{"id":"18f2292e-f810-4b61-b924-f4be4d675cc7","arxiv_id":"2509.06110","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Anisotropic capillary Gauss curvature flows converge to smooth solutions of capillary L_p Minkowski problems for even data with p > -n-1 and for non-even data with p > n+1.","lead":"New geometric flow equations for capillary surfaces in a half-space are introduced, and the flows are shown to converge to solutions of capillary L_p Minkowski type boundary value problems. The paper gives a flow-based route to existence results in convex geometry and nonlinear PDE, covering all even exponents p > -n-1 and non-even p > n+1.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The negative-p C0 estimate in Lemma 4.1 is delegated to an unstated external compactness lemma, and the symmetrized density constructed for that application is only continuous; if the lemma does not cover this case, the flow convergence proof for p in (-n-1,0) collapses.","rationale":"The reader's weakest_assumption correctly identifies the reliance of Lemma 4.1 on [HI25b, Lem. 2.6] as the main risk. My stress-test agrees and sharpens it: not only is the lemma external and from the same author group, but its application here is not immediate because the symmetrized density tilde f is only continuous across the two interfaces, and the full-sphere inequality (4.1) involves contributions from the band where the support function of Xi_tau is not controlled by the capillary support function h. Since Lemmas 4.2-4.5 and the final convergence proof all depend on Lemma 4.1, any gap in this step invalidates Theorem 1.1 for the negative-p range, which is the paper's main new range. The paper has genuine independent structure: monotonicity of J and tilde J, self-contained C0 arguments for p>=0, and a direct C2 estimate via curvature bounds. The concern is therefore not about the overall method but about a specific unproved dependency, so the reader's CONDITIONAL verdict should stand unchanged. No ad hominem is intended; the issue is that the proof as written does not supply enough detail to verify the key compactness step for the hardest parameter range.","tokens_in":16708,"tokens_out":17464,"duration_ms":156357,"concrete_test":"Take [HI25b, Lem. 2.6] and check its hypotheses against the objects in Lemma 4.1: (i) is tilde f required to be smooth, or merely positive and continuous on S^n? (ii) does the lemma cover p in (-n-1,0) with the exact normalization constant e^{-(n+1)J(0)}? (iii) does it apply to the reflection-symmetrized body Xi_tau, whose support function on the band is not controlled by h composed with T? Concretely, write out the proof of Lemma 2.6 for n=2, theta=pi/4, p=-1 with the tilde f defined in the manuscript; if the proof uses integration by parts or C^2 regularity of tilde f at the circles u_{n+1}=+/-cos theta, it fails, and mollifying tilde f changes the constant in (4.1), breaking the inequality. Alternatively, add a self-contained covering argument for p<0 in the spirit of Cases 2-3 of Lemma 4.1; if it succeeds without Lemma 2.6, the dependency is removed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central existence claim for p in (-n-1,0) hinges on Lemma 4.1, Case 1. After monotonicity gives (4.1), the proof symmetrizes the capillary body to Xi_tau and invokes [HI25b, Lem. 2.6] to conclude that the capillary support function is uniformly bounded above and below away from zero. This is the only place where the negative-p C0 estimate is established, and Lemmas 4.2-4.5 (gradient, Gauss curvature, and principal curvature bounds) all start from Lemma 4.1. The paper neither states Lemma 2.6 nor reproduces its proof, and the application is not routine: the density tilde f built on S^n is defined to be constant in the band -cos theta <= u_{n+1} <= cos theta, so it is generally only continuous, not C^1, across the circles u_{n+1} = +/- cos theta. If Lemma 2.6 requires smooth positive data, or if its normalization controls only the cap integral rather than the symmetrized full-sphere integral appearing in (4.1), then the C0 bound for p<0 is not established and the flow convergence proof fails for exactly the range advertised as the main novelty. This is a load-bearing external dependency, not a cosmetic omission.