{"id":"fb9d4707-076e-472a-a935-839079ced0db","arxiv_id":"2507.13189","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Plateau-quasi-minimizers in co-dimension one are characterized, up to the boundary, by bi-John domains with Ahlfors regular boundaries.","lead":"This paper proves that quasi-minimizers of perimeter with a fixed boundary constraint, called Plateau-quasi-minimizers, are exactly the sets equivalent to bi-John domains with Ahlfors regular boundaries. The result is the optimal regularity statement up to the boundary, extending the interior theory of David and Semmes to Plateau's problem in co-dimension one.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.4's proof applies Lemma 2.6 to the non-open set B(x,s)\\Ω0^(0) and uses an equality of perimeters that is not justified and can fail; the boundary Ahlfors regularity and the direct implication of Theorem 5.1 are therefore not established as written.","rationale":"The reader's weakest assumption identifies exactly the same gap: Lemma 3.4 uses a local form of the convex-intersection perimeter inequality on a non-open set, without proof. My analysis confirms the step is not a minor omission: the displayed equality of perimeters is equivalent to a balance of two H^{N-1} terms that the hypotheses do not control, and the direction needed is fragile. I also checked the surrounding argument: Proposition 3.3 lower bound depends on Lemma 3.4, Theorem 3.1 depends on Proposition 3.3, and the direct implication of Theorem 5.1 depends on Theorem 3.1. Thus the concern is load-bearing for the main characterization. I found no independent reason to reject the theorem; the gap appears repairable by proving the local inequality under the no-interface condition or by replacing the step with a two-sided estimate. Therefore the reader's CONDITIONAL verdict is appropriate, and no change is needed.","tokens_in":32959,"tokens_out":17411,"duration_ms":190355,"concrete_test":"Re-derive the disputed step in Lemma 3.4 using U = B(x,s)\\Ω0^(0) and the identity P(E,U)=H^{N-1}(∂*E∩U). Concretely, in R^2 take C={x2<0}, x=0, r=1, s=2, and construct a competitor Ω0 so that Ω0=1 on B(0,1)∩C and on B(0,1)∩{x2>0}, and Ω0=0 just outside B(0,1) near ∂B(0,1)∩{x2>0}, with E0 chosen accordingly so that Σ avoids the open ball B(0,1). Then ∂B(0,1)∩{x2>0} is in ∂*Ω0 and hence in U, while ∂C∩B(0,1) is a density-1 set and hence also in U. Computing both sides gives P(B(0,1)∩C,U) = H^1(∂B(0,1)∩{x2<0}∩U) + H^1(∂C∩B(0,1)) = π + 2, whereas P(B(0,1),U) = H^1(∂B(0,1)∩U) = π. If this configuration satisfies the standing hypotheses of Lemma 3.4, the asserted equality is false and the lemma needs an additional estimate; if it does not, state explicitly which hypothesis excludes it and prove that exclusion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is in Lemma 3.4, Case 2 (pp. 17–18). The author sets U := B(x,s)\\Ω0^(0) and uses the displayed equality P(B(x,r)∩C, U) = P(B(x,r), U). This is not a consequence of Lemma 2.6, which is proved only for an open set Ω; U is not open and no approximation argument is supplied. For arbitrary measurable U the identity is false: P(B∩C,U) = H^{N-1}(∂B∩C∩U) + H^{N-1}(B∩∂C∩U), while P(B,U) = H^{N-1}(∂B∩U), so the asserted equality is equivalent to H^{N-1}(B∩∂C∩U) = H^{N-1}(∂B∩(R^N\\C)∩U). The hypothesis Σ∩B(x,r)=∅ only gives ∂*Ω0∩(B(x,r)\\C)=∅; it does not force Ω0^(0) to avoid ∂C, nor does it control ∂B∩(R^N\\C)∩U. Because the term is subtracted before applying Lemma 2.6, the direction needed for (3.11) is P(B∩C,U) ≤ P(B,U), and the equality is not merely cosmetic. A set U with positive H^{N-1} on B∩∂C and negligible on ∂B∩(R^N\\C) violates this. Proposition 3.3 and Theorem 3.1 rely directly on (3.11), and Theorem 5.1's direct implication relies on Theorem 3.1, so the central characterization is not established as written unless the local inequality is proved under additional assumptions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Plateau-quasi-minimizers: finite-perimeter competitors for a De Giorgi-type Plateau problem in a bounded domain D, with a prescribed set E0 outside a convex