{"id":"8bfc70e7-fba4-4394-b0b2-a4fdd4987bf5","arxiv_id":"2608.08901","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Graphical mean curvature flow with zero boundary data is locally well-posed for approximable Lipschitz graphs, globally well-posed for small initial slopes, and decays exponentially on bounded domains.","lead":"Mathematicians proved that the mean curvature flow of graphs can start from rough, non-smooth initial shapes when the graph vanishes on the boundary and can be approached by smooth shapes. They also showed that small enough initial wiggles keep the flow going forever and, on bounded regions, make it flatten out exponentially fast.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The bounded-domain and rough-data arguments rest on the imported whole-space Schauder estimate Theorem 2.1 from Part I [7], which is not re-derived; if its proof has a gap the central theorems fail. A focused check of [7]'s estimate for the Laplacian case would settle this.","rationale":"The reader's weakest assumption identifies the same load-bearing point: Theorem 2.1 from Part I [7] is imported without proof and is used critically for interior patches and for passing to rough data. The bounded-domain local well-posedness depends on (4.25), which is exactly an application of Theorem 2.1; if that theorem is wrong or its hypotheses are not met, Theorem 1.2 fails, and Theorem 1.1 is also affected through the kernel estimates and odd-extension step. I found no circularity and no outright contradiction in the bounded-domain contraction argument: the inequalities ∥f_in∥ ≤ (σ + ∥f−φ∥)/10 and ∥f_i∥ ≤ (σ + ∥f−φ∥)/10 do imply ∥f−φ∥ ≤ σ/4, so the a priori closing step is not invalid despite the reader's secondary reservation. The contraction estimate similarly absorbs the ∥f∥ term by choosing ε, σ, and T small, which is standard. The half-space model estimates in Section 2 and the boundary identities in Appendix 5 appear coherent and are genuinely new; the specific vulnerability is the imported whole-space Schauder estimate, its high-derivative hypothesis, and the exponential factor. Since this is a dependency that can be checked independently, the appropriate verdict remains CONDITIONAL, unchanged from the reader's assessment.","tokens_in":63760,"tokens_out":27787,"duration_ms":279916,"concrete_test":"Independently verify Theorem 2.1 of [7] for the constant-coefficient case L = -Δ with m = 1, κ = 3/4, γ = 2, and f ∈ C^κ data satisfying the stated Data norm. Compute the Duhamel term u(t) = ∫_0^t e^{(t-τ)Δ} Δ f(τ) dτ and determine whether sup_{0<t≤T} t^{(1+κ)/2}∥u(t)∥_{C^{1+κ}} ≤ C e^{C T log T}(∥u0∥_{L∞} + Data) holds with a constant independent of T, especially at the endpoint τ→t where (t-τ)^{-1} appears. If the estimate fails for the Laplacian, Theorem 2.1 is false and the interior-patch estimate (4.25) collapses. If it holds, repeat for L = A[∇φ]:D² and check whether (2.2) can be satisfied with M controlled by ∥φ∥_{C^{2m+3}} rather than by derivatives of order d+m+6.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's main theorems are conditional on Theorem 2.1 (arXiv:2604.10682, Part I), an imported time-weighted whole-space Schauder estimate for variable-coefficient parabolic systems. It is used in three essential places: (i) the interior-patch estimate (4.25) for bounded domains, (ii) the definition of the solution map Sg on nonsmooth g by smooth approximation, and (iii) the higher-order odd-extension estimate (3.14). The manuscript itself flags the dependency: Section 2.2 says the interior estimates follow from Part I, and Lemma 2.2's proof is deferred to [7]. No circularity is involved, but the central claim is not self-contained: a gap in [7] would propagate directly to Theorems 1.1 and 1.2. A further concrete mismatch is that (2.2) requires symbol regularity through order d+m+4 in x and ξ, which for the differential operator L = A[∇φ]:ξξ^T forces derivatives of φ up to order d+m+6, whereas Theorems 1.1 and 1.2 state dependence only on the C^{2m+3} norm of φ. The exponential factor e^{C T log T} in (2.3) is also imported into the finite-time local theory and, through the iteration up to T*, into the global argument on bounded domains. The new self-contained ingredients—boundary identities in Lemmas 5.1–5.4 and the half-space model estimates in Lemma 2.4—appear internally consistent; the vulnerability is specifically the imported Theorem 2.