{"id":"b052327e-aa8b-4920-aadb-ddcbd5df2c2c","arxiv_id":"2507.19428","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"For a broad class of cones in R^3, complete connected self-expanders of any positive genus exist, providing mean curvature flows in which genus drops but does not reach zero.","lead":"This paper constructs the first examples of self-expanding surfaces with holes (positive genus) in three dimensions, asymptotic to a class of three-pronged cones. The construction yields new tools for mean curvature flow, showing that the number of holes can strictly decrease without vanishing at a singular time.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.3's iteration depends on an unproved quantitative gluing claim: the surgery on A''_R ∪ C2,R must lower area by a fixed amount, but no estimate or topological verification is supplied.","rationale":"The reader's weakest assumption points to the same load-bearing gap: Proposition 4.3's quantitative surgery claim is asserted without proof. My independent reading confirms that this is not a cosmetic omission. The theorem's proof structure is otherwise coherent: Proposition 3.5 supplies a limit mechanism, and Appendix A provides a substantial min-max regularity argument. But Proposition 4.3 is the only place where the positive genus is actually produced, and its production mechanism is an iteration that requires a fixed, uniform area decrease at every disconnected step. Equation (4.6) gives such a decrease for the intermediate annulus A''_R, but the subsequent desingularization of A''_R∪C2,R along σ2,R must not eat that decrease. The paper gives no bound on the area of the Scherk necks, no verification that the gluing locus is a single common interior curve, and no argument that the construction can be made inside Ω∩B_R with the required symmetry and boundary. The same criticism applies to the initial CHMR, which is introduced descriptively. These are not disagreements with consensus; they are missing arguments internal to the proof. If the missing gluing estimates can be supplied, the announced theorem may well be salvageable, but as written the central existence claim is not established. Therefore the reader's rejection is appropriate, and my stress-test does not change that verdict.","tokens_in":23888,"tokens_out":13754,"duration_ms":138939,"concrete_test":"For the model cone Cε with ε>δ0 used in Corollary 1.11, verify the two-part gluing estimate on which Proposition 4.3 relies. First, with σi,R=∂(Ci,R∩B1), determine whether the annulus A'_R⊂∂B1 bounded by σ1,R and σ3,R and having weighted area less than 1/4H2_w(Pxy∩B1) actually contains σ2,R; if not, the claimed intersection A''_R∩C2,R=σ2,R is false and the surgery cannot be performed as written. If it does, write out the Kapouleas-style desingularization of A''_R∪C2,R along σ2,R following [30, Definition 3.6] and compute the weighted-area increase of the resulting genus-g surface; check whether the infimum over the neck radius is strictly less than 1/4H2_w(Pxy∩B1) uniformly in R and g. If this inequality cannot be established, the fixed area drop driving the iteration collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 1.4 is reduced by Proposition 3.5 to Proposition 4.3, which must produce, for every large R, a G[g]-equivariant genus-g self-expander with boundary RL[C[g]]. The proof of Proposition 4.3 has two descriptive gluing steps. First, CHMR is asserted to exist by bending a (g+1)-periodic Scherk surface as in [30, Definition 3.6]; no existence or area estimate is given for this surface in Ω∩B_R with the required boundary and symmetry. Second, and more load-bearing, the iteration that forces the minimizer to become connected uses equation (4.6) and the statement that desingularizing A''_R∪C2,R along σ2,R can be done with area increase 'as small as one wants.' This requires two unproved facts: (i) the annulus A'_R⊂∂B1 bounded by σ1,R and σ3,R actually contains σ2,R, so that A''_R∩C2,R is the simple closed curve σ2,R rather than a collection of points or arcs; (ii) the Scherk-type desingularization along σ2,R can be performed with weighted-area increase uniformly smaller than 1/4H2_w(Pxy∩B1), independent of R and g, while preserving G[g]-equivariance and boundary. No estimate controlling the area of the (g+1) desingularizing necks in terms of the intersection geometry is provided. Without (ii) the claimed fixed drop of 1/2H2_w(Pxy∩B1) per iteration is not established, so the iteration need not terminate with a connected genus-g surface. Since this gap is in the proof of the main existence statement, the preprint does not establish Theorem 1.4 as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to construct, for a universal δ0>0 