{"id":"1deb37b0-cfa1-4ec5-b2a9-fdc9c33d3a29","arxiv_id":"2411.14376","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Minimizing sequences for the area of Lipschitz graphs from R^3 to R^2 can converge to Cartesian currents with large interior vertical, non-minimal patches, in the smallest possible dimension and codimension.","lead":"The paper constructs examples where minimizing the area of a graph in higher codimension leads, in the limit, to an object with a large interior vertical piece that is not area-minimizing. These are the first such examples in the smallest possible dimension (3) and codimension (2), answering open questions in the theory of Cartesian currents.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The abstract's central claim requires the graph of u to be a limit of a minimizing sequence of Lipschitz graphs, but the sequence is only asserted ('easy to see') and the gluing near ∂U is not demonstrated.","rationale":"The reader's CONDITIONAL verdict is appropriate; I identify a different load-bearing gap than the reader's weakest assumption. The Cauchy-Kovalevskaya extension and sign estimates, while terse, are actually proved in Section 4.1 and appear sound after checking the tangential-derivative cancellations. The genuinely unproved step is the 'easy to see' minimizing sequence. Without it, Theorem 1.2 shows only that the graph of u is a lower bound for Lipschitz graph area, not that it arises as a limit of minimizing sequences. The paper's Remark 4.3 claims the same argument gives minimization among Cartesian currents, but again the link to Lipschitz approximability is not detailed. This is a fillable but nontrivial gap, so CONDITIONAL rather than ACCEPT is correct.","tokens_in":15439,"tokens_out":40847,"duration_ms":374127,"concrete_test":"Construct explicitly, for each k, a Lipschitz map v_k: U → R^2 with v_k = u on ∂U such that v_k equals (after a suitable reparametrization) the rotated graph of w_k = w+(x1/k,0) outside a τ_k-collar of ∂U, and prove that |area(graph v_k) - area(Gr(u))| → 0 as k→∞, τ_k→0. A natural choice is to define v_k on the collar by linear interpolation between u and the reparametrized u_k, and to estimate the area of the interpolation region using the smallness of |Dw| and the C^0-closeness of u_k to u near ∂U. If such a sequence exists, the abstract's 'limits of minimizing sequences' claim is justified; if the area error cannot be made to vanish, the central claim must be reformulated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper proves (Theorem 1.2) that the graph of u has smaller area than any Lipschitz graph from U to R^2 with the same boundary data. This establishes a lower bound, but the central claim in the abstract—that limits of minimizing sequences of Lipschitz graphs can have large interior vertical and non-minimal portions—requires an explicit minimizing sequence (v_k) of Lipschitz maps with boundary data u|∂U whose graphs converge to the Cartesian current Gr(u) and whose areas converge to area(Gr(u)). The paper asserts this is 'easy to see' by rotating the graphs of w+(x1/k,0) and gluing near ∂U. This is not carried out: the rotated maps u_k are graphs over U_k:=y_k(Ω), which differs from U, so they are not defined on U; one must reparametrize or modify them in a collar of ∂U, and the area of this modification must be shown to vanish. No such construction or estimate is given. If the gluing cannot be done with area loss →0, the infimum among Lipschitz graphs could be strictly larger than area(Gr(u)), and the example would not be a limit of minimizing sequences, invalidating the claimed answer to the Giaquinta-Modica-Souček questions. This gap is logically independent of the local Cauchy-Kovalevskaya construction and is the least secure link between Theorem 1.2 and the paper's stated conclusion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a C^{1,1} map w from a bounded convex domain Ω⊂R^3 to R^2 satisfying the minimal surface system outside an analytic convex subdomain Ω0, with ∂1w1≥0 and {∂1w1=0}=closure(Ω0). By swapping the x1 and w1 axes, the authors obtain a map u on U=y(Ω) that is analytic outside the surface Σ=y(Ω0) and develops a three-dimensional vertical patch over Σ. They prove by a calibration argument (Theorem 1.2) that the graph of u has area no larger than that of any Lipschitz graph over U with the same boundary data, and they conclude that limits of minimizing sequences of Lipschitz graphs can have large interior vertical and non-minimal portions, giving negative answers to questions of Giaquinta-Modica-Souček. The