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a boundary Schauder theory for the graphical mean curvature flow in arbitrary codimension, subject to homogeneous Dirichlet boundary conditions, on the half-space and on smooth bounded domains. It claims local well-posedness at the scaling-critical Lipschitz regularity for initial graphs that are W^{1,∞}-limits of smooth boundary-compatible profiles, global existence when the Lipschitz seminorm is sufficiently small, and exponential convergence to the flat graph on bounded domains. The proof combines coefficient freezing with odd/even parity extensions on the half-space, boundary identities specific to the graphical system, localization and boundary flattening on curved domains with anisotropic estimates, and an energy-based bootstrap for global decay. A central whole-space time-weighted Schauder estimate is imported from the authors' Part I preprint [7].","tokens_in":64041,"tokens_out":14474,"duration_ms":146932,"significance":"If the results are correct, they provide a fixed-boundary, critical-regularity theory for graphical mean curvature flow in arbitrary codimension, complementing the whole-space theory of Koch and Lamm and extending earlier boundary results. The paper's original contributions include the flat-boundary model estimates (Lemmas 2.4 and 2.6), the parity-based boundary identities (Lemmas 5.1–5.4), and the recursive recovery of normal derivatives on curved boundaries. The exposition is detailed and the structural arguments are coherent. However, the two main theorems rest on the imported whole-space Schauder estimate Theorem 2.1 from [7], which is not re-derived here; in addition, there is a concrete mismatch between the symbol regularity demanded by that theorem and the regularity stated in Theorems 1.1 and 1.2. These issues are load-bearing and require attention before the claims can be regarded as established.","major_comments":[{"comment":"In the contraction proof for Theorem 1.2(i), the interior estimate gives ∥f^in∥_{Z^m_{T,1}} ≲ (1+T)^m(σ+T^{κ/2}∥φ∥)(1+...)(∥f∥_{Z^m_T}+∥g∥_{Z^m_T}). The text then states: 'Since ∥f^in∥_{Z^m_{T,1}} ≤ ∥f∥_{Z^m_T}, we can take σ and T small enough to ensure ∥f^in∥_{Z^m_{T,1}} ≤ 2^{-10dm}∥g∥_{Z^m_T}.' This inference is invalid as written: the small factor multiplies (∥f∥+∥g∥), not ∥g∥ alone, and the inequality ∥f^in∥ ≤ ∥f∥ does not permit discarding the ∥f∥ term on the right. The absorption of the ∥f∥ term must be performed after combining the interior and boundary estimates, with the full norm on the left-hand side. As written, the proof of contractivity on bounded domains is incomplete.","section":"Section 4.2, contraction step"},{"comment":"Assumption (2.2) in Theorem 2.1 requires sup_{0≤j,l≤d+m+4} |ξ|^{l-2}|∇^j_x∇^l_ξ L(t,x,ξ)| ≤ M. For the symbol L = A[∇φ]:ξξ^T, the j-th x-derivative involves derivatives of φ up to order j+1, so the assumption forces φ to belong to C^{d+m+5}. Theorems 1.1 and 1.2, however, state that the lifespan and constants depend only on ∥φ∥_{C^{2m+3}_b} (respectively ∥e^{ε1∆}f0∥_{C^{2m+3}}). For m < d+2 (which includes d=2, m=1,2,3 and d=3, m=1,2,3, among others), d+m+5 > 2m+3, so the stated regularity dependence is not justified by the quoted Theorem 2.1. This is load-bearing: either the main theorems must be stated with dependence on a higher norm, or Theorem 2.1 must be proved or quoted with symbol regularity only through order 2m+3, or a separate argument must show that the actual symbol A[∇g] supplies the needed regularity under the hypotheses of the theorems.","section":"Section 2.2 and Theorems 1.1–1.2"},{"comment":"The main theorems rely crucially on Theorem 2.1, imported from the authors' Part I preprint [7], and on Lemma 2.2, whose proof is deferred to [7]. Theorem 2.1 is used in essential