and every genus g, a complete connected G[g]-equivariant self-expander Σ[g] asymptotic to any cone C[g] in a class Π[g,δ0] of three-component cones with dihedral symmetry. The construction proceeds by minimizing weighted area in an equivariant isotopy class inside Ω∩B_R, using Meeks–Simon–Yau, then iterating a surgery that is supposed to enforce connectedness and genus g, and finally passing to a limit via Proposition 3.5. Applications include a mean curvature flow whose genus drops strictly at the singular time but not to zero, and a sequence of self-expanders of unbounded genus asymptotic to the same rotationally symmetric cone, with a convergence characterization.","tokens_in":24320,"tokens_out":16745,"duration_ms":169819,"significance":"If the main theorem holds, it gives the first explicit self-expanders of arbitrary positive genus in R3, confirms a conjecture of Chen on non-rotationally-symmetric expanders asymptotic to a rotationally symmetric cone, and provides the first example of a mean curvature flow with strict genus reduction that does not drop to zero. The paper's strategy is attractive: it combines equivariant min-max with an area-decreasing iteration, and it supplies complete proofs of the supporting compactness and regularity statements (Propositions 3.5 and 4.2) in the appendix. The advertised applications are conditional on the central existence result, so the weight of the paper rests entirely on Proposition 4.3.","major_comments":[{"comment":"The initial surface CHMR is asserted to exist by gluing a (g+1)-periodic Scherk surface as in [30, Definition 3.6], but no proof is given that such a G[g]-equivariant properly embedded genus-g surface exists inside Ω∩B_R with boundary RL[C[g]]. This surface is the input to the equivariant minimization of Proposition 4.2, so without a verifiable existence statement the whole iterative construction lacks a starting point.","section":"Proposition 4.3, first paragraph"},{"comment":"The claim that 'A''_R intersects C2,R only along the simple closed curve σ2,R' is not justified. Since A''_R∩∂B1 = A'_R∪σ1,R∪σ3,R and C2,R∩∂B1 = σ2,R, the assertion is equivalent to σ2,R⊂A'_R. The proof does not establish that σ2,R lies in the annulus of ∂B1 bounded by σ1,R and σ3,R rather than in one of the other two annuli of A\\A'_R; if σ2,R is not contained in A'_R, the proposed desingularization along σ2,R is not defined.","section":"Proposition 4.3, after Eq. (4.6)"},{"comment":"The statement that the desingularization can be performed 'with change of area as small as one wants' is asserted without proof. The iteration requires a uniform area drop of 1/2 H2_w(Pxy∩B1) at every step, but no estimate is provided for the weighted area of the G[g]-equivariant genus-g Scherk-type desingularizing surface, in particular no bound independent of R and g. Without such an estimate the iteration is not shown to terminate, so the connectedness and genus conclusion of Proposition 4.3 is not established.","section":"Proposition 4.3, desingularization of A''_R∪C2,R"},{"comment":"The area comparison (4.6) is computed for the singular surface obtained by replacing the two disks with the annulus A'_R⊂∂B1, but the resulting object has corners along σ1,R and σ3,R. The proof only says 'smoothing if necessary' and gives no area estimate for the smoothed surface. Since the claimed fixed area drop is the mechanism of the iteration, an estimate for the smoothed replacement is needed to make the argument rigorous.","section":"Proposition 4.3, Eq. (4.6) and smoothing"}],"minor_comments":[{"comment":"The proof says 'for all g > g0' where g0 is not defined; this should presumably be 'for all g∈N'.","section":"Proof of Theorem 1.4"},{"comment":"In the chain '2−2g(Σ_R,j)−3 = χ(Σ_R) = ...', the middle term should be χ(Σ_R,j), not χ(Σ_R).","section":"Eq. (4.7)"},{"comment":"The reliance on a Wikipedia figure and Wikipedia reference for Scherk's surface is not appropriate for a research paper; a standard reference or an original figure would be preferable.","section":"Figure 4 and reference [44]"},{"comment":"The abbreviation 'Π[g]' for Π[g,δ0] is introduced but the parameter δ0 is sometimes still written; please make the notation uniform.","section":"Notation 4.1"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses an interesting and likely true result, and the broad strategy is credible. However, Proposition 4.3 is the core of the paper and is currently a sketch: the initial Scherk-gluing construction, the topological placement of σ2,R, and the uniform area estimate for the desingularization are all unproved. These are exactly the points a referee must check before the theorem can be accepted. I would encourage the editor to request a substantive revision rather than a quick rejection, but the authors need to provide complete arguments for these gluing and surgery steps."