paper also constructs a point-singularity example in the minimal dimension n=3 and codimension m=2, and shows that a previous special Lagrangian example does not minimize area among Lagrangian graphs.","tokens_in":15683,"tokens_out":14405,"duration_ms":142490,"significance":"If the construction is correct, the result is significant: it provides the first examples showing that non-parametric area minimization in higher codimension can produce interior vertical patches of positive area that are not minimal, in sharp contrast with the codimension-one theory. The free-boundary construction via Cauchy-Kovalevskaya, together with the new calibration argument based on the convexity of the area integrand, is an interesting and potentially reusable technique. The point-singularity example is also valuable as the first singular solution to the minimal surface system in the minimal dimension and codimension. The paper is carefully written and the main algebraic steps are reproducible, though some estimates are compressed.","major_comments":[{"comment":"The assertion that the graph of u is the limit of a minimizing sequence of Lipschitz graphs is not demonstrated. The natural approximants are obtained by rotating the graphs of w+(x1/k,0), but these are maps defined on U_k=y_k(Ω), not on U. The paper does not explain how to reparametrize or glue them to u near ∂U, nor does it estimate the area of the modification. Theorem 1.2 is only a comparison inequality against all Lipschitz graphs; it does not by itself provide a sequence attaining the infimum. Without such a sequence, the negative answers to the Giaquinta-Modica-Souček questions stated in the introduction do not follow from the calibration theorem alone. This is a load-bearing gap and should be fixed, either by giving the missing gluing construction with a vanishing area-loss estimate or by explicitly reformulating the conclusions for minimizers among Cartesian currents, which are already addressed in Remark 4.3.","section":"Introduction (paragraph after Thm. 1.2) and §4.3"},{"comment":"The proof that ∂ν ṽ1_1 > 0 on ∂Ω0\\Γ and ∂νν ṽ1_1 > 0 on Γ is too compressed to be readily verified. In particular, after differentiating the equations in the e1 direction at p∈Γ, the claim that 'the first and second terms can be rewritten as expressions involving the curvature' and hence vanish using D2v1 = D2ṽ1 at p is not written out. These estimates are exactly what guarantee that {w1_1 = 0} = closure(Ω0), and therefore that the rotated map u has the claimed vertical patch. Please provide a complete derivation of the two sign estimates, including the role of the chosen coordinates and the geometry of ∂Ω0.","section":"§4.1, sign estimates after equation (10)"}],"minor_comments":[{"comment":"The symbol U is used both for Ω×R^m in the Cartesian-current preliminaries and for y(Ω) in the main construction. Please use a different letter for one of these to avoid ambiguity.","section":"§2.3 and §4.2"},{"comment":"Before defining the form ω̃, please make explicit the coordinate identification after rotation: (y1,y2,y3,u1,u2) corresponds to (w1,x2,x3,x1,w2), so that the variables z1 and z2 in the calibration argument are w1 and w2 respectively. Without this, the expressions dz^1∧dx^2∧... are difficult to follow.","section":"§4.3, beginning"},{"comment":"The statement that taking ǫ and then δ small makes |Dw| arbitrarily small in Ω is plausible but not proved. Since the calibration in §4.3 requires |Dw| < δ(n,m), please spell out the parameter choices.","section":"Remark 4.2"},{"comment":"There are several typographical errors from the arXiv source, including 'surf ace' in the title, 'th e' in the abstract, and 'the the' in Remark 1.3. These should be corrected in the final version.","section":"Throughout"},{"comment":"The expansion (11) is stated with O(λ^2)O(|x|^2); it would be clearer to write a single term O(λ^2 |x|^2), and later a similar notational cleanup would help in the displayed estimate for the integral over Ω0.","section":"§5, after equation (11)"}],"recommendation":"major_revision","confidential_remarks":"The paper contains an interesting and likely correct construction, but the missing minimizing-sequence argument is a genuine gap between Theorem 1.2 and the paper's central claim. It is probably fixable with a standard collar-gluing argument, so I would not reject on that basis. The sign estimates in §4.1 also need to be expanded for the paper to be verifiable. I recommend major_revision rather than reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read Mooney-Savin. The headline: this is a genuinely new construction in the minimal dimension and codimension, and if it holds up it settles three questions from Giaquinta-Modica-Souček in the negative—limits of area-minimizing Lipschitz graphs can develop large interior vertical parts with positive mass that are not minimal. That is a real result, not a marginal one.