{"id":"13ed074d-900d-4381-90ac-6e6b03fa522d","arxiv_id":"2502.07098","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On any closed 3-7 dimensional manifold that contains a strictly stable minimal surface, a large class of prescribed-mean-curvature functions admit infinitely many distinct almost embedded hypersurfaces.","lead":"This paper proves that if a curved space contains one special, stable soap-film surface, then it contains infinitely many different surfaces whose mean curvature is fixed in advance. It is the first construction of infinitely many prescribed-mean-curvature surfaces on closed manifolds, a key step toward a broad conjecture in geometry.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tethering step (Prop. 2) lacks a verified comparison between h_epsilon and the actual g_epsilon-mean curvature of the slices on the long collar; as written this is the main load-bearing gap.","rationale":"The reader identifies the tethering-to-the-core step (Proposition 2) as the weakest assumption, and I agree that this is the load-bearing part of the proof. My concern sharpens the reader's point: even granting the uniform diameter bound of Proposition 1, the maximum-principle contradiction in Proposition 2 compares |h_epsilon| with H_t(s), the mean curvature of the slices in the original metric g, while the actual ambient metric on the approximating manifolds is g_epsilon. On the long collar the g_epsilon-mean curvature of the slices decays like 1/v_epsilon, so the required strict inequality involves a rate competition between h_epsilon -> 0 and H^{g_epsilon}_{slice} -> 0. The paper's Section 2.4 only records C^1 convergence of h_epsilon to 0 on the collar, not the sharper comparison needed for the maximum principle. This is a genuine gap in the written proof, but it is likely repairable: h_epsilon has compact support and vanishes near Sigma, so one can choose h_epsilon identically zero on the epsilon-collar and then perturb it to be 'good' with arbitrarily small C^1 norm. Making that choice explicit and verifying the strict inequality against the g_epsilon leaf mean curvature would close the gap. Because the issue is a missing estimate rather than a demonstrated contradiction, the appropriate verdict remains CONDITIONAL, matching the reader's assessment. I do not see a reason to move to ACCEPT or REJECT on the basis of what is written.","tokens_in":42962,"tokens_out":16666,"duration_ms":162555,"concrete_test":"In the explicit model of §2.3, take the slices {t = const} with metric g_epsilon = g_t \\oplus (v_epsilon(t)dt)^2 and compute their mean curvature H^{g_epsilon}_t(s). In particular, on [0,z_epsilon] where v_epsilon is constant and tends to infinity, verify whether H^{g_epsilon}_t(s) = H^{g}_t(s)/v_epsilon(t) up to controlled error. Then check whether the approximating functions h_epsilon constructed in §2.4 (or a modified choice, e.g. h_epsilon identically 0 on the epsilon-collar) satisfy |h_epsilon(s,t)| < H^{g_epsilon}_t(s) for all t in [0,delta_epsilon], uniformly in epsilon. If this inequality holds with explicit rates, Proposition 2 is repaired; if it fails for the natural choice of h_epsilon, the tethering argument as written is incomplete and Theorems 1.4-1.6 require an additional estimate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.5 depends on Proposition 2, which asserts that every component of the min-max PMC Y_epsilon,p in (U_epsilon,g_epsilon) reaches the fixed level set t = \\hat t and therefore stays away from the boundary. The argument is: if t_epsilon = max t < \\hat t, then at a maximizer p_epsilon the surface is tangent to a slice, and Assumption 2.2 plus the C^1-smallness of h_epsilon on the collar give |h_epsilon|(s,t_epsilon) < H_t(s), contradicting the maximum principle. The gap is that the maximum principle must be applied in the metric g_epsilon, where the relevant comparison is the mean curvature of the slice with respect to g_epsilon, not the original mean curvature H_t(s) of the foliation in g. On the modified collar, g_epsilon = g_t \\oplus (v_epsilon(t)dt)^2 with v_epsilon very large on [0,z_epsilon], so the g_epsilon-mean curvature of a slice scales like H_t/v_epsilon and tends to 0 as epsilon -> 0. Condition 3 of §2.4 only says h_epsilon -> 0 in C^1 on the collar; it does not by itself establish |h_epsilon| < H^{g_epsilon}_{slice} uniformly in epsilon. If the constructed h_epsilon cannot be made smaller than this rescaled leaf mean curvature, the strict inequality fails and Y_epsilon,p may touch the boundary, making the limiting object a free-boundary PMC or a varifold with boundary components. The written proof does not compute