposition 3.13 says 'Riemannia metric' instead of 'Riemannian metric'.
[Lemma 2.1, equation (2.7)] In the displayed formula for the third derivative, the last term has the repeated index ∂^3u/∂x^a∂x^b∂x^b; it should be ∂^3u/∂x^a∂x^b∂x^c for a correct chain-rule expression.
[Around equation (5.34)] The stereographic metric formula has the inverse exponent, which appears to be a typo but should be fixed consistently in the text because it is used in the conformal factor computation.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the Yamabe theorem follows from external PMT and prior regularity results, not from its own conclusion.

full rationale

The paper's derivation chain is not circular. Theorem A is reduced to: (i) the rough conformal Laplacian theory, whose central regularity input Theorem 3.1 is quoted from the first author's earlier paper [6]; this is a genuine self-citation, but [6] is an independent parameter-free regularity statement for W^{2,q} metrics and does not assume the Yamabe conclusion, so it is not a reduction to the target result; (ii) the Green function blow-up analysis of Theorem B, which is carried out in Section 4 from the elliptic estimates; (iii) the Lee–LeFloch positive mass theorem [44], an external result; and (iv) the standard Aubin bubble test function. The constant A in the Green expansion is identified with half the ADM mass in Proposition 5.8 and its sign is supplied by the external PMT; no fitted parameter is renamed as a prediction, and no equation reduces by construction to constant scalar curvature. The A=0 branch relies on Theorem 5.10, whose proof contains an internally questionable harmonicity claim ('using that φ is an isometry' for a conformal map), but that is a correctness risk in a proof step, not a circularity: it does not make the target result an input. The self-citations lower self-containedness but do not make the argument circular; hence score 1.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted parameters or new physical entities. It relies on standard analytical theorems and the external Lee-LeFloch PMT. The constants A and B in the Green function expansion are determined by the metric, not chosen freely.

assumptions (4)

domain assumption Lee-LeFloch positive mass theorem for manifolds with distributional curvature (Theorem 5.7, cited from [44, Theorem 1.1])
Used in Proposition 5.8 to conclude the generalized ADM mass is nonnegative and zero only for Euclidean space; it is not proved in this paper.
domain assumption First author's prior elliptic regularity for the rough Laplace-Beltrami operator (Theorem 3.1, cited from [6, Corollary 4.2 and 4.4])
Foundational for the regularity theory in Section 3; the present paper extends it but does not re-prove it.
standard math Standard elliptic estimates and De Giorgi-Nash / Trudinger Harnack inequality [29, 72]
Invoked for strong maximum principle and regularity of weak solutions, e.g., in Corollary 3.10 and Proposition 5.12.
standard math Conformal covariance of the conformal Laplacian (equation 3.4)
Used to reduce Theorem B to the good conformal gauge and to transfer Green functions between conformal metrics.

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Cite this review

Pith. "Pith review of Conformal Green functions and Yamabe metrics of Sobolev regularity." pith.science (2026). https://pith.science/paper/KANCYSPU

@misc{pith2026250701674,
  author       = {Pith},
  title        = {Pith review of: Conformal Green functions and Yamabe metrics of Sobolev regularity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KANCYSPU}},
  note         = {Machine review of arXiv:2507.01674}
}

abstract

We provide a full resolution of the Yamabe problem on closed 3-manifolds for Riemannian metrics of Sobolev class $W^{2,q}$ with $q > 3$. This requires developing an elliptic theory for the conformal Laplacian for rough metrics and establishing existence, regularity and a delicate blow-up analysis for its Green function. Most of the analytical work is carried out in dimensions $n \geq 3$ and for $W^{2,q}$ Riemannian metrics with $q>\tfrac{n}{2}$ and should be of independent interest.

