{"id":"d751aa12-0f9e-4997-a9ec-bf9835545d76","arxiv_id":"2507.18825","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For every half-integer J and sufficiently large integer m, the authors construct an embedded self-shrinker with 2J+1 ends and genus 2J(m-1) by gluing stacked planes with m catenoidal bridges between adjacent levels.","lead":"This paper constructs new self-shrinking surfaces in three-dimensional space with any chosen number of ends, built by stacking flat planes and connecting them with many small catenoidal necks. The result provides the first examples of self-shrinkers with arbitrarily many ends that lack continuous symmetry, relevant for understanding singularities of mean curvature flow.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The horizontal-balancing estimates Lemmas 4.31 and 4.32 are imported from [30] for a case the paper admits was not fully treated in [14]; formulas (5.32) and Lemma 6.28 depend on them.","rationale":"The reader's weakest assumption is exactly the load-bearing concern: the dislocation estimates in Lemmas 4.31 and 4.32 are the only bridge between the LD solutions on different circles and the horizontal balancing formulas (5.32). The paper itself flags that the relevant case was not fully studied in [14] and appeals to the thesis [30], which is not self-contained here. I found no independent contradiction in the rest of the construction: the linear theory on R², the construction of LD solutions from Green's functions, the initial surface gluing, the approximate solution operator, and the Schauder fixed-point scheme are structurally consistent with the established Linearised Doubling framework. The main worry is therefore not an internal inconsistency but an unverified imported estimate. Because the consequence would be failure of Lemma 6.28, which is indispensable for the parameter correction in Theorem 8.1, the paper should be accepted only after that estimate is either proved in the text or the cited thesis passage is transparently verified. This matches the reader's CONDITIONAL verdict, so no change is recommended.","tokens_in":38698,"tokens_out":14220,"duration_ms":162137,"concrete_test":"Verify Lemma 4.31 in the parameter regime actually needed, r0 = r1 = rmu (allowed by (5.15) up to O(m^{-2})). Using the explicit average formula (4.12)-(4.17) and the Green's function estimate (4.1), re-derive the limit of Texp(ω)(Φ - φ[φ1, mħ(r)]) near q ∈ L1 on the cylinder and confirm that it equals G∞ with the stated constants, in particular the second derivative ∂²G∞/∂ŝ²(0, π) = 1/4 that enters Lemma 4.32. If this derivation requires a new estimate not present in Sections 3-4, then the proof of (5.32) is not self-contained; if the limit or the second-derivative constant differs, then formula (5.32) and the invertibility of A in Lemma 6.28 fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is the unproved dislocation analysis in Section 4, specifically Lemmas 4.31 and 4.32. Lemma 4.31 asserts that near a singularity of the other circle the LD solution converges, after rescaling, to the cylinder Green's function G∞; Lemma 4.32 uses the second derivative of G∞ to control the radial shift of the singularity. The paper's proof of Lemma 4.31 is one sentence referring to [14, Lemma 9.26], and the text explicitly states that the relevant case, m_i = 2m, 'was not fully studied' in [14] and defers to the thesis [30, Section 5.2.5]. No proof is reproduced. These estimates are not decorative: Lemma 5.28 derives the horizontal balancing formulas (5.32) by plugging (5.15) into the expansion from Lemma 4.32. Formula (5.32) is then the sole input that makes the linear map A in Lemma 6.28 invertible, and the inverse of A is exactly what is needed for the parameter-correction map Z in the Schauder fixed-point argument of Theorem 8.1. If Lemma 4.32 has a wrong constant or an unaccounted O(1) term, the matching analysis for μ'_j,± collapses, the invertibility of Z is no longer justified, and the fixed point step that produces the self-shrinker does not go through. This is a verification gap rather than a demonstrated contradiction, but it is the point where the central existence theorem is least self-contained.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for each half-integer J and all sufficiently large integers m, a G[m,J]-invariant embedded self-shrinker in R^3 with 2J+1 ends and genus 2J(m-1). The construction stacks 2J+1 approximately planar levels connected by 2Jm catenoidal bridges in the Gaussian metric, following the Linearised Doubling (LD) methodology of Kapouleas. The main theorem (Theorem 8.1) asserts existence for large m, varifold convergence to (2J+1)R^2 as m tends to infinity, and an explicit asymptotic cone given as a union of graphs. The proof proceeds by constructing LD solutions with logarithmic singularities, matching them via catenoids, solving balancing and unbalancing equations, and then applying a Schauder