open set C, satisfying the asymmetric comparison H^{N-1}((∂*Ω0\\∂*Ω)∩D) ≤ Q H^{N-1}((∂*Ω\\∂*Ω0)∩D). Under Hypothesis H (Ahlfors regularity and Condition B' for Σ=∂E0∩(D\\C), plus isoperimetric domain assumptions), the main theorem (Theorem 5.1) asserts that Ω0 is a Plateau-quasi-minimizer if and only if it is equivalent to a bi-John domain with Ahlfors regular boundary. The proof proceeds in three stages: Ahlfors regularity up to the boundary (Section 3), uniform rectifiability/BPLG via Condition B (Section 4), and the bi-John characterization (Section 5), with a weighted Plateau problem for the converse. Section 6 gives a Lipschitz-graph example; Section 7 extends the result to bi-Lipschitz images of C.","tokens_in":33277,"tokens_out":32509,"duration_ms":367426,"significance":"The result, if established, would be a natural and valuable extension of David–Semmes and Rigot to boundary-value (Plateau-type) quasi-minimizers, giving optimal up-to-the-boundary regularity and a converse geometric characterization. The manuscript is largely self-contained, re-proves standard GMT preparatory lemmas, and provides a concrete example and a clean bi-Lipschitz invariance argument. The main theorem is precise and falsifiable. However, the proof of the crucial boundary Ahlfors regularity lemma (Lemma 3.4) contains an unjustified local use of a global convex-intersection inequality; since both directions of Theorem 5.1 rely on Lemma 3.4, the main claim is not established as written.","major_comments":[{"comment":"The proof applies Lemma 2.6 to the measurable set U = B(x,s) ∩ Ω0^(0) and asserts the displayed equality P(B(x,r)∩C, U) = P(B(x,r), U) immediately before (3.11). Lemma 2.6 is a global inequality for open sets; for an arbitrary measurable U the analogous local inequality P(E∩K,U) ≤ P(E,U) is false (for example, take U = E ∩ ∂K, which gives a positive left-hand side and zero right-hand side). The hypothesis Σ ∩ B(x,r) = ∅ only gives ∂*Ω0 ∩ B(x,r) ⊂ C; it does not control Ω0^(0) ∩ ∂C or Ω0^(1) ∩ ∂C, which are exactly the sets responsible for the difference between P(B(x,r)∩C, ·) and P(B(x,r), ·). Concretely, the asserted equality is equivalent to H^{N-1}(B(x,r)∩∂C∩U) = H^{N-1}(∂B(x,r)∩(R^N\\C)∩U), and nothing in the assumptions forces this balance. The proof therefore needs a genuinely local estimate of the form P(B(x,r)∩C, W) ≤ P(B(x,r), W) for W = B(x,s)∩(Ω0^(1)∪∂*Ω0), or an additional argument showing the two subtracted terms cancel; none is provided. Since (3.11) is used in Proposition 3.3, Theorem 3.1, Theorem 4.4 and the direct implication of Theorem 5.1, this gap is load-bearing.","section":"Lemma 3.4, Case 2 (pp. 17–18)"},{"comment":"The lower-semicontinuity argument states that “U, D\\U and U\\∂Ω0 are open sets” and applies lower semicontinuity of perimeter to the term H^{N-1}(∂*Ω_n ∩ (D\\U)). Since U is open, D\\U is closed, and perimeter on a closed set is not lower semicontinuous under L1 convergence. The preceding choice of U with H^{N-1}(∂*Ω_n∩∂U)=0 and H^{N-1}(∂*Ω∩∂U)=0 appears to permit a repair by replacing D\\U with D\\overline{U}, but as written the existence proof for minimizers of the weighted problem (5.6), which is needed in the converse implication, is incomplete. This issue is local and likely fixable, unlike the gap in Lemma 3.4.","section":"Proposition 5.8, Section 5.2"}],"minor_comments":[{"comment":"The heading “Charectization by bi-John domain” should read “Characterization by bi-John domain”.","section":"Section 7.3"},{"comment":"The sentence “The first inequality from the previous calculation is represented in Figure 6” refers to an equality, not an inequality; the wording should be corrected.","section":"Lemma 3.4, after the main computation"},{"comment":"The displayed inclusion ∂∗F ⊂ ∂∗F ⊂ ∂F appears to contain a typo; presumably one occurrence should be ∂*F (the essential boundary), otherwise the inclusion is circular.","section":"Remark 1.9"},{"comment":"The definition D = λC with λ>1 ensures C ⊂ D only when the dilation is centered at a point of C; since C is an arbitrary convex set, this should be stated explicitly or the dilation