\n\nWhat is actually new: the point-singularity solution in n=3, m=2 (Section 3) is the first in the minimal dimension/codimension, and the large-singularity example (Section 4) produces a Cartesian current whose vertical patch projects to an interior analytic surface and is non-minimal. The construction is coherent: build v via Cauchy data so that f = ∂ᵢ(√det g g^{ij} ∂ⱼ v¹) = ∂₁H, take Ω₀ = {H > 0}, extend v across ∂Ω₀ by Cauchy-Kovalevskaya, and prove sign estimates on normal derivatives of ṽ¹₁ so that {w¹₁ = 0} is exactly Ω̄₀. The sign computation in 4.1 is terse but the pieces are present. The calibration in 4.3 is the right tool: the convexity inequality (5) gives a form with the correct tightness, and the Stokes argument handles graphical competitors cleanly. Section 5's demonstration that their earlier special Lagrangian example fails the same area property is a useful contrast. Citations are appropriate; the only self-citation (the special Lagrangian example [13]) is used for comparison, not as a load-bearing step.\n\nThe soft spot is the one the stress-test flags, and it is real. The abstract's central claim—that Gr(u) is the limit of a minimizing sequence of Lipschitz graphs—rests on one 'easy to see' sentence. The rotated maps live on U_k = y_k(Ω), not U, and no gluing argument is given to transfer them to U with boundary data u|∂U and area loss going to zero. I believe the gluing works: the boundary values match at corresponding points, the domains differ by O(1/k), and u is smooth near ∂U. But it is load-bearing for the interpretation, and a referee should ask for it to be written out. The reader's other worry, the Cauchy-Kovalevskaya radius, is more minor: the sign estimates are strict at ∂Ω₀, so they persist on a fixed neighborhood, and CK existence in a neighborhood is standard for this kind of construction. Also minor: the projection argument for competitors outside the cylinder is sketched, and a few computations are asserted rather than shown.\n\nWho this is for: geometric measure theory and calculus of variations people who care about Cartesian currents, the minimal surface system, and free boundary problems. The calibration inequality alone is a nice tool. I would send it to a serious referee. Verdict: conditional accept with revisions; the main theorem is likely correct, but the minimizing-sequence gap should be closed before publication.","headline":"Genuinely new construction in minimal dimension/codimension answering three GMS questions; main proof is coherent but the 'easy to see' minimizing sequence is a real gap that needs filling.","tokens_in":16280,"tokens_out":13903,"would_cite":true,"duration_ms":112679,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q05","53A10","35J47"],"pacs":[],"model":"deepseek-v4-flash","headline":"Area-minimizing limits of Lipschitz graphs can develop large interior vertical patches that are not minimal.","keywords":["minimal surface system","Cartesian currents","higher codimension","area minimization","free boundary problem","calibration","vertical tangent planes","singular solutions"],"falsifier":"Numerically compute the Cauchy–Kovalevskaya extension in Section 4.1 for small $\\varepsilon$ and $\\delta$ and check that $\\partial_\\nu \\tilde{v}^1_1 > 0$ on $\\partial\\Omega_0 \\setminus \\Gamma$ and $\\partial^2_{\\nu\\nu} \\tilde{v}^1_1 > 0$ on $\\Gamma$; any violation would destroy the exact identity $\\{\\partial_1 w_1 = 0\\} = \\overline{\\Omega_0}$. Alternatively, compute the area of the graph of an analytic competitor with the same boundary data as $u$ (for instance the small solution whose existence is cited in Remark 4.2) and compare it with the completed graph of $u$; if the competitor has area no larger, Theorem 1.2 is false.","tokens_in":15184,"feed_emoji":"📐","tokens_out":10514,"duration_ms":84090,"temperature":0.7,"pith_summary":"This paper constructs solutions to the minimal surface system whose graphs develop large interior vertical and non-minimal portions when one passes to the limit of minimizing sequences. The authors show that limits of Lipschitz graphs with fixed boundary data—objects called Cartesian currents—can contain a positive-area vertical piece projecting onto an interior analytic