H^{g_epsilon}_{slice} or specify the rate of vanishing of h_epsilon on the collar, so this load-bearing comparison is unverified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":43284,"tokens_out":25723,"duration_ms":235938,"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":[{"comment":"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.","section":"§3.4.3, Lemma 3.5"}],"minor_comments":[{"comment":"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'.","section":"§1.1 and §4"},{"comment":"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.","section":"References and §1.1"},{"comment":"There is a typo: 'embbeding' should be 'embedding'.","section":"§2.3, Lemma 3"},{"comment":"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.","section":"§1.2"},{"comment":"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.","section":"§3.1, equation (15)"}],"recommendation":"major_revision","confidential_remarks":"The central idea is plausible and the paper is well-structured, but the tethering step in Proposition 2 is a genuine load-bearing gap: the maximum-principle comparison has not been verified in the warped collar metric. I believe this is repairable by imposing a rate condition in the construction of h_epsilon or by moving the maximum-principle argument entirely to the unmodified part of the manifold. The no-pinching lemma also needs a cleaner case split. If the authors close these gaps, I would be inclined to accept; the paper would then be a significant contribution to the PMC min-max program."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is the first construction of infinitely many distinct prescribed-mean-curvature hypersurfaces on closed manifolds, and it deserves a serious referee. The authors take Song's cylindrical-end gap, Dey's suspension construction, and Sun–Wang–Zhou's free-boundary min-max and combine them into a coherent machine. The genuinely new quantitative pieces—the uniform diameter bound for PMCs in the collapsing collar and the tethering argument—are real additions, and the paper is honest about scope: the prescribing function must have small L1 norm, vanish to first order on the boundary surface, and the construction does not handle constant mean curvature.\n\nI read the proof looking for an internal contradiction and did not find one. The reliance on deep external theorems (Song, Sun–Wang–Zhou, Chambers–Marx–Kuo) is explicit and appropriate. The main soft spot is Proposition 2 in §3.3. The proof compares h_epsilon with the mean curvature H_t of the original foliation, but the maximum principle is applied in (U_epsilon,g_epsilon), where the comparison leaves are the slices with the modified metric. On the long collar, g_epsilon scales the normal direction by v_epsilon, so the slice mean curvature in g_epsilon is roughly H_t/v_epsilon and tends to zero. Condition 3 in §2.4 only says h_epsilon → 0 in C^1 on the collar; it does not specify a rate. Without a rate, the strict inequality |h_epsilon| < H^{g_epsilon}_{slice} that Proposition 2 needs is not established. This is load-bearing: if it fails, Y_{epsilon,p} may touch ∂U_epsilon and the limit could be a free-boundary PMC or a varifold with boundary components, which would invalidate Theorems 1.4–1.6 as stated. I expect the gap is fixable—one should be able to choose h_epsilon with support away from the collar, or at least with vanishing rate o(v_epsilon^{-1})—but the written proof needs that computation.\n\nMinor: in Corollary 1.6.1 the phrase 'if Σ is degenerate' looks like a slip; the nondegeneracy of a stable minimizer under a bumpy metric is what gives strict stability. Also, the area interval in the abstract and theorem is wide, but that is not a flaw.\n\nVerdict: this is a serious paper by people who know the literature. Send it to referees. The referee should focus on Proposition 2 and the passage from Y_{epsilon,p} to the limit in §3.4. If the tethering comparison is filled in, the main theorem stands.","headline":"First construction of infinitely many distinct PMCs on closed manifolds, with a load-bearing comparison missing in the tethering step; deserves refereeing.","tokens_in":43858,"tokens_out":4631,"would_cite":true,"duration_ms":42870,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53A10","53C42","49Q20","58E12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a strictly stable minimal surface forces infinitely many prescribed-mean-curvature hypersurfaces.","keywords":["prescribed mean curvature","min-max theory","strictly stable minimal surface","cylindrical end","volume spectrum","multiplicity-one","almost embedded hypersurface","free boundary"],"falsifier":"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.","tokens_in":42728,"feed_emoji":"📐","tokens_out":6111,"duration_ms":53863,"temperature":0.7,"pith_summary":"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.","feed_headline":"One stable minimal surface yields infinitely many PMC surfaces","feed_subtitle":"First proof of infinitely many prescribed-mean-curvature hypersurfaces on closed manifolds.