places: the interior-patch estimate (4.25) in the bounded-domain proof, the definition of the solution map Sg on nonsmooth g by smooth approximation in Section 3.2 and Section 4.2, and the higher-order odd-extension estimate (3.14). Since [7] is an unpublished preprint, a gap in its proof would invalidate Theorems 1.1 and 1.2. The manuscript should either include a complete proof of the needed whole-space estimate, or state explicitly that the main theorems are conditional on [7], or cite a published version. Without this, the central claim is not self-contained.","section":"Section 2.2 and proofs of Theorems 1.1 and 1.2"}],"minor_comments":[{"comment":"The statement says 'there exists κ∈(0,κ)' but the displayed estimate (2.9) uses the same symbol κ on both sides; the proof introduces a different κ = κ/(2(1+κ)) for the exponents. Please clarify the notation to avoid ambiguity.","section":"Section 2.3, Lemma 2.3(2)"},{"comment":"The sentence 'This section collects the notation and linear Schauder estimates that will be used throughout the nonlinear analysis.' appears twice, once at the beginning of Section 2 and again at the beginning of Section 2.1.","section":"Section 2.1"},{"comment":"The chain in (2.23) uses κ for two different exponents (κ and κ−κ) in the same displayed estimate, which is hard to read; consider renaming one of the exponents.","section":"Equation (2.23)"},{"comment":"The notation R_i[h,α] is defined with a bracket but then used with arguments (χ_i f, ∇g̃_i(∇Φ_i)^{-1}); the typesetting of the second argument should be checked for consistency.","section":"Section 4.1, equation (4.6)"}],"recommendation":"major_revision","confidential_remarks":"The main theorems are conditional on the authors' Part I preprint [7], and the symbol-regularity mismatch in the application of Theorem 2.1 is a concrete technical gap. The contraction-step issue in Section 4.2 appears fixable, but the symbol-regularity issue may require either strengthening the theorem statements or proving a reduced-regularity version of Theorem 2.1. If the editors are willing to consider a revised version that resolves these points, the paper could become a valuable contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. This paper proves the missing fixed-boundary theory for graphical mean curvature flow at the critical Lipschitz level in arbitrary codimension: local well-posedness for data in the W^{1,∞}-closure of smooth boundary-compatible profiles, global existence for small Lipschitz seminorm, and exponential decay on bounded domains. The new analytic content is a boundary Schauder theory—coefficient freezing with parity extensions, boundary identities for the graphical operator, anisotropic estimates after flattening—and it is worked out in serious detail.\n\nThe boundary identities in Lemmas 5.1–5.4 are genuinely new and internally consistent. The half-space estimates in Section 2 are derived line by line. The paper is also honest about its limitations: the approximation assumption is flagged as a real restriction (Remark 1.3), flat-boundary identities are shown not to survive on curved domains without extra forcing, and the compatibility counterexample in Remark 4.5 is correct. I do not see circularity: the imported Theorem 2.1 from Part I concerns different whole-space systems and is legitimate framework reuse.\n\nThe soft spots are fixable but need attention before the paper is final. (1) The main theorems depend on Theorem 2.1, an imported whole-space Schauder estimate that is not re-derived. The text flags this, but it means a gap in arXiv:2604.10682 would propagate directly. (2) There is a concrete mismatch between the symbol class (2.2), which needs derivatives of ∇φ through order d+m+4, and the stated dependence on ∥φ∥_{C^{2m+3}} in Theorems 1.1 and 1.2. This looks like an imprecise statement rather than a fatal error, but the authors should fix it. (3) In the bounded-domain contraction proof, after bounding the interior patch by C(...)