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces two anisotropic capillary Gauss curvature flows in the Euclidean half-space with Robin boundary conditions, one volume-normalized (1.3) and one anisotropic (1.5), and studies their long-time behavior. The main results, Theorems 1.1 and 1.2, assert that for an even smooth positive density f on the capillary cap C_theta and an even smooth strictly convex capillary initial hypersurface, the normalized flow (1.3) exists for all times and subconverges in C^infty to a capillary hypersurface whose support function solves the capillary even L_p Minkowski equation for every p > -n-1; for p > n+1 the evenness assumption is removed for the flow (1.5). The proof combines monotone functionals with a priori estimates, including a C^0 estimate that is split into three regimes, maximum-principle bounds on gradient, Gauss curvature, and principal curvatures, and a final parabolic compactness argument.","tokens_in":16867,"tokens_out":13392,"duration_ms":115279,"significance":"If the theorems are correct, the paper provides a flow-based existence proof for smooth solutions of the capillary even L_p Minkowski problem in the full range p > -n-1, and of the non-even problem for p > n+1, complementing the elliptic and iterative approaches of [MWW25a, MWW25b, HI25b]. The monotone functionals J and J-tilde are defined directly from the flow and the target equation emerges from the equality case, so the core derivation is not circular. The maximum-principle computations in Lemmas 4.3-4.5 and the C^0 arguments for p >= 0 are self-contained and internally consistent. However, the advertised extension to negative p rests on an unverified external compactness lemma and on an inequality that is not justified in the manuscript, so the central claim is not yet established as written.","major_comments":[{"comment":"The transition from the monotonicity bound (4.1) to the stated normalized bound for the symmetrized body Xi_tau is not justified. Since the symmetrized density tilde f is positive on the whole band -cos theta <= u_{n+1} <= cos theta and the support function hat h_{Xi_tau} is positive there, the full-sphere integral int_{S^n} tilde f hat h_{Xi_tau}^p du strictly exceeds the capillary integral int_{C_theta} f h^p dxi. Because p < 0, the exponent -(n+1)/p is positive, so the claimed inequality V(Xi_tau) (int_{S^n} tilde f hat h^p)^{-(n+1)/p} >= e^{-(n+1)J(0)} does not follow from (4.1); the displayed inequality is in fact weaker than what (4.1) directly controls. Since Lemmas 4.2-4.5 all start from the C^0 bound of Lemma 4.1, the proof for the range -n-1 < p < 0 has a load-bearing gap as written.","section":"Section 4.1, Lemma 4.1, after (4.1)"},{"comment":"The use of [HI25b, Lem. 2.6] is not verifiable from the manuscript. The lemma is neither stated nor proved, and its hypotheses are not checked. In particular, the symmetrized density tilde f built on S^n is defined to be constant in the band -cos theta <= u_{n+1} <= cos theta and is generally only continuous, not C^1, across the circles u_{n+1} = +/- cos theta; if Lemma 2.6 requires smooth positive data, or if its normalization controls only the capillary integral rather than the full-sphere integral appearing in the manuscript, then the C^0 bound for p in (-n-1,0) is not established. This is the only mechanism producing the C^0 estimate in the advertised negative-p range, so the authors must either state and prove the needed compactness lemma or replace it with a self-contained argument.","section":"Section 4.1, Lemma 4.1, after (4.1)"}],"minor_comments":[{"comment":"The phrase 'for all p in (-n-1, infinity)' should be qualified as the even case (and with theta in (0, pi/2)) to agree with Theorem 1.1; the non-even statement in Theorem 1.2 requires p > n+1.","section":"Abstract"},{"comment":"The constant C_theta introduced in (4.2) has the same symbol as the capillary cap C_theta; renaming one of them would avoid ambiguity.","section":"Section 4.1, equation (4.2)"},{"comment":"The gradient bound is imported from [HIS25, Lem. 4.8] without statement; since this is another self-citation to an unpublished preprint, the authors should quote the lemma or give a proof.","section":"Section 4.1, Lemma 4.2"},{"comment":"The sentence combining [Don88, Theorems 6.1, 6.4, 6.5] and [Lie96, Theorem 14.23] with the uniform estimates would benefit from stating precisely which theorem yields the extinction V(M_t) -> 0 and hence tau(t) -> infinity.","section":"Section 5"},{"comment":"The informal phrases 'max K >> 1' and 'max Q approx max K' should be replaced by quantitative