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Works this paper leans on

77 extracted references · 76 canonical work pages · cited by 1 Pith paper

[6]

R. Avalos. Sobolev regularity of compactified 3-manifolds and the ADM Center of Mass . 2024. arXiv: 2403.04034 [math.DG]

arXiv 2024
[44]

The positive mass theorem for manifolds with distributional curvature

D. A. Lee and P. G. LeFloch. “The positive mass theorem for manifolds with distributional curvature”. In: Commun. Math. Phys. 339.1 (2015), pp. 99–120

work page 2015
[1]

R. A. Adams. Sobolev spaces. Vol. 65. Pure Appl. Math., Academic Press. New York, NY, 1975

work page 1975
[2]

R. A. Adams and J. J. F. Fournier. Sobolev spaces. 2nd ed. Vol. 140. Pure Appl. Math., Academic Press. New York, NY, 2003

work page 2003
[3]

Equations diff´ erentielles non lin´ eaires et probl` eme de Yamabe con- cernant la courbure scalaire

T. Aubin. “Equations diff´ erentielles non lin´ eaires et probl` eme de Yamabe con- cernant la courbure scalaire”. In: J. Math. Pures Appl. (9) 55 (1976), pp. 269– 296

work page 1976
[4]

Probl` emes isop´ erim´ etriques et espaces de Sobolev

T. Aubin. “Probl` emes isop´ erim´ etriques et espaces de Sobolev”. In:Journal of Differential Geometry 11.4 (1976), pp. 573 –598

work page 1976
[5]

T. Aubin. Some nonlinear problems in Riemannian geometry . Springer Monogr. Math. Berlin: Springer, 1998

work page 1998
[7]

A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature

R. H. Bamler. “A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature”. In: Math. Res. Lett. 23.2 (2016), pp. 325–337

work page 2016

Show all 77 references

[8]

The mass of an asymptotically flat manifold

R. Bartnik. “The mass of an asymptotically flat manifold”. In: Commun. Pure Appl. Math. 39 (1986), pp. 661–693

1986
[9]

Multiplication in Sobolev spaces, revisited

A. Behzadan and M. Holst. “Multiplication in Sobolev spaces, revisited”. In: Ark. Mat. 59.2 (2021), pp. 275–306

2021
[10]

Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited

A. Behzadan and M. Holst. “Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited”. In: Mathematics 10.3 (2022)

2022
[11]

Scalar curvature rigidity of convex polytopes

S. Brendle. “Scalar curvature rigidity of convex polytopes”. In: Invent. Math. 235.2 (2024), pp. 669–708. REFERENCES 76

2024
[12]

H. Brezis. Functional analysis, Sobolev spaces and partial differential equations . New York, NY: Springer, 2011

2011
[13]

On a conjecture of J. Serrin

H. Brezis. “On a conjecture of J. Serrin”. In: Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 19.4 (2008), pp. 335–338

2008
[14]

Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents

H. Brezis and L. Nirenberg. “Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents”. In: Commun. Pure Appl. Math. 36 (1983), pp. 437–477

1983
[15]

Asymptotics for elliptic equations involving critical growth

H Brezis and L. Peletier. “Asymptotics for elliptic equations involving critical growth”. In: Partial differential equations and the calculus of variations. Essays in Honor of Ennio De Giorgi, 149-192 (1989). (1989)

1989
[16]

On some nonlinear equations with critical exponents

H. Brezis and M. Willem. “On some nonlinear equations with critical exponents”. In: J. Funct. Anal. 255.9 (2008), pp. 2286–2298. issn: 0022-1236

2008
[17]

ADM mass for C 0 metrics and distortion under Ricci- DeTurck flow

P. Burkhardt-Guim. “ADM mass for C 0 metrics and distortion under Ricci- DeTurck flow”. In: J. Reine Angew. Math. 806 (2024), pp. 187–245

2024
[18]

Asymptotic symmetry and local behav- ior of semilinear elliptic equations with critical Sobolev growth