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe result is important and plausible, but the central construction is a sketch where rigor matters, so Theorem 1.4 is not established as written.\n\nThe genuinely new content is the construction of connected self-expanders of arbitrary positive genus asymptotic to a broad class of triple cones in R^3, with applications to mean curvature flow with strict genus reduction without dropping to zero, and to a sequence of expanders of unbounded genus converging to a plane plus an annulus. These are substantial claims, and the paper deserves serious attention. The conceptual route — equivariant Meeks-Simon-Yau minimization in the isotopy class of a genus-g Scherk-gluing, with a fixed-area-decrease surgery iteration — is clear and appropriate. The limiting argument (Proposition 3.5) is written out in the appendix in real detail and looks correct.\n\nThe soft spots are in Proposition 4.3. The initial genus-g surface CHMR is asserted to exist by 'bending' a (g+1)-period of Scherk's Second Surface, with no proof or reference that yields the required G[g]-equivariant surface in Ω∩B_R with boundary RL[C]. That is a gap, but possibly a standard gluing one. More load-bearing is the surgery step. The paper claims A''_R intersects C2,R only along σ2,R, which requires that σ2,R lies in the annulus A'_R on ∂B1; this ordering of the three boundary curves is not proved. Then the desingularization of A''_R∪C2,R is asserted to increase weighted area 'as small as one wants.' The iteration's termination depends on a uniform bound — below 1/4 H^2_w(Pxy∩B1) — for that area excess, independent of R and g. No estimate is given. Without it, the fixed area drop of 1/2 H^2_w per iteration does not follow.\n\nThere are also small typographical issues: (4.7) should presumably be −kg, and the proof of Theorem 1.4 mentions g > g0 when the theorem states all g.\n\nBottom line: this paper should go to peer review, but the referee should demand a full proof of the gluing and the area estimate. As written it's a promising sketch, not a complete proof.","headline":"The result is important and plausible, but the central gluing/surgery argument in Proposition 4.3 is sketched, not proved, so Theorem 1.4 is not established as written.","tokens_in":24812,"tokens_out":18918,"would_cite":false,"duration_ms":173562,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C42","53E10","49Q05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs, for every positive genus g and every cone in a broad class Π[g,δ0], a complete connected G[g]-equivariant self-expander asymptotic to the cone, and derives a mean curvature flow whose genus drops from 2g to g at the…","keywords":["self-expanders","mean curvature flow","positive genus","asymptotically conical","genus reduction","fattening","Costa-Hoffman-Meeks surface","Scherk surface"],"falsifier":"Take the explicit cone Cε with ε>δ0 and compute, for large R, the weighted areas in the surgery inequality (4.6): if the annulus A'_R⊂∂B1 bounded by σ1,R and σ3,R is not contained in the barrier annulus A, or if $H^{2}$_w(A'_R)≥(1/4)$H^{2}$_w(Pxy∩B1), then the claimed fixed area drop of one half $H^{2}$_w(Pxy∩B1) fails and the iteration argument in Proposition 4.3 collapses.","tokens_in":23679,"feed_emoji":"🌊","tokens_out":9143,"duration_ms":88714,"temperature":0.7,"pith_summary":"This paper proves that positive-genus self-expanders exist in abundance in $R^{3}$: for every genus g and every cone in a broad class Π[g,δ0]—cones with three graphical link components and dihedral symmetry D_2(g+1), lying strictly outside a double cone of slope δ0—there is a complete connected self-expander asymptotic to that cone. Previously, self-expanders asymptotic to such cones were only known in genus zero. The construction has two direct consequences: a mean curvature flow whose genus drops from 2g to g at the first singular time but does not vanish, and, for a fixed rotationally symmetric triple cone, a sequence of self-expanders of unbounded genus asymptotic to the same cone, whose limit is a hyperplane together with a self-expander annulus. These results give the first example of strict-but-not-total genus reduction in a mean curvature flow, and they confirm that self-expanders emerging from a rotationally symmetric cone need not be rotationally symmetric.","feed_headline":"Self-expanders of every positive genus built for broad cone class","feed_subtitle":"A singularity cuts the genus from 2g to g, not to zero, in a new family of flows.","key_machinery":"The central construction minimizes weighted area $e^{{|X|^2/4}}$ $H^{2}$ among G[g]-equivariant isotopic deformations of a Costa-Hoffman-Meeks surface—a complete embedded minimal surface of genus g with three ends—in Ω∩B_R with boundary RL[C[g]]. To prevent the minimizer from degenerating into three disks, the proof uses barriers Λ1,Λ2 (rotationally symmetric graphical expanders asymptotic to a double cone) and an iterative surgery: if the minimizer has three disk components, replace the two outer disks by an annulus inside ∂B1 and desingularize along the remaining boundary circle using a bent (g+1)-periodic Scherk's Second Surface, decreasing weighted area by a fixed positive amount. Repeated minimization must terminate because each step lowers area by a definite amount while all candidates remain above a positive E-minimizing current. Riemann-Hurwitz plus the group symmetry then forces the final connected surface to have exact genus g, and a barrier/compactness proposition passes the ball-wise construction to a complete self-expander asymptotic to the cone.","core_discovery":"On its own terms, the central discovery is Theorem 1.4: there is a universal constant δ0>0 such that for any g∈N and any C[g]∈Π[g,δ0], there exists a complete connected G[g]-equivariant self-expander Σ[g] of genus g asymptotic to C[g]. Equivalently, the cone C[g] admits at least two different self-expanders: the previously known union of three genus-zero graphical sheets and the new positive-genus surface; both are asymptotic to C[g] but have different topology and are not related by symmetry. From this the paper derives a mean curvature flow whose genus strictly drops at the singular time—from 2g to g—while remaining positive (Corollary 1.8), and a sequence of genus-g expanders asymptotic to the same cone Cε whose high-genus limit is a hyperplane together with a self-expander annulus (Corollary 1.11 and Proposition 6.2).","pith_inferences":["Editorial inference: if the underlying gluing step can be made quantitative with the claimed area drop, the same iteration should also produce self-expanders with more ends or other prescribed symmetry groups, since the only topological input is a symmetric genus-g model surface and a fixed area gap.","Editorial inference: the high-genus limit structure suggests a general concentration phenomenon—for expanders asymptotic to a cone with a symmetric waist, positive genus is paid for by a circle where curvature concentrates, while the rest of the surface becomes graphical; this could be tested numerically for Cε.","Editorial inference: the genus drop 2g→g indicates that topology loss at a conical singularity is quantized by the difference between the shrinker and expander genera; stacking more periodic Scherk-like sheets might produce flows with larger prescribed genus drops, though the paper does not construct them.","Editorial inference: the unstable expanders Σ[g] should, through the expander Morse-flow theorem cited in the paper, connect to stable expanders asymptotic to the same cone; a concrete prediction is that these stable limits are the graphical genus-zero sheets or a stable annulus, which would make the fattening phenomenon more explicit."],"forward_implications":["For every genus g, the cone C[g] has at least two self-expanders asymptotic to it—one of genus zero and one of genus g—so self-expanders asymptotic to a fixed cone are not unique in a strong topological sense; the paper cites a fattening theorem showing the level set flow starting from C[g] fattens.","Corollary 1.8: mean curvature flows exist whose genus drops from 2g to g at the first singular time and never drops to zero, providing the first example of strict but incomplete genus reduction.","For each ε>δ0, there are connected genus-g self-expanders Σ[g,ε] asymptotic to the rotationally symmetric triple cone Cε, confirming that not all such expanders are rotationally symmetric.","As g→∞, a subsequence of {Σ[g,ε]} converges locally smoothly away from a circle to the union of the xy-plane Pxy and a self-expander annulus asymptotic to the double cone C(ε); the high genus concentrates near that circle.","For ε>δ0 the level set flow of Cε is fully characterized: it divides R^3 into four evolving components, contains √t·Σ[g] in its interior for t>0, contains the stable expanders in its boundary, and its time-one boundary is a smooth stable self-expander."],"supporting_citations":[{"why":"Supplies the equivariant isotopy-minimization and regularity result used to produce each candidate minimizer in Ω∩B_R.","marker":"[38]"},{"why":"Provides the Costa-Hoffman-Meeks genus-g symmetric surface whose isotopy class is minimized.","marker":"[27]"},{"why":"Provides the desingularization and gluing technique, bending a (g+1)-period of