surface, and this piece is not a minimal submanifold. This gives negative answers to three structural questions about such limits posed in the Cartesian-currents literature, and it highlights a sharp contrast with codimension one, where interior discontinuities cannot occur. The construction is explicit and lives in the smallest possible dimensions: domain dimension $n = 3$ and codimension $m = 2$.","feed_headline":"Minimal-graph limits can turn vertical in the interior","feed_subtitle":"In the smallest dimension and codimension, area-minimizing limits can carry positive-area non-minimal vertical patches.","key_machinery":"The central mechanism is the axis swap $y(x) = (w_1(x), x_2, x_3)$ together with a calibration form $\\omega$ built from the area integrand $F(M) = \\sqrt{\\det(I + M^T M)}$ and its gradient. The axis swap converts a region where $\\partial_1 w_1$ vanishes into a vertical patch of positive area projecting to the analytic surface $\\Sigma$, while preserving the area of the graph. The calibration, constructed from $w$ and from the free-boundary equation $\\partial_i(\\sqrt{\\det g}\\, g^{ij} \\partial_j w_\\alpha) = \\partial_1 H \\, \\chi_{\\{H>0\\}}$ (with $\\alpha = 1$, and zero for $\\alpha = 2$), satisfies $|\\omega(T)| \\le 1$ on every unit $n$-plane with equality only on the tangent plane to the graph of $w$, so Stokes' theorem gives the global area comparison.","core_discovery":"The paper's central claim is that there exists a Lipschitz map $w = (w_1, w_2)$ from a bounded convex domain $\\Omega \\subset \\mathbb{R}^3$ to $\\mathbb{R}^2$, analytic off an analytic convex subdomain $\\Omega_0$ compactly contained in $\\Omega$, solving the minimal surface system in $\\Omega \\setminus \\Omega_0$ and not solving it in $\\Omega_0$, with $\\partial_1 w_1 \\ge 0$ and $\\{\\partial_1 w_1 = 0\\} = \\overline{\\Omega_0}$. Swapping the $x_1$ and $w_1$ axes turns the graph of $w$ into the graph of a map $u$ defined on $U \\setminus \\Sigma$, where $\\Sigma = y(\\Omega_0)$ is an analytic embedded surface resembling a potato chip; the graph of $u$, completed by a vertical patch over $\\Sigma$, is a Cartesian current obtained as a limit of Lipschitz graphs. Theorem 1.2 states that this completed graph has smaller area than the graph of every Lipschitz map from $U$ to $\\mathbb{R}^2$ with the same boundary data, even though the vertical patch is not minimal. The same mechanism yields the first singular solution to the minimal surface system in the minimal dimension $n = 3$ and codimension $m = 2$.","pith_inferences":["The same axis-swap and calibration strategy may transfer to other variational problems whose integrand has the convexity and growth properties of the area functional, producing vertical patches for more general quasiconvex or polyconvex energies in higher codimension.","Because the construction uses Cauchy–Kovalevskaya, the vertical patch is placed very close to the boundary; a natural next step is to see whether the free-boundary minimization described in the paper can push the patch deep into the interior, which would indicate the phenomenon is stable rather than a boundary artifact.","The comparison with the special Lagrangian example suggests that among Lagrangian competitors the area comparison may fail, so one could test whether the new example admits a Lagrangian approximation without increasing area; the paper's Section 5 indicates this is not the case for the earlier construction.","The explicit point-singularity example gives a concrete numerical target: verifying the expansion $\\partial_1 w_1 = 3x_1^2 + x_2^2 + x_3^2 + O(|x|^3)$ and the resulting $C^{1/3}$ regularity of $u$ would independently confirm the claimed optimal regularity."],"forward_implications":["The three structural questions about Cartesian currents—mass-minimality, vanishing vertical mass, and generalized mean curvature zero—all receive negative answers in higher codimension.","Area minimizers among Cartesian currents can have vertical parts of the highest possible dimension (three in this case), and these parts can project onto an analytic surface inside the domain.","Such minimizers need not be minimal as geometric objects: the vertical patch carries positive area and is not a minimal submanifold, so the classical monotonicity formula cannot hold for these minimizers.","Graphicality itself acts as an interior free-boundary constraint in higher codimension, in contrast with codimension one where non-graphical behavior can only appear at the boundary.","The dimension and codimension are optimal: when $n = 2$ or $m = 