L. A. Caffarelli, B. Gidas, and J. Spruck. “Asymptotic symmetry and local behav- ior of semilinear elliptic equations with critical Sobolev growth”. In: Commun. Pure Appl. Math. 42.3 (1989), pp. 271–297

1989
[19]

Intermediate spaces and interpolation, the complex method

A. P. Calder´ on. “Intermediate spaces and interpolation, the complex method”. In: Stud. Math. 24 (1964), pp. 113–190

1964
[20]

M. P. do Carmo. Riemannian geometry. Translated from the Portuguese by Fran- cis Flaherty . Boston, MA etc.: Birkh¨ auser, 1992

1992
[21]

Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian 3-manifolds

O. Chodosh et al. “Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian 3-manifolds”. In: Commun. Pure Appl. Math. 74.4 (2021), pp. 865– 905

2021
[22]

Elliptic Systems in Hs,δ Spaces on Manifolds Which are Euclidean at Infinity

Y. Choquet-Bruhat and D. Christodoulou. “Elliptic Systems in Hs,δ Spaces on Manifolds Which are Euclidean at Infinity”. In: Acta Mathematica 146 (1981), pp. 129–150

1981
[23]

Choquet-Bruhat and C

Y. Choquet-Bruhat and C. DeWitt-Morette. Analysis, manifolds and physics. Part II. Revised and enl. ed. Amsterdam: North-Holland, 2000

2000
[24]

Ricci-DeTurck flow from rough metrics and applications to scalar curvature problems

J. Chu and M-C. Lee. “Ricci-DeTurck flow from rough metrics and applications to scalar curvature problems”. In: J. Funct. Anal. 289.2 (2025), p. 41

2025
[25]

Some regularity theorems in Riemannian geometry

D. M. DeTurck and J. L. Kazdan. “Some regularity theorems in Riemannian geometry”. In: Ann. Sci. ´Ec. Norm. Sup´ er. (4)14 (1981), pp. 249–260

1981
[26]

Yamabe classification and prescribed scalar curvature in the asymptotically Euclidean setting

J. Dilts and D. Maxwell. “Yamabe classification and prescribed scalar curvature in the asymptotically Euclidean setting”. In: Commun. Anal. Geom. 26.5 (2018), pp. 1127–1168

2018
[27]

Druet, E

O. Druet, E. Hebey, and F. Robert. Blow-up theory for elliptic PDEs in Rie- mannian geometry . Vol. 45. Math. Notes. Princeton, NJ: Princeton University Press, 2004

2004
[28]

Stability of the Pohoˇ zaev obstruction in dimension 3

O. Druet and P. Laurain. “Stability of the Pohoˇ zaev obstruction in dimension 3”. In: J. Eur. Math. Soc. (JEMS) 12.5 (2010), pp. 1117–1149

2010
[29]

Gilbarg and N

D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Reprint of the 1998 ed. Class. Math. Berlin: Springer, 2001. REFERENCES 77

1998
[30]

Convex polytopes, dihedral angles, mean curvature and scalar cur- vature

M. Gromov. “Convex polytopes, dihedral angles, mean curvature and scalar cur- vature”. In: Discrete Comput. Geom. 72.2 (2024), pp. 849–875

2024
[31]

Dirac and Plateau billiards in domains with corners

M. Gromov. “Dirac and Plateau billiards in domains with corners”. In: Cent. Eur. J. Math. 12.8 (2014), pp. 1109–1156

2014
[32]

Four lectures on scalar curvature

M. Gromov. “Four lectures on scalar curvature”. In: Perspectives in scalar cur- vature. In 2 volumes . Singapore: World Scientific, 2023, pp. 1–514

2023
[33]

Asymptotic expansions of solutions of the Yamabe equation and the σk-Yamabe equation near isolated singular points

Q. Han, X. Li, and Y. Li. “Asymptotic expansions of solutions of the Yamabe equation and the σk-Yamabe equation near isolated singular points”. In: Com- mun. Pure Appl. Math. 74.9 (2021), pp. 1915–1970