statements, for instance by noting that if Q is bounded then K is bounded because of the formula Q = (alpha f h^p K - h)/(h - epsilon_0).","section":"Section 4.1, Lemmas 4.3 and 4.4"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the decisive C^0 estimate for the negative-p range depends on two self-citations to unpublished preprints ([HI25b, Lem. 2.6] and, for the gradient bound, [HIS25, Lem. 4.8]); this is acceptable only if the authors reproduce the required statements. The apparent inequality-direction problem in the symmetrization step of Lemma 4.1 is the more serious issue and should be addressed head-on; if the authors can supply a short proof of the claimed normalized bound for Xi_tau, the paper should be reconsidered. I do not see evidence of misconduct; the issue is verification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know two things about this paper. First, the anisotropic capillary Gauss curvature flows (1.3) and (1.5) with their monotone functionals are real additions to the toolset: they give a parabolic route to solutions of the capillary L_p Minkowski problem and a more direct C^2 estimate than the elliptic continuity method. Second, the most interesting range, -n-1 < p < 0, is also the most fragile part of the proof: the C^0 bound in Lemma 4.1, Case 1, is not proved in the paper. It is imported from [HI25b, Lem. 2.6], a companion paper by the same group, and the way it is applied here involves a symmetrized density \\tilde f on S^n that is only continuous across the circles u_{n+1} = \\pm cos θ. If that external lemma requires smooth positive data—or if its normalization controls only the cap integral rather than the symmetrized full-sphere integral used in (4.1)—then the C^0 estimate fails, and the convergence argument for exactly the range the abstract advertises collapses.\n\nI don't think this is a fatal objection to the whole paper. For p ≥ 0 the C^0 bound is self-contained (geometric covering plus ODE comparison), and once Lemma 4.1 is granted, the gradient, Gauss curvature, and principal curvature estimates in Lemmas 4.2–4.5 are internally consistent; the monotonicity arguments in Section 3 are clean, and the convergence follows by standard compactness. The authors are also honest in the introduction that most existence ranges were already settled in [MWW25a], [MWW25b], [HI25b]. The genuinely new content is the flow-based proof, not the existence statement itself, and the abstract could be read as overclaiming on that point.\n\nBut the negative-p gap is load-bearing, and it needs to be fixed before I would trust Theorem 1.1 as stated. The authors should either state and prove the needed compactness lemma under the continuity hypothesis, or verify that [HI25b, Lem. 2.6] applies verbatim to their symmetrized density, and say explicitly what regularity it requires. As written, the paper is a well-structured contribution with an important open problem at its center.\n\nWho gets value? Geometric analysts working on Monge-Ampère equations and curvature flows, and anyone who wants to see how the parabolic method behaves with Robin boundary conditions. The paper deserves a serious referee, not a desk reject—the core method is sound in spirit and the gap is concrete and fixable. But my own verdict is conditional: I would not cite the negative-p theorem until the external dependency is resolved.\n\nBest,\n[Your name]","headline":"A genuinely useful flow proof for the capillary L_p Minkowski problem, but the advertised negative-p range rests on an unverified external compactness lemma applied to a density that is only continuous.","tokens_in":17549,"tokens_out":4833,"would_cite":false,"duration_ms":39429,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","35J66","35K55","52A20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a volume-normalized anisotropic Gauss curvature flow for capillary hypersurfaces in the Euclidean half-space converges to a strictly convex solution of the capillary even $L_p$ Minkowski problem for every $p>-n-1$…","keywords":["capillary L_p Minkowski problem","capillary Gauss curvature flow","Monge-Ampere equation","Robin boundary condition","support function","convex hypersurface","long-time existence","maximum principle"],"falsifier":"A concrete test is to search for a sequence of even, smooth, strictly convex capillary hypersurfaces $\\Sigma_i$ with fixed volume and a fixed even positive $f$ for which the normalized