1$, a maximum principle for the gradient rules out the existence of such interior vertical patches."],"supporting_citations":[{"why":"Supplies the non-existence and irregularity results for the Dirichlet problem in higher codimension that motivate the question addressed here.","marker":"[12]"},{"why":"Provides the theory of Cartesian currents and the structural questions about mass-minimality, vertical mass, and mean curvature that the paper answers negatively.","marker":"[7]"},{"why":"Gives the calibrated-geometry framework, including the area-minimizing cone example and the mean-curvature formula used in the special Lagrangian comparison.","marker":"[9]"},{"why":"Supplies the existence of a small analytic solution with the same boundary data, used in Remark 4.2 to build the competing graph in the calibration argument.","marker":"[19]"},{"why":"Analyzes the Hopf-map boundary data in the model case, including the zero-area vertical 'piece' that motivates the notion of vertical parts of Cartesian currents.","marker":"[1]"}],"fun_headline_variants":["Area-minimizing limits may carry vertical non-minimal patches","Limits of Lipschitz graphs can develop vertical, non-minimal regions","Smallest case reveals non-minimal vertical patches in area limits","Vertical anomalies in minimal-graph limits for n=3, m=2","Area limits can turn vertical and non-minimal in smallest case"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction hinges on a local extension step: the minimal surface system can be solved from prescribed first-order data on the free boundary, with strict sign control on the derivatives of the first-component derivative just outside; if that solvability or those sign estimates failed, the vertical patch would not sit exactly over the intended interior surface.","fun_headline_variants_meta":{"raw":{"variants":["Area-minimizing limits may carry vertical non-minimal patches","Limits of Lipschitz graphs can develop vertical, non-minimal regions","Smallest case reveals non-minimal vertical patches in area limits","Vertical anomalies in minimal-graph limits for n=3, m=2","Area limits can turn vertical and non-minimal in smallest case"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001032,"raw_usage":{"total_tokens":4338,"prompt_tokens":927,"completion_tokens":3411,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":3320}},"tokens_in":543,"tokens_out":3411,"duration_ms":22615,"temperature":1.0,"reasoning_tokens":3320,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:16:21.602508+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically compute the Cauchy–Kovalevskaya extension in Section 4.1 for small $\\varepsilon$ and $\\delta$ and check that $\\partial_\\nu \\tilde{v}^1_1 > 0$ on $\\partial\\Omega_0 \\setminus \\Gamma$ and $\\partial^2_{\\nu\\nu} \\tilde{v}^1_1 > 0$ on $\\Gamma$; any violation would destroy the exact identity $\\{\\partial_1 w_1 = 0\\} = \\overline{\\Omega_0}$. Alternatively, compute the area of the graph of an analytic competitor with the same boundary data as $u$ (for instance the small solution whose existence is cited in Remark 4.2) and compare it with the completed graph of $u$; if the competitor has area no larger, Theorem 1.2 is false.","supporting_citations":[{"cited_title":"B.; Osserman, R","cited_arxiv_id":null,"evidence_quote":"Supplies the non-existence and irregularity results for the Dirichlet problem in higher codimension that motivate the question addressed here."},{"cited_title":"Cartesian curren ts, weak diffeomorphisms and exis- tence theorems in nonlinear elasticity","cited_arxiv_id":null,"evidence_quote":"Provides the theory of Cartesian currents and the structural questions about mass-minimality, vertical mass, and mean curvature that the paper answers negatively."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the calibrated-geometry framework, including the area-minimizing cone example and the mean-curvature formula used in the special Lagrangian comparison."},{"cited_title":"The Dirichlet problem for the minimal surfa ce system in arbitrary dimensions and codimensions","cited_arxiv_id":null,"evidence_quote":"Supplies the existence of a small analytic solution with the same boundary data, used in Remark 4.2 to build the competing graph in the calibration argument."},{"cited_title":"Resolving the singularity of minim al Hopf cones","cited_arxiv_id":null,"evidence_quote":"Analyzes the Hopf-map boundary data in the model case, including the zero-area vertical 'piece' that motivates the notion of vertical parts of Cartesian currents."}],"review_version":1}