2021
[34]

E. Hebey. Sobolev spaces on Riemannian manifolds. Vol. 1635. Lect. Notes Math. Berlin: Springer, 1996

1996
[35]

Holst, D

M. Holst, D. Maxwell, and G. Tsogtgerel. A Scaling Approach to Elliptic Theory for Geometrically-Natural Differential Operators with Sobolev-Type Coefficients
[36]

Rough solutions of the Einstein con- straint equations on Closed Manifolds without Near CMC Conditions

M. Holst, Gabriel N., and G. Tsogtgerel. “Rough solutions of the Einstein con- straint equations on Closed Manifolds without Near CMC Conditions”. In:Comm. Math. Phys. 288 (2009), pp. 547–613

2009
[37]

The Lichnerowicz Equation on Compact Manifolds with Boundary

M. Holst and G. Tsogtgerel. “The Lichnerowicz Equation on Compact Manifolds with Boundary”. In: Class. Quantum Grav. 30 (2013), p. 205011

2013
[38]

An isoperimetric concept for mass and quasilocal mass

G. Huisken. “An isoperimetric concept for mass and quasilocal mass”. In: Ober- wolfach Reports, European Mathematical Society (EMS), Z¨ urich3.1 (2006), pp. 87– 88

2006
[39]

An isoperimetric concept for the mass in general relativity

G. Huisken. “An isoperimetric concept for the mass in general relativity”. In: Oberwolfach Reports, European Mathematical Society (EMS), Z¨ urich3.3 (2006), pp. 1898–1899

2006
[40]

Lower semicontinuity of mass under C 0 conver- gence and Huisken’s isoperimetric mass

J. L. Jauregui and D. A. Lee. “Lower semicontinuity of mass under C 0 conver- gence and Huisken’s isoperimetric mass”. In: J. Reine Angew. Math. 756 (2019), pp. 227–257

2019
[41]

p-harmonic coordinates for H¨ older met- rics and applications

V. Julin, T. Liimatainen, and M. Salo. “ p-harmonic coordinates for H¨ older met- rics and applications”. In: Commun. Anal. Geom. 25.2 (2017), pp. 395–430

2017
[42]

T. Kato. Perturbation theory for linear operators . Vol. 132. Grundlehren Math. Wiss. Springer, Cham, 1966

1966
[43]

Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem

T. K¨ onig and P. Laurain. “Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem”. In: Ann. Inst. Henri Poincar´ e C, Anal. Non Lin´ eaire 41.5 (2024), pp. 1239–1287

2024
[45]

The Yamabe problem

J. M. Lee and T. H. Parker. “The Yamabe problem”. In: Bull. Am. Math. Soc., New Ser. 17 (1987), pp. 37–91

1987
[46]

A polyhedron comparison theorem for 3-manifolds with positive scalar curvature

C. Li. “A polyhedron comparison theorem for 3-manifolds with positive scalar curvature”. In: Invent. Math. 219.1 (2020), pp. 1–37

2020
[47]

The dihedral rigidity conjecture for n-prisms

C. Li. “The dihedral rigidity conjecture for n-prisms”. In: J. Differ. Geom. 126.1 (2024), pp. 329–361. REFERENCES 78

2024
[48]

Fredholm properties of a class of elliptic operators on non- compact manifolds

R. B. Lockhart. “Fredholm properties of a class of elliptic operators on non- compact manifolds”. In: Duke Mathematical Journal 48.1 (1981), pp. 289–312

1981
[49]

Loewner and L

C. Loewner and L. Nirenberg. Partial differential equations invariant under con- formal or projective transformations . Contribut. to Analysis, Collect. of Papers dedicated to Lipman Bers, 245-272 (1974). 1974

1974
[50]

Isolated singularities of solutions to the Yamabe equation

F. C. Marques. “Isolated singularities of solutions to the Yamabe equation”. In: Calc. Var. Partial Differ. Equ. 32.3 (2008), pp. 349–371

