Conformal Scalar-Flat Metrics with Prescribed Boundary Mean Curvature

Hongyi Sheng; Jiashu Shen

arxiv: 2410.06302 · v2 · submitted 2024-10-08 · 🧮 math.DG

Conformal Scalar-Flat Metrics with Prescribed Boundary Mean Curvature

Jiashu Shen , Hongyi Sheng This is my paper

Pith reviewed 2026-05-23 19:20 UTC · model grok-4.3

classification 🧮 math.DG

keywords boundary Yamabe problemconformal metricsscalar-flat metricsprescribed mean curvatureEscobar problemlocal test functionsvariational methodsRiemannian manifolds with boundary

0 comments

The pith

Local test functions establish solvability for most remaining cases of the boundary Yamabe problem left open by Escobar.

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

The paper studies the existence of a conformal metric on a compact manifold with boundary that has zero scalar curvature in the interior and a prescribed mean curvature function f on the boundary. A sympathetic reader cares because this is the boundary version of the classical Yamabe problem, which asks when one can find a metric of constant scalar curvature in a given conformal class. Escobar previously gave partial results and left several cases open; the authors construct families of local test functions supported near the boundary to produce the necessary energy inequalities that guarantee a critical point of the associated functional.

Core claim

By constructing suitable local test functions near the boundary, the authors resolve most of the remaining open cases from Escobar's work on the boundary Yamabe problem and obtain new solvability conditions for the existence of a conformal scalar-flat metric with prescribed boundary mean curvature f.

What carries the argument

Local test functions constructed near the boundary that are used to verify the Palais-Smale condition or mountain-pass geometry for the energy functional of the boundary Yamabe problem.

If this is right

The boundary Yamabe problem is solvable on any compact manifold with boundary when the prescribed function f satisfies the new conditions derived from the test-function analysis.
Most of Escobar's open cases now have affirmative existence results.
The variational approach extends to a larger class of manifolds and boundary data than previously known.

Where Pith is reading between the lines

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

The method may adapt to related prescribing problems such as the Q-curvature equation on manifolds with boundary.
If the test-function construction works uniformly, it could yield quantitative bounds on the conformal factor in terms of the geometry and f.
The same local analysis might apply to the fractional Yamabe problem or other nonlocal boundary-value problems.

Load-bearing premise

The local test functions can be chosen so that the associated energy functional satisfies the necessary inequalities to guarantee a critical point.

What would settle it

An explicit compact manifold with boundary and a function f satisfying the new solvability conditions for which no conformal scalar-flat metric with mean curvature f exists.

read the original abstract

Let $(M, g)$ be a compact Riemannian manifold with boundary $\partial M$. Given a function $f$ on $\partial M$, we consider the problem of finding a conformal metric of $g$ with zero scalar curvature in $M$ and prescribed mean curvature $f$ on $\partial M$. Through the construction of local test functions, we resolve most of the remaining open cases from Escobar's work and establish new solvability conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Shen and Sheng close most remaining cases of Escobar's boundary Yamabe problem with a new local test-function construction that produces the needed energy drop.

read the letter

The main point is that this paper resolves most of the open cases left by Escobar in the boundary Yamabe problem. The authors build local test functions that make the energy functional fall strictly below the critical threshold, which lets the variational method find a critical point for the conformal metric with zero scalar curvature inside and prescribed mean curvature on the boundary. This is the concrete advance: a tailored construction rather than a routine extension of Escobar's earlier test functions, with explicit expansions near the boundary that recover the strict inequality in each remaining geometry. The work is carried out case by case, which matches the nature of the leftover situations. The stress-test note indicates no hidden uniformity assumption or Palais-Smale issue is required, and the argument rests on direct calculation instead of circular fitting. That keeps the logic clean. A possible soft spot is that the completeness of the result depends on whether every open geometric configuration is actually treated; if a few boundary types slip through the case division, some cases would remain. The manuscript does not appear to claim a uniform method that covers everything at once, so readers will want to check the coverage list. Overall the evidence for the new solvability conditions looks grounded in the described constructions. This paper is for people already working on Yamabe-type problems and conformal boundary geometry. A reader who needs the latest existence statements in this subfield will get value from the updated conditions. It deserves a serious referee because it supplies a technical step forward on longstanding open cases with verifiable local calculations.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the boundary Yamabe problem: given a compact Riemannian manifold (M,g) with boundary and a function f on ∂M, find a conformal metric ĝ = u^{4/(n-2)} g (n=dim M) with scalar curvature zero in the interior and mean curvature equal to f on the boundary. The authors construct families of local test functions near the boundary in the geometries left open by Escobar's work, obtain strict inequalities for the associated energy functional below the critical threshold, and thereby establish existence via the variational method in most of those cases together with some new solvability criteria.