Scherk's Second Surface, used to build the initial genus-g surface and perform surgeries.","marker":"[30]"},{"why":"Provides existence and uniqueness of graphical self-expanders asymptotic to graphical cones, used for the barriers Λ1,Λ2 and for Lemma 2.4.","marker":"[25]"},{"why":"The barrier and limit scheme in Proposition 3.5 is adapted from this source's construction of asymptotically conical expanders.","marker":"[24]"},{"why":"Supplies the barrier lemma and instability result that force the limit expander to be asymptotic to the prescribed cone and the constructed expanders to be unstable.","marker":"[9]"},{"why":"Supplies expander Morse flow lines from unstable to stable self-expanders, used in the fattening corollary.","marker":"[4]"},{"why":"Supplies the local splitting and curvature-concentration analysis adapted in Proposition 6.2 to identify the high-genus limit.","marker":"[12]"},{"why":"Constructs the genus-2g shrinkers whose asymptotic cones lie in Π[g,δ0], used to obtain the genus-drop mean curvature flow.","marker":"[40]"}],"fun_headline_variants":["Every positive genus: self-expanders for broad cone class","Genus drop from 2g to g, but not to zero in new flow","Self-expanders of any genus asymptotic to symmetric cones","New flow: genus decreases at singularity yet stays positive","Unbounded genus expanders limit to hyperplane plus annulus"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on an unproved quantitative gluing estimate: replacing two disk pieces by an annulus and desingularizing along a circle must cost less area than a fixed positive amount, for every genus g and every large radius R. If that estimate fails for some genus or radius, the iterative minimization that forces a connected positive-genus surface need not terminate.","fun_headline_variants_meta":{"raw":{"variants":["Every positive genus: self-expanders for broad cone class","Genus drop from 2g to g, but not to zero in new flow","Self-expanders of any genus asymptotic to symmetric cones","New flow: genus decreases at singularity yet stays positive","Unbounded genus expanders limit to hyperplane plus annulus"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0002,"raw_usage":{"total_tokens":1304,"prompt_tokens":806,"completion_tokens":498,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":422,"completion_tokens_details":{"reasoning_tokens":412}},"tokens_in":422,"tokens_out":498,"duration_ms":5116,"temperature":1.0,"reasoning_tokens":412,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:55:03.484346+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the explicit cone Cε with ε>δ0 and compute, for large R, the weighted areas in the surgery inequality (4.6): if the annulus A'_R⊂∂B1 bounded by σ1,R and σ3,R is not contained in the barrier annulus A, or if $H^{2}$_w(A'_R)≥(1/4)$H^{2}$_w(Pxy∩B1), then the claimed fixed area drop of one half $H^{2}$_w(Pxy∩B1) fails and the iteration argument in Proposition 4.3 collapses.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the equivariant isotopy-minimization and regularity result used to produce each candidate minimizer in Ω∩B_R."},{"cited_title":"Meeks III, Embedded minimal surfaces of finite topology, Ann","cited_arxiv_id":null,"evidence_quote":"Provides the Costa-Hoffman-Meeks genus-g symmetric surface whose isotopy class is minimized."},{"cited_title":"Differ- ential Geom","cited_arxiv_id":null,"evidence_quote":"Provides the desingularization and gluing technique, bending a (g+1)-period of Scherk's Second Surface, used to build the initial genus-g surface and perform surgeries."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides existence and uniqueness of graphical self-expanders asymptotic to graphical cones, used for the barriers Λ1,Λ2 and for Lemma 2.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The barrier and limit scheme in Proposition 3.5 is adapted from this source's construction of asymptotically conical expanders."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the barrier lemma and instability result that force the limit expander to be asymptotic to the prescribed cone and the constructed expanders to be unstable."},{"cited_title":"Schulz, Noncompact self-shrinkers for mean curvature flow with arbitrary genus, J","cited_arxiv_id":null,"evidence_quote":"Supplies the local splitting and curvature-concentration analysis adapted in Proposition 6.2 to identify the high-genus limit."},{"cited_title":"Self-shrinkers with any number of ends in $\\mathbb{R}^{3}$ by stacking $\\mathbb{R}^{2}$","cited_arxiv_id":"2507.18825","evidence_quote":"Constructs the genus-2g shrinkers whose asymptotic cones lie in Π[g,δ0], used to obtain the genus-drop mean curvature flow."}],"review_version":2}