center should be chosen inside C.","section":"Section 6.1"},{"comment":"The boundary condition E0 = A+ is defined on the side shell of D\\C, but the top and bottom caps of the dilated cylinder also belong to D\\C and are not described; the example should specify E0 there.","section":"Section 6.1, Definition of E0"},{"comment":"When applying (2.5) to P(Ω\\ω), the term H^{N-1}({νΩ = -νω}) is omitted without comment; it vanishes for open ω ⊂ Ω because the reduced-boundary normals coincide rather than oppose, but this justification should be stated.","section":"Proposition 5.5"}],"recommendation":"major_revision","confidential_remarks":"The central issue is confined to the proof of Lemma 3.4; the surrounding architecture is coherent and the result is plausible. I believe the gap can be repaired by proving a local convex-intersection estimate under the boundary condition or by choosing the competitor differently, but that repair is essential. The Proposition 5.8 issue is also real but local. No concerns about attribution or scope beyond the technical gap."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's contribution is real. It adds a boundary condition to the David-Semmes quasi-minimizer framework and characterizes Plateau-quasi-minimizers as bi-John domains with Ahlfors regular boundary. The converse direction, using a weighted Plateau problem, is clean and convincing. The technical work in Sections 3–5 is mostly careful, and the example plus bi-Lipschitz extension are useful additions.\n\nThe soft spot is Lemma 3.4, and the stress-test concern holds up. In Case 2, the author replaces P(B(x,r)∩C, U) with P(B(x,r), U) for U = B(x,s) \\ Ω0^(0). Lemma 2.6 gives P(E∩K, A) ≤ P(E,A) for Borel A (the extension from open sets to Borel sets is standard), but for the subtracted term this is the wrong direction: P(B∩C,U) ≤ P(B,U) makes the bracket larger, not smaller. You need P(B∩C,U) ≥ P(B,U), or an equality, and nothing in the hypothesis Σ∩B(x,r)=∅ forces the extra boundary term H^{N-1}(B∩∂C∩U) to vanish. As written, the boundary Ahlfors regularity and the direct implication of Theorem 5.1 are not established. This looks repairable, but it is a genuine gap, not a cosmetic one.\n\nA minor issue: in the Section 6 example, A+ is defined with r∈]1,λ[, so its essential boundary inside D includes the inner wall r=1. The claim that Σ = ∂*A+∩D is at least imprecise. Not load-bearing, but worth fixing.\n\nWho this is for: geometric measure theory and calculus of variations readers who care about quasi-minimizers and Plateau's problem. The statement is worth having once the gap is patched. I would recommend peer review with a request for revision, and I would not cite the main theorem in its current form.","headline":"A serious, mostly careful paper with a plausible main theorem, but Lemma 3.4 has a real gap that blocks the direct implication as written; send it to a referee, not to the printer.","tokens_in":33872,"tokens_out":8932,"would_cite":false,"duration_ms":97711,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q20","49Q05","28A75"],"pacs":[],"model":"deepseek-v4-flash","headline":"Plateau-quasi-minimizers with prescribed boundary data are, up to an equivalent set, exactly the bi-John domains with Ahlfors regular boundary.","keywords":["Plateau's problem","quasi-minimizers of perimeter","sets of finite perimeter","bi-John domains","Ahlfors regular boundary","uniform rectifiability","boundary regularity","co-dimension one"],"falsifier":"A concrete check is to compute, in the plane, $P(E\\cap K,B(x,r))$ versus $P(E,B(x,r))$ when $E$ is a cusp-shaped set touching the convex set $K$ at a single boundary point with zero density, using boundary data of the type constructed in Section 6; if for some admissible $(D,C,E_0)$ satisfying Hypothesis H there is a Plateau-quasi-minimizer and a boundary ball with $\\liminf_{r\\to0}P(\\Omega_0,B(x,r))/r^{N-1}=0$, then Theorem 3.1 is false, and more narrowly, any example with $P(E\\cap K,B)>P(E,B)+c$ for some $c>0$ would disprove the local Lemma 2.6 step used in Lemma 