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Infinitely Many Surfaces with Prescribed Mean Curvature in the Presence of a Strictly Stable Minimal Surface

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Pith's one-line read The paper proves that a strictly stable minimal surface forces infinitely many prescribed-mean-curvature hypersurfaces.

desk verdict First construction of infinitely many distinct PMCs on closed manifolds, with a load-bearing comparison missing in the tethering step; deserves refereeing. read the letter →

arxiv 2502.07098 v3 pith:VREA766Z submitted 2025-02-10 math.DG

Pedro Gaspar , Jared Marx-Kuo This is my paper · ORCID
classification math.DG MSC 53A1053C4249Q2058E12
keywords prescribedmeancurvaturemin-maxtheorystrictlystableminimalsurfacecylindricalendvolumespectrummultiplicity-onealmostembeddedhypersurfacefreeboundary
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 proves that on any manifold of dimension 3 to 7 whose boundary is a strictly stable minimal surface, or any closed manifold containing one, there are infinitely many distinct hypersurfaces whose mean curvature equals a prescribed smooth function h, provided h is small in L1, vanishes appropriately on the boundary, and is 'good' in a genericity sense. The constructed surfaces are almost embedded, multiplicity one, stay away from the stable minimal surface, and their area grows linearly with the index bound p+1. If correct, this gives the first infinite families of prescribed-mean-curvature hypersurfaces on closed manifolds and a step toward the conjecture that every smooth h admits infinitely many h-PMC hypersurfaces. The proof works by gluing a cylindrical end, applying min-max to suspended sweepouts, and sending the cylinder to infinity while a maximum principle keeps the surfaces tethered to the core.

What carries the argument

The argument runs on a sequence of compact manifolds (U_epsilon, g_epsilon) obtained by cylindrical-end gluing: a collar of Σ is stretched by a warping factor into a piece of a cylinder, so that the p-widths of U_epsilon converge to those of Cyl(M), which grow with linear gap at least A(Σ1). On each U_epsilon, a suspension construction turns a p-sweepout into a (p+1)-sweepout, and free-boundary min-max theory produces an almost embedded free-boundary PMC Y_{epsilon,p} with H=h_epsilon. Uniform diameter estimates from the area and mean-curvature bound, together with the strict inequality |h_epsilon|(s,t)<H_t(s) on the contracting collar, force Y_{epsilon,p} to stay away from ∂U_epsilon and to reach the core, so it is actually closed. As epsilon→0, a maximum principle for stationary varifolds plus a no-pinching monotonicity argument show that no copy of Σ survives in the limit and the remaining varifold is a multiplicity-one almost embedded h-PMC.

What would settle it

Take the theorem's setting with a strictly stable Σ and choose h satisfying all assumptions except that |h|(s0,t0)>H_{t0}(s0) on a small patch of the contracting collar; run the min-max construction on U_epsilon for decreasing epsilon. If the tethering step is load-bearing, some component of Y_{epsilon,p} will meet ∂U_epsilon, producing a free-boundary component in the limit varifold or a positive-multiplicity copy of Σ, contradicting Theorems 1.5 and 1.6.

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Extended reading notes

Core claim

On the paper's own terms: for ($M^{{n+1}}$,g) with 3≤n+1≤7 and Σ=∂M an embedded strictly stable minimal surface, for any h satisfying Assumptions 1 or 2 there exist infinitely many distinct, almost embedded, multiplicity-one hypersurfaces Y_{h,p} with H_{Y_{h,p}}=h, each disjoint from Σ, with Index(R(Y_{h,p}))≤p+1 and area between (p+1)A(Σ1)−2||h||_{$L^{1}$} and p A(Σ1)+W0+A(Σ)+C $p^{{1/(n+1)}}$+2||h||_{$L^{1}$}. The same conclusion holds on a closed manifold containing a strictly stable minimal surface, by cutting along it and lifting h. Consequently, a closed bumpy manifold with H_n(M;Z_2)≠0 or one failing the Frankel property carries infinitely many such PMC hypersurfaces for the allowed prescribing functions. This is the first construction of infinitely many PMC surfaces for any nonzero prescribing function.

