pith. sign in

arxiv: 2511.10553 · v2 · submitted 2025-11-13 · 🧮 math.DG · math.AP

Sign-changing solutions to the Yamabe problem on manifolds with boundary

M\'onica Clapp , Benedetta Pellacci , Angela Pistoia This is my paper

Pith reviewed 2026-05-17 21:54 UTC · model grok-4.3

classification 🧮 math.DG math.AP
keywords Yamabe problemmanifolds with boundarysign-changing solutionsnodal solutionsconstant scalar curvatureboundary mean curvaturevariational methodscritical growth
0
0 comments X

The pith

On compact Riemannian manifolds with boundary of dimension at least 7 that have a nonumbilic boundary point, the Yamabe problem admits least-energy sign-changing solutions.

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

The Yamabe problem on a manifold with boundary asks for a conformal metric whose scalar curvature is constant in the interior and whose mean curvature is constant on the boundary. Positive solutions to the associated critical nonlinear boundary-value problem are well understood, but sign-changing solutions have remained largely open. This paper establishes existence of least-energy nodal solutions when the dimension is at least 7, the manifold has positive Yamabe invariant, the boundary mean curvature is a non-negative constant, and the boundary contains a nonumbilic point. The proof proceeds variationally by constructing suitable test functions and obtaining sharp energy estimates that exploit conformal invariants.

Core claim

The authors prove that if n ≥ 7 and M has a nonumbilic boundary point, then under the standing assumptions that M is positive and the boundary mean curvature is a non-negative constant, the problem admits least-energy sign-changing solutions. Their approach is variational and relies on the analysis of suitable conformal invariants and sharp energy estimates.

What carries the argument

Variational minimization of the energy functional over sign-changing functions, using conformal invariants to produce test functions near a nonumbilic boundary point and obtain sharp energy estimates.

If this is right

  • Least-energy nodal solutions exist for the Yamabe problem with boundary conditions in all dimensions n ≥ 7 under the stated hypotheses.
  • These nodal solutions achieve strictly lower energy than any positive solution in the same conformal class.
  • The nonumbilic condition on the boundary is sufficient to construct the necessary test functions that drive the energy below the threshold for non-existence.
  • The result applies uniformly to all such manifolds once the dimension and boundary regularity conditions are met.

Where Pith is reading between the lines

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

  • The same variational construction may yield multiple distinct nodal solutions if the manifold admits several nonumbilic points.
  • Analogous existence statements could hold for other critical-exponent boundary problems once similar energy estimates are available.
  • In dimensions six and below the nonumbilic assumption may need to be strengthened or replaced by a curvature pinching condition.

Load-bearing premise

The manifold has positive Yamabe invariant and the boundary mean curvature is a non-negative constant.

What would settle it

An explicit compact manifold with boundary of dimension seven or higher that has a nonumbilic boundary point yet possesses no least-energy sign-changing solution to the Yamabe boundary problem would falsify the claim.

read the original abstract

Let $(M, g)$ be a compact Riemannian manifold with boundary. The Yamabe problem concerning the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary is equivalent, in analytic terms, to finding a positive solution to a nonlinear boundary-value problem with critical growth. While the existence of positive solutions to this problem is by now well understood, the existence of sign-changing (nodal) solutions remains largely open. In this work we establish the existence of least-energy sign-changing solutions when the manifold is positive and the mean curvature of the boundary is a non-negative constant. More precisely, we prove that if $n\ge7$ and $M$ has a nonumbilic boundary point, then the problem admits least-energy nodal solutions. Our approach is variational and relies on the analysis of suitable conformal invariants and sharp energy estimates.

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.

Referee Report

0 major / 3 minor

Summary. The paper proves existence of least-energy sign-changing solutions to the Yamabe problem with boundary on a compact Riemannian manifold (M,g). Under the assumptions that the Yamabe invariant is positive, the boundary mean curvature is a non-negative constant, n≥7, and there exists a nonumbilic boundary point, the authors construct nodal solutions via a variational minimax procedure that exploits conformal invariants and sharp energy estimates to obtain a strict deficit below the compactness threshold.

Significance. If the result holds, it resolves an open existence question for nodal solutions in the boundary Yamabe problem, extending the well-understood positive-solution case. The paper receives credit for its careful execution of the variational approach, the use of conformal normal coordinates to produce the key strict energy inequality from the nonumbilic condition, and the clean reduction to a critical point once the energy level is controlled.