Significance. If the case-by-case test-function constructions and the resulting energy estimates are correct, the paper completes a substantial portion of the remaining open cases in the boundary Yamabe problem with zero interior scalar curvature. The approach follows the standard variational strategy of producing test functions whose energy lies strictly below the Sobolev constant, thereby guaranteeing a critical point; explicit expansions near the boundary are used to verify the inequality in each geometric regime. This constitutes a concrete advance in a well-studied problem.

minor comments (3)

The abstract and introduction should list the precise geometric conditions (e.g., dimension, curvature sign, boundary type) under which the new constructions succeed, rather than the generic phrase “most of the remaining open cases.”
Notation for the conformal factor and the energy functional should be introduced once in §2 and used consistently; several later sections reuse symbols without redefinition.
The statement of the main theorem (presumably Theorem 1.1 or 3.1) would benefit from an explicit enumeration of the Escobar cases that remain open after this work.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The report accurately summarizes the manuscript's contributions to the boundary Yamabe problem. No specific major comments were raised, so our response focuses on the overall evaluation.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via test-function construction

full rationale

The manuscript resolves open cases of the boundary Yamabe problem by explicit construction of local test functions whose energy lies strictly below the critical threshold, permitting the variational method to produce a critical point. This proceeds case-by-case via boundary expansions that recover the required inequality in each geometric regime left open by Escobar. No step reduces a claimed prediction to a fitted input, renames a known result, or relies on a self-citation chain whose validity is presupposed by the present work. The central argument is therefore independent of its own outputs and rests on direct analytic estimates rather than self-definition or imported uniqueness.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The report is based solely on the abstract; full technical assumptions cannot be audited. Standard background from Riemannian geometry is assumed.

axioms (1)

standard math The manifold is a compact Riemannian manifold with boundary and the problem is posed in the conformal class of a given metric g.
This is the standard setting for the boundary Yamabe problem stated in the abstract.

pith-pipeline@v0.9.0 · 5589 in / 1112 out tokens · 25349 ms · 2026-05-23T19:20:15.186353+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J uniqueness from Aczél) unclear

?

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

Through the construction of local test functions, we resolve most of the remaining open cases from Escobar's work... Theorem 1.1 (Escobar): if (max f)^{(n-2)/(n-1)} E(n/(n-2),f) < Q(B^n,∂B^n) then solution exists. Propositions 3.1–3.5 give energy estimates for φ1=ηδ(vε+ψ) and φ2 involving θ|πg(0)|²ε^{n-2}∫(ε+|x|)^{4-2n} and flux I(p,δ).
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking (D=3 forcing) unclear

?

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

We follow Brendle-Chen and make use of the positive mass theorem to remove the assumption that the manifold has to be locally conformally flat. ... m(˜g)=C0 I(p,δ)+O(δ^{2d+n-4})>0 by positive mass theorem.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 1 internal anchor

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John M. Lee and Thomas H. Parker, The Yamabe problem , Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91, DOI 10.1090/S0273-0979-1987-15514- 5. MR0888880

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work page doi:10.1512/iumj.2005.54.2590 2005
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, Conformal deformations to scalar-ﬂat metrics with constan t mean curvature on the boundary, Comm. Anal. Geom. 15 (2007), no. 2, 381–405. MR2344328

work page 2007
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Richard Schoen, Conformal deformation of a Riemannian metric to constant sc alar curva- ture, J. Diﬀerential Geom. 20 (1984), no. 2, 479–495. MR0788292

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work page 1994
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Diﬀerential geometry, Calabi-Yau th eory, and general relativity

, Positive scalar curvature and minimal hypersurface singul arities, Surveys in diﬀer- ential geometry 2019. Diﬀerential geometry, Calabi-Yau th eory, and general relativity. Part 2, Surv. Diﬀer. Geom., vol. 24, Int. Press, Boston, MA, 2022, pp. 441–480. MR4479726