Load-bearing premise

The construction requires that, on the thin collar where the metric is being stretched into a cylinder, the prescribed mean curvature h satisfies |h| strictly less than the mean curvature of the foliating slices; if that strict inequality fails, the approximating surfaces could touch the boundary of the approximating manifold and the limit would cease to be a closed prescribed-mean-curvature surface.

Editorial extensions

If this is right

  • For every p there is an h-PMC with Index(R(Y_{h,p}))≤p+1 and area in the stated linear interval, so the surfaces are distinct because their areas grow linearly with p.
  • Each constructed PMC is disjoint from the strictly stable minimal surface Σ, so the boundary-manifold construction transfers directly to closed manifolds by cutting along Σ.
  • On any closed bumpy manifold with H_n(M;Z_2)≠0, or any closed bumpy manifold failing the Frankel property, there are infinitely many prescribed-mean-curvature hypersurfaces for the allowed functions h.
  • The compact-support condition on h can be relaxed to functions that vanish to first order on Σ and are 'good' away from it, which is the content of Assumption 1 and Theorem 1.4.
  • The weak index bound of the min-max construction is inherited by the regular set of the almost embedded limit, giving a quantitative Morse-index control on each constructed surface.

Reading between the lines

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

  • Not in the paper, but a natural testable consequence is that the linear lower bound (p+1)A(Σ1)−2||h||_{L^1} should force the constructed surfaces to have area ratios approaching A(Σ1) as p grows, which could be checked numerically in the round 3-sphere with a small prescribing function centered near an equator.
  • The proof's reliance on a contracting neighborhood, rather than on strict stability itself, suggests that the same infinitude should hold for any minimal surface admitting such a neighborhood, even in non-bumpy metrics, as the authors themselves remark.
  • The strict inequality |h|<H_t on the collar likely yields a quantitative positive lower bound on the distance from the constructed PMCs to Σ, and the appendix's quantitative maximum principle formalizes this; this distance could be tracked as a function of area, index, and ||h||_{C^1} in small perturbations.
  • If the tethering step is the only obstruction, then relaxing Assumption 2 so that |h| exceeds the leaf mean curvature on a small patch should produce free-boundary PMCs or positive-multiplicity copies of Σ in the limit, which would mark the boundary of validity of the construction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper constructs infinitely many distinct, almost embedded, multiplicity-one hypersurfaces with prescribed mean curvature (PMC) on a compact manifold with boundary, assuming the boundary is a strictly stable minimal surface, and transfers the result to closed manifolds containing such a surface. The construction glues a cylindrical end a la Song, applies Dey's suspension construction and the free-boundary PMC min-max theory of Sun-Wang-Zhou, and then uses diameter estimates and maximum-principle/tethering arguments to rule out free-boundary and boundary-pinching phenomena. The main results are Theorem 1.5 (compactly supported prescribing functions satisfying Assumption 2), Theorem 1.4 (smooth h with vanishing boundary data satisfying Assumption 1), and Theorem 1.6 (closed manifolds), together with area and index bounds for each constructed surface.

Significance. If correct, this is a substantial step toward Conjecture 1.2.1: it appears to be the first construction of infinitely many PMC hypersurfaces for a fixed nonzero prescribing function on closed manifolds, and it gives quantitative area bounds and index bounds. The paper is commendably explicit: it states the auxiliary lemmas, proves the monotonicity formula and the quantitative maximum principle in appendices, and carefully identifies the external inputs from Song, Dey, Zhou-Zhu, and Sun-Wang-Zhou. The main caveat is that the proof depends on several deep external results and on one key comparison inequality in the tethering step that is not fully verified as written.