fixed point argument using a global linear theory on the initial surface.","tokens_in":39007,"tokens_out":4374,"duration_ms":43691,"significance":"If the construction is correct, the paper settles the existence of embedded self-shrinkers in R^3 with any prescribed number of ends at least two, a qualitative advance over the previously known classes with one, two, or three ends. The stacking ansatz is new in the self-shrinker setting, and the paper provides explicit parameter formulas, a detailed analysis of the linearized problem, and a clear fixed-point framework. The authors also give credit for the systematic LD machinery from [11,13,14,17,30] and adapt it to a multilevel stacking configuration. However, the paper is not self-contained at several load-bearing points: the dislocation estimates in Section 4 are imported from a thesis with the critical case not fully covered in a published reference, and the derivation of the balancing formulas (5.18)-(5.19) is explicitly omitted. These gaps are verification gaps rather than demonstrated errors, but they make the central existence theorem depend on unpublished or insufficiently documented material.","major_comments":[{"comment":"The dislocation estimates in Lemmas 4.31 and 4.32 are load-bearing: Lemma 4.31 is stated to follow from [14, Lemma 9.26] with one sentence, yet the text immediately notes that the relevant case m_i=2m 'was not fully studied' in [14], deferring to the thesis [30, Section 5.2.5]. No proof is reproduced. Lemma 4.32 then derives the horizontal balancing formulas (5.32) using the second derivative of G_∞ at (0,π), and (5.32) is the sole input for the invertibility of the linear map A in Lemma 6.28, which is needed for the fixed-point step in Theorem 8.1. If Lemma 4.32 has a wrong constant or an unaccounted O(1) term, the matching analysis for μ'_{j,±} collapses and the fixed-point argument no longer goes through. The authors should reproduce the proof of Lemma 4.31 in the m_i=2m case, or at least provide a detailed derivation of Lemma 4.32 that does not depend on the unpublished case, so that the central existence theorem is verifiable from the paper alone.","section":"Section 4, Lemmas 4.31 and 4.32"},{"comment":"The paper explicitly states 'We omit this part here and check both balancings in 5.28' before introducing the waist and height formulas (5.18)-(5.19). This is an admitted omission of the derivation of the vertical balancing conditions. Lemma 5.28 uses (5.18)-(5.19) as input to compute the mismatches, rather than deriving these formulas from the matching equations or showing that they are uniquely determined up to the unbalancing freedom. The formulas are compared to similar ones in [29] and [3], but no derivation is given. Since the fixed-point argument relies on the explicit structure of these parameters, the authors should include the linearized vertical balancing analysis (e.g., the eigenvalue problem in A.1) or otherwise demonstrate how (5.18)-(5.19) arise from the matching conditions.","section":"Section 5, equations (5.18)-(5.19)"},{"comment":"The global linear theory on the initial surface is stated with 'The proof is the same as [14, Proposition 4.18]', with only the remark that the subscript j for multiple levels replaces the ± notation. This proposition is central to the Schauder fixed-point argument, and the initial surface here has 2J+1 levels and a more complicated catenoid region structure. It is not immediate that the proof in [14] carries over verbatim; in particular, the decomposition into low/high modes on the catenoids, the handling of the kernel, and the constant estimates may depend on the multilevel geometry. The authors should spell out at least the modifications and identify the precise statements in [14] that are used, or give a more complete proof if the differences are nontrivial.","section":"Proposition 7.18"}],"minor_comments":[{"comment":"The notation φj is used both for the LD solutions in (5.20) and for the graph functions φgl_j in Definition 6.10; this is confusing and should be changed, for example by renaming the latter to ψj or g_j.","section":"Notation throughout"},{"comment":"The correspondence between the index sets L_0,L_1 and the notation L_± used later (e.g., in Definition 4.6 and (5.20)) is not stated explicitly; please clarify which set plays the role of L_+ and which plays the role of L_- depending on the level index.","section":"Definitions 4.3 and 4.6"},{"comment":"In the proof of Lemma 4.11, the limits ε1→0 and ε2→0 are taken in a specific order, but the justification for interchanging limits is not given; a brief remark on the logarithmic singularities would make the argument rigorous.","section":"Lemma 4.11"},{"comment":"In the formula for the asymptotic cones, the case j=0 when J is an integer gives a cone that is O(˘τ^{1+α/4}_{1/2})r, which is sublinear; please clarify whether this is a degenerate cone or whether the leading term vanishes in this case.","section":"Theorem 8.1"},{"comment":"The notation for the constructed surface is inconsistent: the abstract uses ˘M[J,m] while the body (e.g., Theorem 8.1) uses ˘M[m,J]; please standardize the order of the arguments.","section":"Title and abstract"},{"comment":"Reference [30] is a PhD thesis that is not publicly available in the usual channels; the authors should include the relevant chapter or a preprint version in the bibliography, since a key lemma is deferred to this thesis.