2008
[51]

A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature

D. Maxwell. “A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature”. In: Math. Res. Lett. 16 (2009), pp. 627– 645

2009
[52]

Rough solutions of the Einstein constraint equations on compact manifolds

D. Maxwell. “Rough solutions of the Einstein constraint equations on compact manifolds”. In: J. Hyperbolic Differ. Equ. 2.2 (2005), pp. 521–546

2005
[53]

Solutions of the Einstein constraint equations with apparent hori- zon boundaries

D. Maxwell. “Solutions of the Einstein constraint equations with apparent hori- zon boundaries”. In: Commun. Math. Phys. 253.3 (2005), pp. 561–583

2005
[54]

Constant scalar curvature metrics with isolated sin- gularities

R. Mazzeo and F. Pacard. “Constant scalar curvature metrics with isolated sin- gularities”. In: Duke Math. J. 99.3 (1999), pp. 353–418

1999
[55]

Moduli spaces of singular Yamabe metrics

R. Mazzeo, D. Pollack, and K. Uhlenbeck. “Moduli spaces of singular Yamabe metrics”. In: J. Am. Math. Soc. 9.2 (1996), pp. 303–344

1996
[56]

The behavior of the laplacian on weighted sobolev spaces

R. C. McOwen. “The behavior of the laplacian on weighted sobolev spaces”. In: Communications on Pure and Applied Mathematics 32.6 (1979), pp. 783–795

1979
[57]

Charles Bradfield jun. Morrey. Multiple integrals in the calculus of variations . Vol. 130. Grundlehren Math. Wiss. Springer, Cham, 1966

1966
[58]

The null spaces of elliptic partial differential operators in Rn

L. Nirenberg and H. F. Walker. “The null spaces of elliptic partial differential operators in Rn”. In: Journal of Mathematical Analysis and Applications 42.2 (1973), pp. 271–301

1973
[59]

R. S. Palais. Foundations of global non-linear analysis . Math. Lect. Note Ser. The Benjamin/Cummings Publishing Company, Reading, MA, 1968

1968
[60]

The connections between the order of smooth- ness of a surface and its metric

I. Kh. Sabitov and S. Z. Shefel’. “The connections between the order of smooth- ness of a surface and its metric”. In: Sib. Math. J. 17 (1977), pp. 687–694

1977
[61]

Conformal deformation of a Riemannian metric to constant scalar curvature

R. Schoen. “Conformal deformation of a Riemannian metric to constant scalar curvature”. In: J. Differ. Geom. 20 (1984), pp. 479–495

1984
[62]

Conformally flat manifolds, Kleinian groups and scalar curvature

R. Schoen and S-T. Yau. “Conformally flat manifolds, Kleinian groups and scalar curvature”. In: Invent. Math. 92.1 (1988), pp. 47–71

1988
[63]

Schoen and S-T

R. Schoen and S-T. Yau. Lectures on differential geometry . Vol. 1. Conf. Proc. Lect. Notes Geom. Topol. Cambridge, MA: International Press, 1994

1994
[64]

The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation

R. M. Schoen. “The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation”. In: Commun. Pure Appl. Math. 41.3 (1988), pp. 317–392

1988
[65]

Pathological solutions of elliptic differential equations

J. Serrin. “Pathological solutions of elliptic differential equations”. In: Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 18 (1964), pp. 385–387

1964
[66]

Best constant in Sobolev inequality

G. Talenti. “Best constant in Sobolev inequality”. In: Ann. Mat. Pura Appl. (4) 110 (1976), pp. 353–372. REFERENCES 79

1976
[67]

Existence and regularity of isometries

M. Taylor. “Existence and regularity of isometries”. In: Trans. Am. Math. Soc. 358.6 (2006), pp. 2415–2423

2006
[68]