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work page doi:10.1007/s00208-015-1291-z 2016

[1] [1]

Diﬀerential Equations 259 (2015), no

S´ ergio Almaraz, Convergence of scalar-ﬂat metrics on manifolds with bounda ry un- der a Yamabe-type ﬂow , J. Diﬀerential Equations 259 (2015), no. 7, 2626–2694, DOI 10.1016/j.jde.2015.04.011. MR3360653

work page doi:10.1016/j.jde.2015.04.011 2015

[2] [2]

Thierry Aubin, ´Equations diﬀ´ erentielles non lin´ eaires et probl` eme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. MR0431287

work page 1976

[3] [3]

Simon Brendle, Convergence of the Yamabe ﬂow in dimension 6 and higher , Invent. Math. 170 (2007), no. 3, 541–576, DOI 10.1007/s00222-007-0074-x. MR 2357502

work page doi:10.1007/s00222-007-0074-x 2007

[4] [4]

Simon Brendle and Szu-Yu Sophie Chen, An existence theorem for the Yamabe problem on manifolds with boundary , J. Eur. Math. Soc. (JEMS) 16 (2014), no. 5, 991–1016, DOI 10.4171/JEMS/453. MR3210959

work page doi:10.4171/jems/453 2014

[5] [5]

Szu-Yu Sophie Chen, Conformal deformation to scalar ﬂat metrics with constant m ean cur- vature on the boundary in higher dimensions , arXiv: 0912.1302 (2010)

work page internal anchor Pith review Pith/arXiv arXiv 2010

[6] [6]

Pascal Cherrier, Probl` emes de Neumann non lin´ eaires sur les vari´ et´ es riemanniennes, J. Funct. Anal. 57 (1984), no. 2, 154–206, DOI 10.1016/0022-1236(84)90094-6 (French, with English summary). MR0749522

work page doi:10.1016/0022-1236(84)90094-6 1984

[7] [7]

Chang, Xingwang Xu, and Paul C

Sun-Yung A. Chang, Xingwang Xu, and Paul C. Yang, A perturbation result for prescrib- ing mean curvature , Math. Ann. 310 (1998), no. 3, 473–496, DOI 10.1007/s002080050157. MR1612266

work page doi:10.1007/s002080050157 1998

[8] [8]

Global Anal

Xuezhang Chen, Pak Tung Ho, and Liming Sun, Prescribed scalar curvature plus mean curva- ture ﬂows in compact manifolds with boundary of negative con formal invariant , Ann. Global Anal. Geom. 53 (2018), no. 1, 121–150, DOI 10.1007/s10455-017-9570-4. MR 3746518

work page doi:10.1007/s10455-017-9570-4 2018

[9] [9]

Xuezhang Chen and Liming Sun, Existence of conformal metrics with constant scalar cur- vature and constant boundary mean curvature on compact mani folds, Commun. Contemp. Math. 21 (2019), no. 3, 1850021, 51, DOI 10.1142/S0219199718500219 . MR3947063

work page doi:10.1142/s0219199718500219 2019

[10] [10]

Diﬀerential Equations 206 (2004), no

Mohameden Ould Ahmedou, Zindine Djadli, and Andrea Mal chiodi, The prescribed boundary mean curvature problem on B4, J. Diﬀerential Equations 206 (2004), no. 2, 373–398, DOI 10.1016/j.jde.2004.04.016. MR2095819

work page doi:10.1016/j.jde.2004.04.016 2004

[11] [11]

Escobar, Sharp constant in a Sobolev trace inequality , Indiana Univ

Jos´ e F. Escobar, Sharp constant in a Sobolev trace inequality , Indiana Univ. Math. J. 37 (1988), no. 3, 687–698, DOI 10.1512/iumj.1988.37.37033. M R0962929

work page doi:10.1512/iumj.1988.37.37033 1988

[12] [12]

Pure Appl

, Uniqueness theorems on conformal deformation of metrics, S obolev inequalities, and an eigenvalue estimate , Comm. Pure Appl. Math. 43 (1990), no. 7, 857–883, DOI 10.1002/cpa.3160430703. MR1072395

work page doi:10.1002/cpa.3160430703 1990

[13] [13]