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claim is significant and the overall strategy is coherent, but the load-bearing estimates in Section 4 and the omitted derivation of the balancing formulas in Section 5 make the manuscript difficult to verify in its current form. The reliance on the first author's thesis [30] for a case that is explicitly said not to be fully treated in the published reference [14] is a particular concern; the authors should be encouraged either to include the full proof or to make the relevant part of the thesis available. The comparison with the shrinkers of Ilmanen-White and Ketover in the introduction is speculative and could be softened or supported with a reference."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. This paper does give something genuinely new: for every half-integer J and large enough m, a self-shrinker with 2J+1 ends and genus 2J(m-1), matching the plane-stacking picture. That settles which end counts are achievable among noncompact self-shrinkers without continuous symmetry: every integer at least 2. The previous catalogue stopped at three ends. The new ingredient is a horizontal balancing condition for the stacked catenoidal bridges, and the paper manages the multiple families of LD solutions cleanly.\n\nWhat I like: the LD-to-stacking extension is not a formality. The paper introduces the horizontal balancing explicitly, sets up a parameter space of dimension 4J, and proves the relevant linear map is invertible via an explicit tridiagonal structure. The matching analysis in Lemma 5.28 is detailed, and the paper is honest about what it is not proving.\n\nThe soft spots are real but not fatal on their face. Lemmas 4.31 and 4.32, which control the dislocation of an LD solution near the singularities of the other circle, are load-bearing for the horizontal balancing formulas (5.32) and hence for the invertibility of the map A in Lemma 6.28. The proof of 4.31 is one sentence referring to [14, Lemma 9.26], and the paper states that the relevant case m_i = 2m was not fully studied in [14], deferring to the thesis [30]. That is a genuine verification gap. If the constant in 4.32 or the O(1) term is wrong, the matching analysis collapses and the fixed point argument in Theorem 8.1 does not go through. This is not a demonstrated contradiction, and the structure of the argument is coherent, but a referee needs to be able to check these estimates. Proposition 7.18 is also asserted to have the same proof as [14, Prop 4.18]; that is probably fine but not self-contained. And the derivation of (5.18)-(5.19) is omitted, though the paper says the balancing conditions are checked in 5.28.\n\nWho this is for: anyone working on gluing constructions for self-shrinkers or minimal surfaces. It deserves serious refereeing. My recommendation: send it to peer review, but make sure the referee has access to the thesis, or require the authors to include a full proof of Lemmas 4.31 and 4.32 in a revision. The central existence claim is plausible and important.","headline":"New construction of self-shrinkers with any prescribed number of ends; the main theorem is plausible and important, but a load-bearing dislocation estimate is deferred to a thesis and needs checking.","tokens_in":39597,"tokens_out":3103,"would_cite":true,"duration_ms":31715,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C42","53E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For each half-integer $J$ and large $m$, an embedded self-shrinker with $2J+1$ ends and genus $2J(m-1)$ is built by stacking copies of the plane.","keywords":["self-shrinkers","mean curvature flow","PDE gluing","linearised doubling","stacking constructions","catenoidal bridges","ends of surfaces","asymptotic cones"],"falsifier":"Compute the asserted expansion in Lemma 4.32 numerically for a concrete case, say $J=1/2$ with $m$ large and $\\Delta r$ small, by solving the two-dimensional linearised equation with the explicit logarithmic Green's function; if $\\frac1m \\frac{\\partial_{\\omega}\\tilde{\\Phi}_i}{\\partial r}(\\tilde{q})$ does not equal $\\frac1m \\frac{\\partial_{\\omega}\\Phi_i}{\\partial r}(q)-\\frac{\\Delta r}{r^2}(\\tfrac14+o(1))+O(\\Delta r^2)$, then the horizontal balancing equations (5.32) are wrong. A purely algebraic check is to evaluate the determinant of the $4J\\times 4J$ matrix $A$ defined in Lemma 6.28 for $J=1/2$ and $J=1$; it must be nonzero uniformly as $m\\to\\infty$ for the parameter map to be invertible.","tokens_in":38407,"feed_emoji":"","tokens_out":8123,"duration_ms":83554,"temperature":0.7,"pith_summary":"This paper constructs embedded self-shrinking surfaces for mean curvature flow in $\\mathbb{R}^{3}$. The main theorem says that for every half-integer $J$ and every sufficiently large integer $m$, there is a smooth embedded self-shrinker with $2J+1$ ends and genus $2J(m-1)$, obtained by stacking $2J+1$ parallel copies of the plane and joining adjacent levels by $m$ catenoidal bridges arranged with dihedral symmetry. If correct, this settles the existence of self-shrinkers with any prescribed number $k\\geq 2$ of ends, a question previously open beyond the known one-, two-, and three-ended examples. The surfaces degenerate, as $m\\to\\infty$, to a multiplicity-$(2J+1)$ plane, and their asymptotic cones are explicit graphs with coefficients given by Kummer asymptotics.","feed_headline":"Gluing planes yields self-shrinkers with any number of ends","feed_subtitle":"For every integer k≥2, a shrinker with k ends emerges from stacking k planes joined by catenoidal bridges.","key_machinery":"The load-bearing object is the linearised-doubling solution: a solution of the linearised self-shrinker operator $L=\\Delta-\\tfrac12 X\\cdot\\nabla+\\tfrac12$ on $\\mathbb{R}^{2}$ with prescribed logarithmic singularities at $D_{m}$-symmetric points on two concentric circles. Around this sits a three-part mechanism: rotationally averaged solutions reduce the problem to a one-dimensional ODE whose two independent solutions are Kummer and Tricomi confluent hypergeometric functions; matching conditions express the mismatch of gluing the graphs to catenoids in terms of four parameters per bridge, namely waist $\\tau$, height $h$, radial location $r$, and tilt $\\kappa$; and a $4J$-dimensional parameter map, proved invertible through a tridiagonal Toeplitz eigenvalue lemma, lets the unbalancing parameters absorb the mismatches. The global linear theory combines weighted H\\\"older spaces on the plane with separating-variables analysis on the catenoid, and Schauder's fixed point theorem closes the perturbation.","core_discovery":"On the paper's own terms, the discovery is Theorem 8.1: for each $J\\in\\frac12\\mathbb{N}$ and $m$ large enough, there exists a $G[m,J]$-invariant embedded self-shrinker $\\breve{M}[m,J]\\subset\\mathbb{R}^{3}$ of genus $2J(m-1)$ with $2J+1$ ends. The surface is assembled from $2J+1$ graphs of linearised-doubling solutions over copies of $\\mathbb{R}^{2}$, connected by $2Jm$ catenoidal bridges in the Gaussian metric $g_{\\mathrm{Shr}}$, with $m$ bridges between each adjacent pair of levels. As $m\\to\\infty$ the surfaces converge in the varifold sense to $(2J+1)\\mathbb{R}^{2}$, and the asymptotic cone of $\\breve{M}[m,J]$ is the union of the explicit cones $\\cup_{j}\\breve{C}_{j}$ described in Theorem 8.1. The construction solves the linearised self-shrinker equation with logarithmic singularities at points on concentric circles, estimates these solutions by their rotationally averaged profiles, glues catenoidal bridges, balances the resulting mismatches, and closes the nonlinear problem by a Schauder fixed point argument.","pith_inferences":["A natural next test is whether the stacking construction can be run with unequal numbers of bridges between adjacent levels; the balancing system would then become a general tridiagonal eigenvalue problem rather than the uniform Toeplitz case used here.","If the link the authors cite to blow-ups of mean curvature flow holds, the new family supplies candidates for multi-ended singularity models with any prescribed number of ends, going beyond the usual one-ended or low-multiplicity examples.","The same linearised-doubling and stacking scheme is a plausible template for constructing embedded $f$-minimal surfaces of arbitrary end count for other ambient functions $f$, with the rotationally invariant ODE analysis replaced by the corresponding one-dimensional problem for that $f$.","Because the theorem produces both prism and antiprism symmetries depending on whether $J$ is half-integral or integral, the same construction with a different signature convention would give reflected shrinkers, which the authors note explicitly."],"forward_implications":["For every integer $k\\geq 2$ and every sufficiently large $m$, there is an embedded self-shrinker in $\\mathbb{R}^{3}$ with $k$ ends, so the construction covers all prescribed end counts rather than only the previously known one-, two-, and three-ended examples.","The genus of the shrinker grows linearly with $m$ at fixed $J$, yielding infinite families of high-genus self-shrinkers with the same small number of ends.","As $m$ tends to infinity the surfaces converge as varifolds to the plane with multiplicity $2J+1$, and each member has an explicit asymptotic cone, providing concrete high-multiplicity conical tangent objects for mean curvature