M. E. Taylor. Partial differential equations. I: Basic theory . 2nd ed. Vol. 115. Appl. Math. Sci. New York, NY: Springer, 2011

2011
[69]

M. E. Taylor. Partial differential equations. III: Nonlinear equations. 2nd ed. Vol. 117. Appl. Math. Sci. New York, NY: Springer, 2011

2011
[70]

M. E. Taylor. Tools for PDE. Pseudodifferential operators, paradifferential opera- tors, and layer potentials. Vol. 81. Math. Surv. Monogr. Providence, RI: American Mathematical Society (AMS), 2000

2000
[71]

H. Triebel. Interpolation theory, function spaces, differential operators. 2nd ed. Leipzig: Barth, 1995

1995
[72]

Linear elliptic operators with measurable coefficients

N. S. Trudinger. “Linear elliptic operators with measurable coefficients”. In: Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 27 (1973), pp. 265–308

1973
[73]

Remarks concerning the conformal deformation of Riemannian structures on compact manifolds

N. S. Trudinger. “Remarks concerning the conformal deformation of Riemannian structures on compact manifolds”. In: Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 22 (1968), pp. 265–274

1968
[74]

Isolated singularities of solutions to the Yamabe equa- tion in dimension 6

J. Xiong and L. Zhang. “Isolated singularities of solutions to the Yamabe equa- tion in dimension 6”. In: Int. Math. Res. Not. 2022.12 (2022), pp. 9571–9597

2022
[75]

On a deformation of Riemannian structures on compact manifolds

H. Yamabe. “On a deformation of Riemannian structures on compact manifolds”. In: Osaka Math. J. 12 (1960), pp. 21–37

1960
[76]

The Yamabe problem for distributional curvature

H. Zhang. “The Yamabe problem for distributional curvature”. In: J. Geom. Anal. 33.10 (2023), p. 33. Eberhard Karls Universit¨at T¨ubingen, F achbereich Mathematik, Auf der Morgen- stelle 10, 72076 T¨ubingen, Germany Email address: rodrigo.avalos@mnf.uni-tuebingen.de Email add...

2023
[2023]

arXiv: 2306.15842 [math.AP]

arXiv

Pith tools

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

[1] [6]

R. Avalos. Sobolev regularity of compactified 3-manifolds and the ADM Center of Mass . 2024. arXiv: 2403.04034 [math.DG]

arXiv 2024

[2] [44]

The positive mass theorem for manifolds with distributional curvature

D. A. Lee and P. G. LeFloch. “The positive mass theorem for manifolds with distributional curvature”. In: Commun. Math. Phys. 339.1 (2015), pp. 99–120

work page 2015

[3] [1]

R. A. Adams. Sobolev spaces. Vol. 65. Pure Appl. Math., Academic Press. New York, NY, 1975

work page 1975

[4] [2]

R. A. Adams and J. J. F. Fournier. Sobolev spaces. 2nd ed. Vol. 140. Pure Appl. Math., Academic Press. New York, NY, 2003

work page 2003

[5] [3]

Equations diff´ erentielles non lin´ eaires et probl` eme de Yamabe con- cernant la courbure scalaire

T. Aubin. “Equations diff´ erentielles non lin´ eaires et probl` eme de Yamabe con- cernant la courbure scalaire”. In: J. Math. Pures Appl. (9) 55 (1976), pp. 269– 296

work page 1976

[6] [4]

Probl` emes isop´ erim´ etriques et espaces de Sobolev

T. Aubin. “Probl` emes isop´ erim´ etriques et espaces de Sobolev”. In:Journal of Differential Geometry 11.4 (1976), pp. 573 –598

work page 1976

[7] [5]

T. Aubin. Some nonlinear problems in Riemannian geometry . Springer Monogr. Math. Berlin: Springer, 1998

work page 1998

[8] [7]

A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature

R. H. Bamler. “A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature”. In: Math. Res. Lett. 23.2 (2016), pp. 325–337

work page 2016