major comments (1)
  1. [§3.4.3, Lemma 3.5] The no-pinching lemma asserts that a component Y*_epsilon,p converging to a varifold with a boundary component must have points far from both Sigma and the other limit support, because 'the alpha-neighborhoods do not cover Y*'. This is only justified when the limit has nontrivial mass both on Sigma and away from Sigma. If a component converges entirely to Sigma (the case W_tilde_h = 0), the argument as written does not yield the transition point y_i. The desired contradiction in that case can be obtained from Proposition 2, since every component contains a point with t >= hat t while Hausdorff convergence to Sigma would force all points near Sigma. The proof should separate these two cases explicitly; as written, the lemma is incomplete for the case most relevant to ruling out ai > 0.
minor comments (5)
  1. [§1.1 and §4] The sentence before Theorem 1.4 ends with 'and Index(R(Yh,p))' without specifying the bound; it should read 'and Index(R(Yh,p)) <= p + 1'.
  2. [References and §1.1] In the introduction, the paper attributes the generic regularity result in dimension 8 to 'Li-Wang [6]', but reference [6] appears to list Bellettini-Wickramasekera for a different paper title; the citation should be corrected or disambiguated.
  3. [§2.3, Lemma 3] There is a typo: 'embbeding' should be 'embedding'.
  4. [§1.2] The concluding remark that the construction does not produce c-CMC surfaces is an honest limitation and is consistent with Assumption 2.2 excluding constant prescribing functions; it would be helpful to state this limitation in the abstract or introduction.
  5. [§3.1, equation (15)] The supremum in equation (15) is written over S tilde X without specifying the paired variable (x,t); this should be made explicit for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is conditional on a strictly stable minimal surface and uses independent external tools; the only self-citation supplies a separate diameter bound and is not equivalent to the target result.

full rationale

The paper's central claim is conditional on the existence of a strictly stable minimal surface and on prescribing functions satisfying explicit inequalities (Assumptions 1 or 2). The derivation chain uses Song's cylindrical-end approximation, Dey's suspension construction, Sun-Wang-Zhou's min-max theory for free-boundary prescribed-mean-curvature hypersurfaces, and standard maximum-principle and compactness results. The only self-citation is Chambers-Marx-Kuo [8], used for diameter estimates of submanifolds with bounded mean curvature; this is a separate, parameter-free theorem whose assumptions do not include the existence of infinitely many PMCs, so it qualifies as independent support and does not make the argument circular. The tethering step (Proposition 2) invokes the assumed barrier inequality |h_epsilon| < H_t from Assumption 2.2 and applies the maximum principle; this is a hypothesis of the theorem, not a quantity fitted from the surfaces being predicted. Area and index bounds are inherited from external width estimates and min-max theory, with no parameter fitted to the output. No step was found in which an input is defined in terms of the target, a fitted quantity is renamed as a prediction, or a load-bearing premise is justified solely by a self-citation chain. Any concerns about the verification of the barrier inequality in the rescaled collar metric are correctness or gap issues, not circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim introduces no fitted parameters. All quantitative inputs, such as areas, widths, L1 norms and index bounds, are geometric data or outputs of min-max theorems. The axioms listed are the external results that the proof borrows; none of them asserts the target theorem, so the construction is not circular.

assumptions (7)
  • domain assumption Almgren-Pitts min-max existence and compactness for free-boundary PMCs (Sun-Wang-Zhou, Theorems 2.6, 2.11, 2.12 here)
    Provides the min-max surface Y_epsilon,p on the approximating manifold and controls its limits, multiplicity, regularity and index. Stated in Section 2 as background; proof not reproduced.
  • domain assumption Song's cylindrical Weyl law and width convergence (Theorem 2.15, from Song Theorem 9)
    Gives the linear gap omega_{p+1}-omega_p >= A(Sigma_1) on Cyl(M) and convergence omega_p(U_epsilon,g_epsilon)->omega_p(Cyl(M)), which is the engine for infinitely many distinct p.
  • domain assumption Strict stability implies a contracting neighborhood (Lemma 2.14, from Song Lemma 11)
    The foliation by hypersurfaces with mean curvature vector pointing to Sigma is needed for the maximum-principle tethering and the no-pinching arguments.
  • domain assumption Chambers-Marx-Kuo diameter estimates (Theorem 3.1 and Lemma 3.2)
    Used in Proposition 1 to bound the diameter of Y_epsilon,p uniformly in epsilon. New theorem of the second author, cited as [8].
  • standard math Solomon-White maximum principle for stationary varifolds (Theorem 3.3)
    Used to decompose the limit varifold into boundary components plus a core-supported part in Section 3.4.2.
  • domain assumption Regularity and compactness for stable PMC varifolds of Bellettini-Wickramasekera (Remark 2.4)
    Justifies weakening of convergence assumptions and application of compactness when h is merely smooth away from the boundary.
  • standard math Existence of area-minimizing hypersurfaces in homology classes and bumpy metrics theorem (Section 5, Corollaries 1.6.1 and 1.6.2)
    Used to produce the strictly stable minimal surface Sigma from nonzero H_n or from the non-Frankel property under a bumpy metric.