flow.","When $J=1/2$ and $J=1$, the constructed surfaces are expected to reproduce the previously known two- and three-ended shrinkers, placing them in one unified family."],"supporting_citations":[{"why":"supplies the general linearised-doubling framework for self-shrinkers that this stacking construction adapts.","marker":"[14]"},{"why":"provides the dislocation estimates for the case $m_i=2m$ imported in Lemmas 4.31 and 4.32, on which the horizontal balancing rests.","marker":"[30]"},{"why":"introduces the linearised doubling method and the vertical and horizontal balancing scheme whose matching equations are generalised here.","marker":"[11]"},{"why":"supplies the weighted H\\\"older spaces and the linear theory on $\\mathbb{R}^{2}$ used to solve for the LD solutions and their perturbations.","marker":"[12]"},{"why":"provides weighted $L^2$ estimates and cone H\\\"older spaces used in the global linear theory.","marker":"[5]"},{"why":"contains the rotationally invariant ODE analysis for the linearised operator on $\\mathbb{R}^{2}$ and the construction with fattening asymptotic cones that the averaged solutions build on.","marker":"[19]"},{"why":"introduced stacking constructions by PDE gluing and contains the tridiagonal balancing system that motivates formulas (5.18) and (5.19).","marker":"[29]"},{"why":"gives vertical balancing formulas for disc stackings to which the paper compares its own balancing expressions.","marker":"[3]"},{"why":"supplies Schauder's fixed point theorem used to close the nonlinear perturbation problem in Theorem 8.1.","marker":"[8]"}],"fun_headline_variants":["Stacked planes glued by bridges make any-ended shrinkers","PDE gluing delivers shrinkers with all possible ends","Catenoidal bridges connect planes into arbitrary-ended shrinkers","Shrinkers for any end count via stacked planes and bridges","Every integer end count achieved by gluing planes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on an imported estimate about how one linearised-doubling solution behaves near the singularities of the adjacent level; the paper does not reproduce the proof, and a wrong leading coefficient there would break the balancing of the bridges and with it the fixed-point step.","fun_headline_variants_meta":{"raw":{"variants":["Stacked planes glued by bridges make any-ended shrinkers","PDE gluing delivers shrinkers with all possible ends","Catenoidal bridges connect planes into arbitrary-ended shrinkers","Shrinkers for any end count via stacked planes and bridges","Every integer end count achieved by gluing planes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000931,"raw_usage":{"total_tokens":4010,"prompt_tokens":995,"completion_tokens":3015,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":611,"completion_tokens_details":{"reasoning_tokens":2935}},"tokens_in":611,"tokens_out":3015,"duration_ms":24047,"temperature":1.0,"reasoning_tokens":2935,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:07:39.076212+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the asserted expansion in Lemma 4.32 numerically for a concrete case, say $J=1/2$ with $m$ large and $\\Delta r$ small, by solving the two-dimensional linearised equation with the explicit logarithmic Green's function; if $\\frac1m \\frac{\\partial_{\\omega}\\tilde{\\Phi}_i}{\\partial r}(\\tilde{q})$ does not equal $\\frac1m \\frac{\\partial_{\\omega}\\Phi_i}{\\partial r}(q)-\\frac{\\Delta r}{r^2}(\\tfrac14+o(1))+O(\\Delta r^2)$, then the horizontal balancing equations (5.32) are wrong. A purely algebraic check is to evaluate the determinant of the $4J\\times 4J$ matrix $A$ defined in Lemma 6.28 for $J=1/2$ and $J=1$; it must be nonzero uniformly as $m\\to\\infty$ for the parameter map to be invertible.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the general linearised-doubling framework for self-shrinkers that this stacking construction adapts."},{"cited_title":"Thesis, 2023","cited_arxiv_id":null,"evidence_quote":"provides the dislocation estimates for the case $m_i=2m$ imported in Lemmas 4.31 and 4.32, on which the horizontal balancing rests."},{"cited_title":"Differential Geom","cited_arxiv_id":null,"evidence_quote":"introduces the linearised doubling method and the vertical and horizontal balancing scheme whose matching equations are generalised here."},{"cited_title":"Reine Angew","cited_arxiv_id":null,"evidence_quote":"supplies the weighted H\\\"older spaces and the linear theory on $\\mathbb{R}^{2}$ used to solve for the LD solutions and their perturbations."},{"cited_title":"J.170 (2021), no","cited_arxiv_id":null,"evidence_quote":"provides weighted $L^2$ estimates and cone H\\\"older spaces used in the global linear theory."},{"cited_title":"Differential Geom.114 (2020), no","cited_arxiv_id":null,"evidence_quote":"introduced stacking constructions by PDE gluing and contains the tridiagonal balancing system that motivates formulas (5.18) and (5.19)."}],"review_version":2}