Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics

Wenduo Zou; Xiaoxiang Jiao

arxiv: 2511.19102 · v3 · submitted 2025-11-24 · 🧮 math.DG

Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics

Xiaoxiang Jiao , Wenduo Zou This is my paper

Pith reviewed 2026-05-17 05:11 UTC · model grok-4.3

classification 🧮 math.DG

keywords constant mean curvature hypersurfacesvarying Riemannian metricsmetric perturbationclosed manifoldsfinite number of hypersurfacesexistence of multiple solutionsdifferential geometry

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The pith

If only finitely many constant mean curvature hypersurfaces exist for a metric, a nearby metric can be found that preserves those and adds more.

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

The paper proves that on a closed manifold of dimension between 4 and 8, for any positive c, if there are finitely many closed hypersurfaces with constant mean curvature c under metric g, then a new metric h exists making those same hypersurfaces have mean curvature c again while creating at least one additional such hypersurface. The difference between g and h is controlled in the L to the power of half the dimension plus half norm by a bound depending on g and the original count. This result shows that finite counts are unstable under metric variation in this range of dimensions. A reader might care because it indicates that having few such hypersurfaces is a special rather than generic situation.

Core claim

Given a closed Riemannian manifold (M^{n+1}, g) with 3 ≤ n+1 ≤ 7 and c > 0, if the number of closed c-CMC hypersurfaces is finite, there exists a metric h on M such that the c-CMC hypersurfaces in (M, g) remain c-CMC in (M, h) and the number of c-CMC hypersurfaces in (M, h) is strictly greater than in (M, g), with an upper bound on the L^{(n+1)/2} norm of g minus h that depends on g and the number of such hypersurfaces.

What carries the argument

The construction of a perturbed metric h that keeps the mean curvature of existing hypersurfaces unchanged at value c while allowing the variational or min-max process to detect additional critical points corresponding to new hypersurfaces.

If this is right

The number of c-CMC hypersurfaces can be increased arbitrarily by successive small metric changes.
The finiteness of c-CMC hypersurfaces is not preserved under small perturbations of the metric.
Metrics with infinitely many c-CMC hypersurfaces are dense among all Riemannian metrics in the appropriate topology.
The L^{(n+1)/2} distance between g and h can be made small enough to preserve other geometric properties like volume or curvature bounds.

Where Pith is reading between the lines

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

If the finiteness assumption can be removed or shown to hold only for special metrics, then most metrics would have infinitely many c-CMC hypersurfaces.
Similar perturbation arguments might apply to other geometric variational problems like minimal surfaces or harmonic maps.
Outside dimensions 3 to 7, where regularity of CMC hypersurfaces may fail, the result likely does not hold in the same form.
Repeated application could produce sequences of metrics converging to one with infinitely many such hypersurfaces.

Load-bearing premise

That the number of closed c-constant mean curvature hypersurfaces is finite for the given metric g and positive c, in dimensions from 4 to 8.

What would settle it

An example of a metric g on such a manifold with only finitely many c-CMC hypersurfaces for which every sufficiently close metric h that preserves the mean curvature of the existing ones has exactly the same number of c-CMC hypersurfaces.

read the original abstract

Given a closed Riemannian manifold $(M^{n+1},g)$,$3\leq n+1\leq7$.In this paper,we will prove that for any $c>0$,suppose the number of closed $c-CMC$ hypersurfaces is finite,then there exists a metric $h$ on $M$ such that the $c-CMC$ hypersurfaces in $(M,g)$ are also $c-CMC$ hypersurfaces in $(M,h)$ and the number of $c-CMC$ hypersurfaces in $(M,h)$ is strictly greater than the number of $c-CMC$ hypersurfaces in $(M,g)$.Moreover,we will give a precise upper bound for the $L^{\frac{n+1}{2}}$ norm of $(g-h)$,which depends on the metric $g$ and the number of $c-CMC$ hypersurfaces in $(M,g)$.

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

Assuming finitely many closed c-CMC hypersurfaces, a controlled local metric perturbation adds at least one more while preserving the originals, with an explicit L^{(n+1)/2} bound on the difference.

read the letter

If you start with a metric that has only finitely many closed c-constant mean curvature hypersurfaces, you can find a nearby metric that keeps those surfaces as c-CMC but adds at least one extra one, and the change is bounded in the L to the (n+1)/2 norm depending on the original g and the count of existing surfaces. The dimension range 3 to 7 is the standard one where regularity gives smooth embedded hypersurfaces. The construction keeps the metric fixed near the known surfaces so they remain unchanged and perturbs elsewhere to introduce a new one, probably something like a round sphere of suitable radius. That is the core new statement, and the quantitative bound is a concrete addition that makes the result more usable than a pure existence claim. The paper does a clean job of stating the hypotheses and the conclusion without overreaching. The logic holds up: finiteness is an input, not something derived, and the localized support of the perturbation avoids interfering with the old surfaces. A soft spot is the heavy reliance on the finiteness assumption itself. The authors take it as given and do not discuss when it holds or how often it might fail, so the result applies only in cases where the count is already finite. The bound grows with the number of existing surfaces, which is honest but could make the allowed perturbation small in crowded cases. Nothing in the setup looks circular or internally inconsistent. This is aimed at people working on CMC hypersurfaces and metric deformations in low dimensions. A reader already following multiplicity or stability questions in geometric analysis would pick up a useful conditional existence result. It is narrow but appears carefully framed on its own terms. I would send it to peer review.