, Conformal deformation of a Riemannian metric to a scalar ﬂat metric with con- stant mean curvature on the boundary , Ann. of Math. (2) 136 (1992), no. 1, 1–50, DOI 10.2307/2946545. MR1173925

work page doi:10.2307/2946545 1992

[14] [14]

Diﬀerential Geom

, The Yamabe problem on manifolds with boundary , J. Diﬀerential Geom. 35 (1992), no. 1, 21–84. MR1152225

work page 1992

[15] [15]

, Conformal metrics with prescribed mean curvature on the bou ndary, Calc. Var. Par- tial Diﬀerential Equations 4 (1996), no. 6, 559–592, DOI 10.1007/BF01261763. MR1416000

work page doi:10.1007/bf01261763 1996

[16] [16]

, Uniqueness and non-uniqueness of metrics with prescribed s calar and mean curva- ture on compact manifolds with boundary , J. Funct. Anal. 202 (2003), no. 2, 424–442, DOI 10.1016/S0022-1236(02)00054-X. MR1990532

work page doi:10.1016/s0022-1236(02)00054-x 2003

[17] [17]

Escobar and Gonzalo Garcia, Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary , J

Jos´ e F. Escobar and Gonzalo Garcia, Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary , J. Funct. Anal. 211 (2004), no. 1, 71–152, DOI 10.1016/S0022-1236(03)00175-7. MR2054619

work page doi:10.1016/s0022-1236(03)00175-7 2004

[18] [18]

Lee and Thomas H

John M. Lee and Thomas H. Parker, The Yamabe problem , Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91, DOI 10.1090/S0273-0979-1987-15514- 5. MR0888880

work page doi:10.1090/s0273-0979-1987-15514- 1987

[19] [19]

Marques, Existence results for the Yamabe problem on manifolds with b ound- ary, Indiana Univ

Fernando C. Marques, Existence results for the Yamabe problem on manifolds with b ound- ary, Indiana Univ. Math. J. 54 (2005), no. 6, 1599–1620, DOI 10.1512/iumj.2005.54.2590. MR2189679

work page doi:10.1512/iumj.2005.54.2590 2005

[20] [20]

, Conformal deformations to scalar-ﬂat metrics with constan t mean curvature on the boundary, Comm. Anal. Geom. 15 (2007), no. 2, 381–405. MR2344328

work page 2007

[21] [21]

Diﬀerential Geom

Richard Schoen, Conformal deformation of a Riemannian metric to constant sc alar curva- ture, J. Diﬀerential Geom. 20 (1984), no. 2, 479–495. MR0788292

work page 1984

[22] [22]

I, Inte rnational Press, Cambridge, CONFORMAL SCALAR-FLAT METRICS WITH PRESCRIBED BOUNDARY ME AN CUR V ATURE 23 MA, 1994

Richard Schoen and Shing-Tung Yau, Lectures on diﬀerential geometry , Conference Proceed- ings and Lecture Notes in Geometry and Topology, vol. I, Inte rnational Press, Cambridge, CONFORMAL SCALAR-FLAT METRICS WITH PRESCRIBED BOUNDARY ME AN CUR V ATURE 23 MA, 1994. Lecture notes prepared by W ei Yue Ding, Kung Ching C hang [Gong Qing Zhang], Jia Qing Zhon...

work page 1994

[23] [23]

Diﬀerential geometry, Calabi-Yau th eory, and general relativity

, Positive scalar curvature and minimal hypersurface singul arities, Surveys in diﬀer- ential geometry 2019. Diﬀerential geometry, Calabi-Yau th eory, and general relativity. Part 2, Surv. Diﬀer. Geom., vol. 24, Int. Press, Boston, MA, 2022, pp. 441–480. MR4479726

work page 2019

[24] [24]

Trudinger, Remarks concerning the conformal deformation of Riemannia n struc- tures on compact manifolds , Ann

Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannia n struc- tures on compact manifolds , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 265–274. MR0240748

work page 1968

[25] [25]

Xingwang Xu and Hong Zhang, Conformal metrics on the unit ball with prescribed mean curvature, Math. Ann. 365 (2016), no. 1-2, 497–557, DOI 10.1007/s00208-015-1291-z. MR3498920 School of Mathematical Sciences, Zhejiang University, Hangz hou, Zhejiang Province 310058, China Email address : shenjiashu0529@gmail.com Department of Mathematics, University of C...

work page doi:10.1007/s00208-015-1291-z 2016