(∥f∥+∥g∥), there is an inference that does not close as written; the higher-order boundary estimates just before it should supply the missing control, so I expect it is repairable, but a referee should ask for the explicit step.\n\nWho this is for: researchers in geometric parabolic PDEs, particularly the Koch–Lamm rough-data circle. It deserves a serious referee. Recommend engaging, with the three points above requested as revisions.","headline":"Serious and mostly correct fixed-boundary critical-Lipschitz theory for graphical MCF in arbitrary codimension; the main caveats are heavy reliance on the authors' Part I estimate and a few fixable gaps in the bounded-domain proof.","tokens_in":64619,"tokens_out":3628,"would_cite":true,"duration_ms":39993,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K20","35K55","53E10","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"Graphical mean curvature flow with a fixed Dirichlet boundary is locally well-posed for Lipschitz initial graphs that are approximated by smooth boundary-compatible profiles, and globally well-posed with exponential decay when the…","keywords":["mean curvature flow","graphical mean curvature flow","arbitrary codimension","Dirichlet boundary condition","critical Lipschitz regularity","time-weighted Schauder estimates","parabolic systems","well-posedness"],"falsifier":"Conduct a numerical test of the imported estimate: take a smooth compactly supported $u_0$ and an $x$-dependent coefficient field such as $L(t,x,\\xi)=(1+\\delta\\cos(x\\cdot\\xi_0))|\\xi|^2$ satisfying (2.2), solve (2.1) accurately, and compare the supremum in (2.3) with the claimed right-hand side. A single violation of the bound, especially of the exponential factor $e^{CT\\log T}$ in the variable-coefficient case, would invalidate the imported theorem on which both main results rely; conversely, a sharp numerical confirmation would corroborate the paper's foundation.","tokens_in":63503,"feed_emoji":"🌊","tokens_out":12619,"duration_ms":126417,"temperature":0.7,"pith_summary":"This paper claims that graphical mean curvature flow in any codimension is well-posed at the scaling-critical Lipschitz regularity when the initial graph vanishes on the boundary of a half-space $\\mathbb{R}^d_+$ or a smooth bounded domain $\\Omega$ and can be approximated in the Lipschitz norm by smooth boundary-compatible profiles. For such data, the flow exists and is unique for a short time, becomes smooth immediately for positive times, and the estimates controlling its higher derivatives stay bounded as $t\\downarrow 0$. If the Lipschitz seminorm of the initial graph is small, the solution exists globally; on bounded domains it converges exponentially to the flat graph. The analytic engine is a new boundary Schauder theory for variable-coefficient parabolic systems, built by freezing coefficients, extending forcing terms oddly across the boundary, and recovering normal derivatives recursively after boundary flattening. A sympathetic reader would care because this supplies the fixed-boundary counterpart of the well-known whole-space Lipschitz theory at the natural critical scaling.","feed_headline":"Rough initial graphs become smooth under boundary mean curvature flow","feed_subtitle":"New fixed-boundary Schauder estimates make the critical Lipschitz class well-posed, with global decay on bounded domains.","key_machinery":"The central object is a time-weighted boundary Schauder theory for variable-coefficient parabolic systems. On the half-space, the method freezes the coefficient matrix at a point and uses a linear transformation preserving $\\mathbb{R}^d_+$ to reduce the frozen operator to the Laplacian; the error is controlled by moment estimates for the Dirichlet heat kernel. The critical forcing is extended oddly across the boundary, which moves derivatives onto the kernel, and structural boundary identities—in particular $\\Delta^k f=0$ on the flat boundary (Lemma 5.3)—propagate the Dirichlet condition to higher order without extra compatibility assumptions. On curved domains, localization and boundary flattening produce anisotropic estimates in which tangential directions gain a full Hölder exponent while one normal derivative