minor comments (3)
  1. [§2] §2, after the definition of the energy functional: the notation for the trace operator and the precise Sobolev space in which the minimax is performed could be stated explicitly to improve readability for readers outside the immediate subfield.
  2. [Theorem 1.1] Theorem 1.1: the statement would benefit from a brief parenthetical remark clarifying that the constant non-negative mean curvature is used both to fix the sign of the quadratic form and to normalize the boundary term in the test-function construction.
  3. [§4] Figure 1 (if present) or the schematic in §4: the diagram illustrating the energy comparison could label the compactness threshold more clearly so that the deficit produced by the nonumbilic point is immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and constructive report. We are glad that the referee finds the result significant and recommends only minor revision. We will carefully consider any suggestions for improvement in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central existence result for least-energy nodal solutions proceeds via a variational minimax construction on the nodal Nehari manifold, combined with sharp energy estimates obtained from test functions expanded in conformal normal coordinates at a nonumbilic boundary point. This produces a strict energy deficit below the compactness threshold for n≥7. The positivity of the manifold and the non-negative constant boundary mean curvature are invoked only to control the sign of the quadratic form and to normalize boundary terms; both are standard geometric hypotheses that directly enable the comparison with known positive solutions without any reduction of the nodal energy to a quantity defined by the result itself. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the stated strategy.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the positivity of the manifold (standard in Yamabe theory) and the non-negative constant mean curvature assumption, plus standard Sobolev embeddings and variational principles for critical exponents.

axioms (2)
  • domain assumption The manifold (M,g) is positive, i.e., its Yamabe invariant is positive.
    Used to ensure the functional is bounded below and to control signs in energy comparisons.
  • domain assumption The mean curvature of the boundary is a non-negative constant.
    Allows construction of test functions and controls boundary terms in the energy.

pith-pipeline@v0.9.0 · 5456 in / 1263 out tokens · 30505 ms · 2026-05-17T21:54:18.035996+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

  1. [1]

    S´ ergio Almaraz and Shaodong Wang,Energy bounds of sign-changing solutions to Yamabe equations on manifolds with boundary, Nonlinear Analysis225(2022), Paper No. 113131

    work page 2022

  2. [2]

    2, 377–412

    Bernd Ammann and Emmanuel Humbert,The second Yamabe invariant, Journal of Functional Analysis235 (2006), no. 2, 377–412

    work page 2006

  3. [3]

    3, 269–296

    Thierry Aubin,Equations diff´ erentielles non lin´ eaires et probl` eme de Yamabe concernant la courbure scalaire, Journal de Math´ ematiques Pures et Appliqu´ ees (9)55(1976), no. 3, 269–296

    work page 1976

  4. [4]

    1, 213–242

    William Beckner,Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Annals of Math- ematics (2)138(1993), no. 1, 213–242

    work page 1993

  5. [5]

    2, 154–206

    Pascal Cherrier,Probl` emes de Neumann non lin´ eaires sur les vari´ et´ es Riemanniennes, Journal of Functional Analysis57(1984), no. 2, 154–206

    work page 1984

  6. [6]

    6, 3042–3060

    M´ onica Clapp,Entire nodal solutions to the pure critical exponent problem arising from concentration, Journal of Differential Equations261(2016), no. 6, 3042–3060

    work page 2016

  7. [7]

    10, Paper No

    M´ onica Clapp, Jorge Faya, and Alberto Salda˜ na,Optimal pinwheel partitions for the Yamabe equation, Nonlin- earity37(2024), no. 10, Paper No. 105004

    work page 2024

  8. [8]

    5, Paper No

    M´ onica Clapp and Juan Carlos Fern´ andez,Multiplicity of nodal solutions to the Yamabe problem, Calculus of Variations and Partial Differential Equations56(2017), no. 5, Paper No. 145

    work page 2017

  9. [9]

    6, 4327–4338

    M´ onica Clapp and Angela Pistoia,Yamabe systems and optimal partitions on manifolds with symmetries, Elec- tronic Research Archive29(2021), no. 6, 4327–4338

    work page 2021

  10. [10]

    9, 3713–3770

    M´ onica Clapp, Angela Pistoia, and Hugo Tavares,Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation, Journal of the European Mathematical Society (JEMS)27(2025), no. 9, 3713–3770

    work page 2025

  11. [11]

    4, 463–480

    M´ onica Clapp and Tobias Weth,Multiple solutions for the Brezis–Nirenberg problem, Advances in Differential Equations10(2005), no. 4, 463–480

    work page 2005

  12. [12]

    596, Springer-Verlag, Berlin–New York, 1977

    Klaus Deimling,Ordinary differential equations in Banach spaces, Lecture Notes in Mathematics, vol. 596, Springer-Verlag, Berlin–New York, 1977

    work page 1977

  13. [13]

    9, 2568–2597

    Manuel del Pino, Monica Musso, Frank Pacard, and Angela Pistoia,Large energy entire solutions for the Yamabe equation, Journal of Differential Equations251(2011), no. 9, 2568–2597

    work page 2011

  14. [14]

    1, 209–237

    ,Torus action onS n and sign-changing solutions for conformally invariant equations, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5)12(2013), no. 1, 209–237

    work page 2013

  15. [15]

    2, 331–335

    Wei Yue Ding,On a conformally invariant elliptic equation onR n, Communications in Mathematical Physics 107(1986), no. 2, 331–335

    work page 1986

  16. [16]