3.4.","tokens_in":32680,"feed_emoji":"📐","tokens_out":7939,"duration_ms":85608,"temperature":0.7,"pith_summary":"This paper tries to prove that quasi-minimal solutions of Plateau's problem in co-dimension one, meaning sets whose boundary perimeter is within a fixed factor $Q$ of every admissible competitor, have optimal regularity all the way to the prescribed boundary. The main theorem characterizes them: a set with given boundary data outside a convex region is a Plateau-quasi-minimizer exactly when it can be replaced, up to a measure-zero change, by a bi-John domain with Ahlfors regular boundary. That matters because it converts a variational condition into a purely geometric one, and the resulting boundary regularity is used in phase-field approximations of Plateau's problem.","feed_headline":"Quasi-minimal surfaces with a boundary are bi-John domains","feed_subtitle":"Optimal regularity: up to measure-zero changes, any quasi-optimal surface with fixed boundary data has Ahlfors regular boundary.","key_machinery":"Three mechanisms carry the argument. Hypothesis H packages the assumptions on the boundary datum: $\\Sigma=\\partial E_0\\cap(D\\setminus C)$ is Ahlfors regular, $E_0$ satisfies Condition B$'$ (balls of $E_0$ and of its complement on both sides of $\\Sigma$), and the two regions outside $C$ are isoperimetric domains. Lemma 3.4 is the key local tool: for balls $B(x,r)$ that avoid $\\Sigma$, it compares $P(\\Omega_0,B(x,s))$ with $Q\\,P(\\Omega_0\\setminus B(x,r),B(x,s))$ by cutting with the convex set $C$ and using the convex-intersection perimeter inequality (Lemma 2.6); this lets interior Ahlfors regularity and Condition B be replayed at boundary balls. The converse uses a weighted Plateau problem (Definition 5.7), whose minimizers are quasi-minimizers, together with a boundary-comparison estimate (Lemma 5.11) showing that two bi-John competitors with regular boundaries and the same boundary data must have the same boundary almost everywhere.","core_discovery":"The central claim is Theorem 5.1. Let $(D,C,E_0)$ satisfy Hypothesis H and let $\\Omega_0$ be a competitor. Then $\\Omega_0$ is a Plateau-quasi-minimizer if and only if there exists an equivalent open set $\\Omega$ that is a bi-John domain with regular boundary in the sense of Definition 1.11, meaning $\\Omega$ is open, $\\operatorname{spt}\\mu_\\Omega=\\partial\\Omega$, and $\\partial\\Omega$ is Ahlfors regular in $D$. Equivalently, every co-dimension one quasi-optimal surface fixed outside a convex set is, up to a negligible modification, a domain whose interior and exterior are both John domains and whose boundary has Hausdorff measure comparable to $r^{N-1}$ in every ball; conversely, every such domain with the same boundary data is a quasi-minimizer. The forward direction is proved by establishing Ahlfors regularity up to the boundary, then uniform rectifiability and the Big Pieces of Lipschitz Graphs property up to the boundary, then isoperimetry for the domain and its complement, and finally invoking the John-domain criterion; the converse is proved through a weighted Plateau problem whose minimizers are automatically quasi-minimizers and which forces the boundary of the given bi-John domain to coincide almost everywhere with the minimizer.","pith_inferences":["Editorial inference: the most delicate point to test is the unproved local form of the convex-cut inequality in Lemma 3.4; if it fails, the theorem might still hold but would need a different argument at boundary balls where $\\partial C$ and $\\partial^*\\Omega_0$ interact.","Editorial inference: the weighted Plateau problem used for the converse suggests a constructive route to quasi-minimizers, namely minimizing perimeter with a large penalty away from a prescribed bi-John boundary and then letting the penalty tend to infinity; the paper does not study this limit.","Editorial inference: the characterization suggests a compactness heuristic, that uniform John and Ahlfors constants with fixed boundary data should prevent degeneration and make sequences of quasi-minimizers subconverge to quasi-minimizers; the paper does not state such a compactness