is lost; Lemma 5.4 expresses even normal derivatives through derivatives of the forcing and lower-normal-order terms, allowing a recursive closure. The whole-space time-weighted Schauder estimate of Part I (Theorem 2.1) is used to control interior patches.","core_discovery":"The paper proves local well-posedness for the graphical mean curvature system $\\partial_t f = A[\\nabla f]:\\nabla^2 f$ in $(0,T)\\times\\Omega$ with $f=0$ on $\\partial\\Omega$ and $f(0)=f_0$, where $\\Omega$ is the half-space or a $C^{2m+3}$ bounded domain and $A[p]=(\\mathrm{Id}+\\sum_\\alpha p_\\alpha\\otimes p_\\alpha)^{-1}$. The admissible initial data lie in the $W^{1,\\infty}$-closure of smooth functions vanishing on the boundary; no higher-order compatibility conditions at $t=0$ are required. The solution is unique in a time-weighted Hölder class and satisfies weighted estimates such as $\\sup_{0<t\\le T_0}(\\|\\nabla f(t)\\|_{L^\\infty}+t^{m+(1+\\kappa)/2}\\|\\nabla^{2m+2}f(t)\\|_{\\dot C^\\kappa}+t^{m+(1+\\kappa)/2}\\|\\nabla^{2m}\\partial_t f(t)\\|_{\\dot C^\\kappa})\\le C\\|f_0\\|_{\\dot W^{1,\\infty}}$ on the half-space, with an analogous anisotropic bound on bounded domains. Under smallness of the Lipschitz seminorm the solution is global; on bounded domains exponential decay to the flat graph follows from an energy inequality, Poincaré's inequality, and interpolation. Positive-time regularization is quantified by weighted Hölder estimates whose weights remain bounded as $t\\downarrow 0$, and on the flat boundary the solution satisfies the extra identities $\\Delta^k f=0$ for $k=1,\\ldots,m$.","pith_inferences":["The same coefficient-freezing, parity-extension, and recursive-normal-derivative scheme is likely to transfer to other quasilinear parabolic systems at their critical scaling with homogeneous Dirichlet conditions, such as surface-diffusion or two-phase free-boundary problems.","The approximability hypothesis is plausibly a genuine restriction: Lipschitz data whose gradient is not uniformly continuous up to the boundary (for instance, an oscillating normal derivative) may fail the required $W^{1,\\infty}$ convergence of the Dirichlet heat semigroup, and the theorem is silent for them—suggesting that at critical regularity the Dirichlet trace behaves as a propagated structu","If the imported whole-space estimate could be proven with a polynomial-in-$T$ factor instead of $e^{CT\\log T}$, the long-time bounds on bounded domains would sharpen; testing the sharpness of that factor on $x$-dependent coefficients is a concrete next step.","The half-space parity argument hints at an underlying reflection principle: odd extensions of the critical forcing preserve Hölder regularity, which may allow defining weak solutions for boundary data in trace classes beyond the approximable set."],"forward_implications":["On both the half-space and bounded domains, any initial graph satisfying the approximation condition is immediately smoothed: for every $t>0$ the solution has the full higher-order Hölder regularity quantified by the weighted estimates, with no compatibility conditions imposed at $t=0$.","Small Lipschitz initial data yield global solutions; on bounded domains the solution converges exponentially to the flat graph, making the flat graph asymptotically stable within the critical class.","On the half-space the solution satisfies the extra boundary identities $\\Delta^k f=0$ on $\\partial\\mathbb{R}^d_+$ for $k=1,\\ldots,m$, so the Dirichlet condition propagates to all orders without extra assumptions.","For bounded domains, the Dirichlet heat semigroup provides admissible approximations whenever it converges in $W^{1,\\infty}$; whenever it does, the theorem supplies a unique short-time solution with the stated weighted bounds."],"supporting_citations":[{"why":"Supplies the whole-space time-weighted Schauder estimate (Theorem 2.1) that controls interior patches and the linear well-posedness; the boundary proof reduces to it.","marker":"[7]"},{"why":"Establishes global small-slope existence, uniqueness and analyticity for entire graphs, the whole-space