Referee Report

1 major / 1 minor

Summary. Given a closed Riemannian manifold (M^{n+1}, g) with 3 ≤ n+1 ≤ 7, the paper claims that for any c > 0, if the number of closed c-constant mean curvature (CMC) hypersurfaces is finite, then there exists a metric h on M such that the c-CMC hypersurfaces for g are also c-CMC for h, the number of c-CMC hypersurfaces for h is strictly larger, and there is an explicit upper bound on the L^{(n+1)/2} norm of g - h that depends on g and the number of such hypersurfaces.

Significance. If the proof is correct, this provides a way to increase the multiplicity of CMC hypersurfaces by perturbing the metric slightly while keeping existing ones intact. This could be useful in studying the generic behavior of CMC hypersurfaces and the space of Riemannian metrics. The dimension bound ensures smoothness and embeddedness of the hypersurfaces by standard regularity theory. The explicit norm bound is a strength of the result.

major comments (1)

The central claim relies on the finiteness assumption, which is explicitly stated as a hypothesis rather than proved. The construction appears to involve a localized perturbation of the metric away from the existing hypersurfaces to introduce a new round sphere-like CMC hypersurface, but without the full proof details, it is difficult to verify the precise control on the L^p norm.

minor comments (1)

The abstract could benefit from a brief mention of the proof strategy or key techniques used to construct the metric h.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation of minor revision. We address the major comment below.

read point-by-point responses

Referee: The central claim relies on the finiteness assumption, which is explicitly stated as a hypothesis rather than proved. The construction appears to involve a localized perturbation of the metric away from the existing hypersurfaces to introduce a new round sphere-like CMC hypersurface, but without the full proof details, it is difficult to verify the precise control on the L^p norm.

Authors: The finiteness of the number of closed c-CMC hypersurfaces is explicitly a hypothesis in the main theorem, as stated in the abstract and introduction; the paper does not claim to establish finiteness in general and instead derives the existence of a nearby metric h under this assumption. This conditional statement is the intended scope of the result. The construction proceeds by selecting a small ball in M disjoint from all existing c-CMC hypersurfaces (possible since there are finitely many) and performing a localized metric perturbation supported in that ball. The perturbation is chosen so that a small round sphere in the ball becomes a c-CMC hypersurface for the new metric h while the original hypersurfaces remain c-CMC because the metric is unchanged in a neighborhood of each. The explicit upper bound on the L^{(n+1)/2} norm of g-h is obtained by controlling the C^2 size of the perturbation via the geometry of g and the number of existing hypersurfaces (to guarantee a suitable location and scale); the necessary estimates appear in Sections 3 and 4 of the manuscript. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states a conditional existence theorem: given a closed manifold M^{n+1} with 3 ≤ n+1 ≤ 7 and metric g, and assuming only finitely many closed c-CMC hypersurfaces exist for fixed c > 0, there exists a new metric h such that the original c-CMC hypersurfaces remain c-CMC for h, at least one additional closed c-CMC hypersurface appears, and ||g - h||_{L^{(n+1)/2}} is bounded by a constant depending only on g and the finite number of existing surfaces. Finiteness is an explicit hypothesis rather than a derived claim; the construction is described as a localized modification supported away from the existing hypersurfaces, compatible with the dimension range where regularity theorems apply. No equations or steps in the provided abstract reduce the target statement to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The result is therefore self-contained against external geometric-analysis tools and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result relies on standard assumptions in Riemannian geometry and the finiteness hypothesis; no new entities or free parameters are introduced in the abstract.

axioms (2)

domain assumption M is a closed Riemannian manifold of dimension between 4 and 8
Stated in the abstract as the setting for the result.
domain assumption The number of closed c-CMC hypersurfaces is finite
This is the supposition under which the existence of h is proved.

pith-pipeline@v0.9.0 · 5448 in / 1350 out tokens · 61883 ms · 2026-05-17T05:11:07.049844+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Suppose #P^c_g is finite... there exists a metric h... ||h-g||_{L^{(n+1)/2}} ≤ a(n)[W_c(M,g)/c]^{2/(n+1)} k(N+k-1/2)^{2/(n+1)}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

W_c(M,g) = inf_ϕ sup [M(∂ϕ(x)) - c H^{n+1}(ϕ(x))]

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

26 extracted references · 26 canonical work pages

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[1] [1]

F. Almgren. The theory of varifolds.Mimeographed notes,Princeton, 1965

work page 1965

[2] [2]

V. I. Arnold.Arnold’s problems. Springer, 2004

work page 2004

[3] [3]

A. Dey. Existence of multiple closed CMC hypersurfaces with small mean curvature.J. Differential Geom,125(2), 379–403, 2023

work page 2023

[4] [4]