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Pith. "Pith review of Infinitely Many Surfaces with Prescribed Mean Curvature in the Presence of a Strictly Stable Minimal Surface." pith.science (2026). https://pith.science/paper/VREA766Z

@misc{pith2026250207098,
  author       = {Pith},
  title        = {Pith review of: Infinitely Many Surfaces with Prescribed Mean Curvature in the Presence of a Strictly Stable Minimal Surface},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VREA766Z}},
  note         = {Machine review of arXiv:2502.07098}
}
abstract

We construct infinitely many distinct hypersurfaces with prescribed mean curvature (PMC) for a large class of prescribing functions when $(M^{n+1}, g)$ is a closed smooth manifold containing a minimal surface that is strictly stable (or more generally, admits a contracting neighborhood). In particular, we construct infinitely many distinct PMCs when $H_n(M, \mathbb{Z}_2) \neq 0$, or if $(M, g)$ does not satisfy the Frankel property. Our construction synthesizes ideas from Song's construction of infinitely many minimal surfaces in the non-generic setting, Dey's construction of multiple constant mean curvature surfaces, and Sun--Wang--Zhou's min-max construction of free boundary PMCs.

Figures

Figures reproduced from arXiv: 2502.07098 by the authors.

Figure 1
Figure 1. Example of a PMC with large (codimension 0) touching set [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Visualization of N, the metric completion of M\Σ Given that h : M → R lifts to Comp(M\Σ), we will impose the following conditions on h when we work in the closed setting. By abuse of notation, let Σ =˜ ∂Comp(M\Σ) and Σ˜ 1 its largest connected component. Theorem 1.6. Suppose that Mn+1 , 3 ≤ n + 1 ≤ 7, is a closed manifold with a closed, embedded, strictly stable minimal surface, Σ. For h ∈ C∞(M), suppose its lift to… view at source ↗
Figure 3
Figure 3. Visualization of the constructed PMC, after gluing Comp( [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Visualization of a dumbell metric which is not Frankel and for which corollary 1.6.2 applies. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Almost embeddedness can occur on PMCs on large sets where [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Visualization of the contracting neighborhood near our strictly stable minimal surface Σ = [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Visualization of limit metric and manifold, Cyl( [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Visualization of the metrics gϵ in terms of scaling function vϵ(t) [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Change of coordinates t → s so that the contracting neighborhood approximates a cylindrical end [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Visualization of the end with a graph of the mean curvature of the slices [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Case 2 visualized. Green denotes distance minimizing geodesic, orange gives the competitor [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]
Figure 12
Figure 12. Figure 12: Maximum principle argument to prevent the presence of a free boundary prescribed mean curva [PITH_FULL_IMAGE:figures/full_fig_p030_12.png]
Figure 13
Figure 13. Figure 13: Visualization of our PMC touching a part of the core which is a [PITH_FULL_IMAGE:figures/full_fig_p030_13.png]
Figure 14
Figure 14. Figure 14: A visualization of what can go wrong as ϵ → 0. Even though all of the Yϵ,p have uniform diameter bounds, because the metric is modified on t ∈ [0, δϵ] and δϵ → 0, the Yϵ,p may accumulate around {t = δϵ}. In the limit, this naively can lead to pinching at a point y∞. L…

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.