result this paper extends to fixed boundaries.","marker":"[22]"},{"why":"Treats nonparametric mean curvature flow with prescribed boundary data, the boundary problem this paper revisits at critical regularity.","marker":"[16]"},{"why":"Provides the classical linear parabolic theory used to produce smooth approximate solutions in the fixed-point construction on bounded domains.","marker":"[26]"},{"why":"Proves instantaneous smoothing for Lipschitz submanifolds with small local Lipschitz norm, the interior smoothing statement generalized here to Dirichlet boundaries.","marker":"[36]"},{"why":"Studies the Dirichlet problem for the higher-codimensional minimal surface system through the graphical parabolic system, the boundary setting this paper extends to rough data.","marker":"[35]"}],"fun_headline_variants":["Critical Lipschitz boundary flow: well-posed","Boundary MCF smoothes rough Lipschitz graphs","Rough data to smooth flow: boundary MCF well-posed","Global decay for small Lipschitz boundary MCF","No compatibility needed: boundary MCF well-posed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof leans on the whole-space time-weighted Schauder estimate imported from the authors' Part I preprint (Theorem 2.1), whose structural assumptions and exponential factor are not re-derived here; any gap there would invalidate both main theorems.","fun_headline_variants_meta":{"raw":{"variants":["Critical Lipschitz boundary flow: well-posed","Boundary MCF smoothes rough Lipschitz graphs","Rough data to smooth flow: boundary MCF well-posed","Global decay for small Lipschitz boundary MCF","No compatibility needed: boundary MCF well-posed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001552,"raw_usage":{"total_tokens":6259,"prompt_tokens":1059,"completion_tokens":5200,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":675,"completion_tokens_details":{"reasoning_tokens":5120}},"tokens_in":675,"tokens_out":5200,"duration_ms":33670,"temperature":1.0,"reasoning_tokens":5120,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:22:59.199231+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Conduct a numerical test of the imported estimate: take a smooth compactly supported $u_0$ and an $x$-dependent coefficient field such as $L(t,x,\\xi)=(1+\\delta\\cos(x\\cdot\\xi_0))|\\xi|^2$ satisfying (2.2), solve (2.1) accurately, and compare the supremum in (2.3) with the claimed right-hand side. A single violation of the bound, especially of the exponential factor $e^{CT\\log T}$ in the variable-coefficient case, would invalidate the imported theorem on which both main results rely; conversely, a sharp numerical confirmation would corroborate the paper's foundation.","supporting_citations":[{"cited_title":"Geometric flows with rough initial data","cited_arxiv_id":"0902.1488","evidence_quote":"Establishes global small-slope existence, uniqueness and analyticity for entire graphs, the whole-space result this paper extends to fixed boundaries."},{"cited_title":"Huisken,Non-parametric mean curvature evolution with boundary conditions, Journal of Dif- ferential Equations 77 (2) (1989), 369–378","cited_arxiv_id":null,"evidence_quote":"Treats nonparametric mean curvature flow with prescribed boundary data, the boundary problem this paper revisits at critical regularity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classical linear parabolic theory used to produce smooth approximate solutions in the fixed-point construction on bounded domains."},{"cited_title":"Wang.The mean curvature flow smoothes Lipschitz submanifolds.Comm","cited_arxiv_id":null,"evidence_quote":"Proves instantaneous smoothing for Lipschitz submanifolds with small local Lipschitz norm, the interior smoothing statement generalized here to Dirichlet boundaries."},{"cited_title":"Wang.The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions.Comm","cited_arxiv_id":null,"evidence_quote":"Studies the Dirichlet problem for the higher-codimensional minimal surface system through the graphical parabolic system, the boundary setting this paper extends to rough data."}],"review_version":1}