theorem."],"forward_implications":["Every Plateau-quasi-minimizer has an equivalent open representative whose topological boundary is Ahlfors regular in $D$, so perimeter in balls scales like $r^{N-1}$ up to the boundary.","The same representative satisfies Condition B and hence has Big Pieces of Lipschitz Graphs, so the quasi-optimal surface is uniformly rectifiable up to the boundary.","The characterization is sharp: cusps or other non-John boundary behavior in a set with the same boundary data force the quasi-minimality condition to fail.","The regularity transfers by bi-Lipschitz maps to non-convex containers such as curved cylinders, because bi-Lipschitz maps preserve essential boundaries, Ahlfors regularity, the BPLG property, and the bi-John property.","Ahlfors regularity up to the boundary underpins a phase-field, $\\Gamma$-convergence type approximation of Plateau's problem."],"supporting_citations":[{"why":"Supplies the interior version of the characterization and the two workhorse results, Condition B plus isoperimetry implying the John property and the maximal-function estimate used in the boundary comparison.","marker":"[DS98]"},{"why":"Supplies the finite-perimeter toolbox: perimeter equals Hausdorff measure of the essential boundary, isoperimetric and relative isoperimetric inequalities, the existence of equivalent Borel sets with support equal to the boundary, and the convex-cut perimeter inequality behind Lemma 2.6.","marker":"[Mag12]"},{"why":"Supplies the proof scheme and lemmas, including the density function $h$, the open equivalent set $O_0$, and Condition B inside the convex set, reused in Section 4.","marker":"[Rig00]"},{"why":"Supplies Condition B and the implication that Condition B yields Big Pieces of Lipschitz Graphs and hence uniform rectifiability.","marker":"[Dav88]"},{"why":"Supplies the area formula and the BV composition rule used in the bi-Lipschitz transfer.","marker":"[AFP00]"},{"why":"Supplies the result that bi-Lipschitz maps send density points to density points, used to transfer essential boundaries.","marker":"[Buc92]"},{"why":"Supplies the equivalence of carrot and cigar John domains, used to place the John center inside the boundary component $A_+$.","marker":"[Väi88]"}],"fun_headline_variants":["Plateau-quasi-minimizers are bi-John domains up to null sets","Up to null sets, quasi-minimizers with boundary are bi-John","Quasi-minimizers: bi-John up to null sets","Quasi-minimizers with boundary are bi-John up to null sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The boundary Ahlfors regularity step assumes a local version of the convex-cut perimeter inequality: replacing a set by its intersection with a convex set inside a ball never increases the perimeter measured in that ball, whereas the paper proves the inequality only for the whole domain.","fun_headline_variants_meta":{"raw":{"variants":["Plateau-quasi-minimizers are bi-John domains up to null sets","Up to null sets, quasi-minimizers with boundary are bi-John","Quasi-minimizers: bi-John up to null sets","Quasi-minimizers with boundary are bi-John up to null sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001796,"raw_usage":{"total_tokens":7050,"prompt_tokens":896,"completion_tokens":6154,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":6081}},"tokens_in":512,"tokens_out":6154,"duration_ms":47743,"temperature":1.0,"reasoning_tokens":6081,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:33:21.421994+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is to compute, in the plane, $P(E\\cap K,B(x,r))$ versus $P(E,B(x,r))$ when $E$ is a cusp-shaped set touching the convex set $K$ at a single boundary point with zero density, using boundary data of the type constructed in Section 6; if for some admissible $(D,C,E_0)$ satisfying Hypothesis H there is a Plateau-quasi-minimizer and a boundary ball with $\\liminf_{r\\to0}P(\\Omega_0,B(x,r))/r^{N-1}=0$, then Theorem 3.1 is false, and more narrowly, any example with $P(E\\cap K,B)>P(E,B)+c$ for some $c>0$ would disprove the local Lemma 2.6 step used in Lemma 3.4.","supporting_citations":[],"review_version":1}