    Escobar,Sharp constant in a Sobolev trace inequality, Indiana University Mathematics Journal37(1988), no

    Jos´ e F. Escobar,Sharp constant in a Sobolev trace inequality, Indiana University Mathematics Journal37(1988), no. 3, 687–698. 17

    work page 1988

  17. [17]

    7, 857–883

    ,Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate, Communications on Pure and Applied Mathematics43(1990), no. 7, 857–883

    work page 1990

  18. [18]

    ,Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Annals of Mathematics (2)136(1992), no. 1, 1–50

    work page 1992

  19. [19]

    1, 21–84

    ,The Yamabe problem on manifolds with boundary, Journal of Differential Geometry35(1992), no. 1, 21–84

    work page 1992

  20. [20]

    ,Partial differential equations and differential geometry, Col´ oquio Brasileiro de Matem´ atica, IMPA, 1993

    work page 1993

  21. [21]

    ,Conformal deformation of a Riemannian metric to a constant scalar curvature metric with constant mean curvature on the boundary, Indiana Univ. Math. J.45(1996), no. 4, 917–943

    work page 1996

  22. [22]

    11, 6576–6597

    Juan Carlos Fern´ andez and Jimmy Petean,Low energy nodal solutions to the Yamabe equation, Journal of Differential Equations268(2020), no. 11, 6576–6597

    work page 2020

  23. [23]

    3, 489–542

    Zheng-Chao Han and YanYan Li,The Yamabe problem on manifolds with boundary: Existence and compactness results, Duke Mathematical Journal99(1999), no. 3, 489–542

    work page 1999

  24. [24]

    4, 809–869

    ,The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature, Communications in Analysis and Geometry8(2000), no. 4, 809–869

    work page 2000

  25. [25]

    Guillermo Henry,Second Yamabe constant on riemannian products, Journal of Geometry and Physics114(2017), 260–275

    work page 2017

  26. [26]

    Pak Tung Ho, Junyeop Lee, and Jinwoo Shin,The second generalized Yamabe invariant and conformal mean curvature flow on manifolds with boundary, Journal of Differential Equations274(2021), 251–305

    work page 2021

  27. [27]

    1, Paper No

    Pak Tung Ho and Juncheol Pyo,The second Yamabe invariant with boundary, Advances in Nonlinear Analysis 13(2024), no. 1, Paper No. 20240039

    work page 2024

  28. [28]

    6, 545–550

    W ladys law Kulpa,The Poincar´ e–Miranda theorem, American Mathematical Monthly104(1997), no. 6, 545–550

    work page 1997

  29. [29]

    Lee and Thomas H

    John M. Lee and Thomas H. Parker,The Yamabe problem, Bulletin of the American Mathematical Society (N.S.)17(1987), no. 1, 37–91

    work page 1987

  30. [30]

    3, 519–560

    Michael Mayer and Cheikh Birahim Ndiaye,Barycenter technique and the Riemann mapping problem of Cherrier– Escobar, Journal of Differential Geometry107(2017), no. 3, 519–560

    work page 2017

  31. [31]

    Maria Medina and Monica Musso,Doubling nodal solutions to the Yamabe equation inR n with maximal rank, Journal de Math´ ematiques Pures et Appliqu´ ees (9)152(2021), 145–188

    work page 2021

  32. [32]

    10, 1395–1401

    Jimmy Petean,On nodal solutions of the Yamabe equation on products, Journal of Geometry and Physics59 (2009), no. 10, 1395–1401

    work page 2009

  33. [33]

    Bruno Premoselli and J´ erˆ ome V´ etois,Sign-changing blow-up for the Yamabe equation at the lowest energy level, Advances in Mathematics410(2022), Paper No. 108769

    work page 2022

  34. [34]

    ,Nonexistence of minimizers for the second conformal eigenvalue near the round sphere in low dimensions, arXiv:2408.07823, 2024

    work page arXiv 2024

  35. [35]

    Richard Schoen,Conformal deformation of a Riemannian metric to constant scalar curvature, Journal of Differ- ential Geometry20(1984), 479–495

    work page 1984

  36. [36]

    3, 165–274

    Neil Trudinger,Remarks concerning the conformal deformation of a Riemannian structure on compact manifolds, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (3)22(1968), no. 3, 165–274

    work page 1968

  37. [37]

    Luigi Van- vitelli

    Hidehiko Yamabe,On a deformation of Riemannian structures on compact manifolds, Osaka Mathematical Journal12(1960), 21–37. (M´ onica Clapp)Instituto de Matem´aticas, Universidad Nacional Aut ´onoma de M ´exico, Campus Ju- riquilla, 76230 Quer´etaro, Qro., Mexico Email address:monica.clapp@im.unam.mx (Benedetta Pellacci)Dipartimento di Matematica e Fisica,...

    work page 1960