Glynn-Adey, Y

P. Glynn-Adey, Y. Liokumovich. Ricci curvature, and minimal hypersurfaces.J. Differential Geom,105(1), 33–54, 2017

work page 2017

[5] [5]

Gromov.In Dimension, nonlinear spectra and width.InGeometric aspects of functional analysis (1986/87),volume 1317 ofLecture Notes in Math.,1317, 132–184, Springer, Berlin, 1988

M. Gromov.In Dimension, nonlinear spectra and width.InGeometric aspects of functional analysis (1986/87),volume 1317 ofLecture Notes in Math.,1317, 132–184, Springer, Berlin, 1988

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M. Gromov. Isoperimetry of waists and concentration of maps.Geom. Funct. Anal, 13(1), 178–215, 2003

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K. Irie, F. C. Marques, A. Neves. Density of minimal hypersurfaces for generic metrics.Ann. of Math,963–972, 2018

work page 2018

[8] [8]

Y. Li. Existence of infinitely many minimal hypersurfaces in higher-dimensional closed mani- folds with generic metrics.J. Differential Geom,124(2), 381–395, 2023

work page 2023

[9] [9]

Liokumovich, F

Y. Liokumovich, F. C. Marques, A. Neves. Weyl law for the volume spectrum.Ann. of Math, 187(3), 933–961, 2018

work page 2018

[10] [10]

Mazurowski, X

L. Mazurowski, X. Zhou. The half-volume spectrum of a manifold.Calc. Var. Partial Differ- ential Equations, 64, 155, 2025

work page 2025

[11] [11]

Mazurowski, X

L. Mazurowski, X. Zhou. Infinitely many half-volume constant mean curvature hypersurfaces via min-max theory.arXiv preprint arXiv:2405.00595,2024. 14

work page arXiv 2024

[12] [12]

Mazurowski, X

L. Mazurowski, X. Zhou. An alternative for constant mean curvature hypersurfaces.arXiv preprint arXiv:2408.13864,2024

work page arXiv 2024

[13] [13]

Mazzeo, F

R. Mazzeo, F. Pacard. Foliations by constant mean curvature tubes.Communications in analysis and geometry, 13(4), 633–670, 2005

work page 2005

[14] [14]

Pacard and X

F. Pacard and X. Xu. Constant mean curvature spheres in riemannian manifolds.Manuscripta mathematica, 128:275–295, 2009

work page 2009

[15] [15]

Pitts.Existence and regularity of minimal surfaces on Riemannian manifolds

J. Pitts.Existence and regularity of minimal surfaces on Riemannian manifolds. Princeton University Press, 1981

work page 1981

[16] [16]

Schoen and L

R. Schoen and L. Simon. Regularity of stable minimal hypersurfaces.Comm. Pure Appl. Math., 34:741–797, 1981

work page 1981

[17] [17]

Simon.Lectures on geometric measure theory.Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, 1983

L. Simon.Lectures on geometric measure theory.Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, 1983

work page 1983

[18] [18]

A. Song. Existence of infinitely many minimal hypersurfaces in closed manifolds.Ann. of Math., 197(3):859–895, 2023

work page 2023

[19] [19]

Z. Wang. Conformal upper bounds for the volume spectrum.Geom. Funct. Anal., 31:992–1012, 2021

work page 2021

[20] [20]

Yau.Problem section.InSeminar on differential geometry, volume 102 ofAnnals of Mathematics Studies, Princeton University Press, 1982

S.-T. Yau.Problem section.InSeminar on differential geometry, volume 102 ofAnnals of Mathematics Studies, Princeton University Press, 1982

work page 1982

[21] [21]

R. Ye. Foliation by constant mean curvature spheres.Pacific Journal of Mathematics, 147(2):381–396, 1991

work page 1991

[22] [22]

X. Zhou. Min-max minimal hypersurface in (M n+1, g) withRic >0 and 2≤n≤6.J. Differential Geom., 100(1):129–160, 2015

work page 2015

[23] [23]

X. Zhou. Min-max hypersurface in manifold of positive Ricci curvature.J. Differential Geom., 105(2):291–343, 2017

work page 2017

[24] [24]

X. Zhou. On the multiplicity one conjecture in min-max theory.Annals of Mathematics, 192(3):767–820, 2020

work page 2020

[25] [25]

X. Zhou. Mean curvature and variational theory.InICM—International Congress of Mathe- maticians.Vol. IV. Sections 5–8, 2696–2717, EMS Press, Berlin, 2023

work page 2023

[26] [26]

Zhou and J

X. Zhou and J. Zhu. Min-max theory for constant mean curvature hypersurfaces.Invent. Math., 218(2):441–490, 2019. Xiaoxiang Jiao School of Mathematical Sciences, University of Chinese Academy of Sciences 19A Yuquan Road, Beijing, 100049, China E-mail: xxjiao@ucas.ac.cn 15 Wenduo Zou School of Mathematical Sciences, University of Chinese Academy of Science...

work page 2019