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the cylindrical-end gluing construction, the convergence of widths, and the linear width gap A(Σ1) used to get infinitely many distinct PMCs.","marker":"[44]"},{"why":"Supplies the suspension construction that turns a p-sweepout into a (p+1)-sweepout and the mountain-pass min-max framework for constant mean curvature surfaces.","marker":"[10]"},{"why":"Provides the free-boundary prescribed-mean-curvature min-max theorem that yields almost embedded PMCs with optimal regularity and weak index bounds on the approximating manifolds.","marker":"[47]"},{"why":"Provides the intrinsic diameter bound for submanifolds with bounded mean curvature and area, which is used to prove the approximating PMCs are tethered to the core.","marker":"[8]"},{"why":"Supplies the strong maximum principle for stationary varifolds used to decompose the limit varifold away from the boundary.","marker":"[43]"},{"why":"Supplies the related maximum principle that rules out contact of the limit varifold with the stable minimal boundary.","marker":"[50]"},{"why":"Provides compactness and regularity for prescribed-mean-curvature hypersurfaces, including the density-at-most-two conclusion away from minimal components.","marker":"[55]"},{"why":"Supplies the Weyl law for the volume spectrum, used in combination with the cylindrical width estimates to produce linear area growth.","marker":"[26]"}],"fun_headline_variants":["One stable minimal surface yields infinitely many PMC hypersurfaces","Infinitely many PMC hypersurfaces from a single stable minimal surface","Stable minimal surface yields infinite prescribed mean curvature hypersurfaces","A single stable minimal surface seeds infinitely many PMC hypersurfaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["One stable minimal surface yields infinitely many PMC hypersurfaces","Infinitely many PMC hypersurfaces from a single stable minimal surface","Stable minimal surface yields infinite prescribed mean curvature hypersurfaces","A single stable minimal surface seeds infinitely many PMC hypersurfaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001219,"raw_usage":{"total_tokens":4984,"prompt_tokens":886,"completion_tokens":4098,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":502,"completion_tokens_details":{"reasoning_tokens":4027}},"tokens_in":502,"tokens_out":4098,"duration_ms":27588,"temperature":1.0,"reasoning_tokens":4027,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T13:48:54.940262+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the cylindrical-end gluing construction, the convergence of widths, and the linear width gap A(Σ1) used to get infinitely many distinct PMCs."},{"cited_title":"Differ- ential Geom","cited_arxiv_id":null,"evidence_quote":"Supplies the suspension construction that turns a p-sweepout into a (p+1)-sweepout and the mountain-pass min-max framework for constant mean curvature surfaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the free-boundary prescribed-mean-curvature min-max theorem that yields almost embedded PMCs with optimal regularity and weak index bounds on the approximating manifolds."},{"cited_title":"Chambers and Jared Marx-Kuo, Diameter, area, and mean curvature , 2024","cited_arxiv_id":null,"evidence_quote":"Provides the intrinsic diameter bound for submanifolds with bounded mean curvature and area, which is used to prove the approximating PMCs are tethered to the core."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the strong maximum principle for stationary varifolds used to decompose the limit varifold away from the boundary."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the related maximum principle that rules out contact of the limit varifold with the stable minimal boundary."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides compactness and regularity for prescribed-mean-curvature hypersurfaces, including the density-at-most-two conclusion away from minimal components."},{"cited_title":"Marques, and Andr´ e Neves, Weyl law for the volume spectrum , Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the Weyl law for the volume spectrum, used in combination with the cylindrical width estimates to produce linear area growth."}],"review_version":1}