quantity $V(\\Xi_i)(\\int_{S^n}\\tilde f\\,\\hat h_i^p)^{-(n+1)/p}$ stays bounded below as in the paper's inequality (4.1) while the support functions $h_i$ fail to admit uniform positive lower and upper bounds; such a sequence would disprove the external compactness lemma and invalidate Lemma 4.1 for $p<0$. A numerical check for $p=-n$ on ellipsoidal cap initial data would test whether $h$ drifts toward zero or infinity under the normalized flow.","tokens_in":16385,"feed_emoji":"📐","tokens_out":7709,"duration_ms":62429,"temperature":0.7,"pith_summary":"The paper studies a family of capillary hypersurfaces in the half-space whose speed is set by a prescribed function, the $p$-th power of the support function, and the Gauss curvature. It establishes long-time existence and smooth convergence for two volume-normalized flows, and identifies the limit as a solution of the capillary $L_p$ Minkowski equation with Robin boundary condition. The even case is solved for all $p>-n-1$; dropping evenness costs a range restriction to $p>n+1$. The flow approach gives a constructive, parabolic route to solutions of a Monge-Ampere equation that earlier elliptic treatments reached only in smaller ranges or with more involved boundary estimates.","feed_headline":"Curvature flow yields capillary Minkowski solutions for all p > -n-1","feed_subtitle":"The flow's limit solves the even L_p Monge-Ampere equation, and works without evenness for p>n+1.","key_machinery":"The load-bearing object is the capillary support function $h$ defined on the spherical cap $C_\\theta$, which encodes the hypersurface through the relation $b_{ij}=\\nabla^2_{ij}h+h\\delta_{ij}$, so the Gauss curvature satisfies $K=1/\\det(\\nabla^2h+hI)$. The normalized flow (1.3) for $h$ is designed to preserve the enclosed capillary volume while decreasing the functional $J$; equality in the monotonicity formula is exactly the target equation (1.4). The proof proceeds through uniform $C^0$, $C^1$, Gauss curvature, and principal curvature bounds, with the boundary condition $\\nabla_\\mu h=\\cot\\theta\\,h$ used repeatedly to rule out boundary maxima.","core_discovery":"The central claim is Theorem 1.1: for an even smooth positive function $f$ on the capillary spherical cap $C_\\theta$ with $\\theta\\in(0,\\pi/2)$ and an even smooth strictly convex capillary initial hypersurface, the normalized flow (1.3) has a smooth strictly convex solution for all $\\tau>0$, with a subsequence converging in $C^\\infty$ to a solution of $\\det(\\nabla^2 h+hI)=\\frac{(n+1)V(\\widehat\\Sigma_0)}{\\int_{C_\\theta} f h^p\\,d\\xi}\\, f h^{p-1}$ in $C_\\theta$ together with $\\nabla_\\mu h=\\cot\\theta\\,h$ on $\\partial C_\\theta$. Theorem 1.2 removes the evenness assumption for $p>n+1$ using the flow (1.5). The paper presents these convergence results as a flow approach to the capillary even $L_p$ Minkowski problem for all $p>-n-1$ and the capillary $L_p$ Minkowski problem for $p>n+1$, with uniform $C^k$ estimates obtained by maximum-principle arguments rather than the boundary double-normal test used in earlier elliptic work.","pith_inferences":["A natural extension the paper does not pursue is to remove the evenness condition for $p\\in(-n-1,n+1]$: the paper states that evenness enters only through the $C^0$ estimate, so any alternative lower and upper bound on the support function would extend Theorem 1.1 to the full non-even range.","The monotone functional $J$ resembles an entropy whose critical points are the desired solutions; a numerical study of its convexity near the limit could upgrade subsequential convergence to full convergence of the flow.","The flow framework likely adapts to $\\theta>\\pi/2$ if the capillary Minkowski existence question in that regime is settled, since the boundary-angle restriction here inherits the current state of the elliptic theory.","Because the hardest negative range relies on an external compactness lemma, a concrete check is to test that lemma for sequences with $p$ approaching $-n-1$; if the lemma has counterexamples there, the range in Theorem 1.1 would shrink accordingly."],"forward_implications":["Smooth strictly convex solutions to the capillary even $L_p$ Minkowski problem exist for every $p>-n-1$ and $\\theta\\in(0,\\pi/2)$, including all negative $p$ in that range.","For $p>n+1$, the evenness assumption is unnecessary: the flow (1.5) produces a solution of (1.1) from any smooth strictly convex capillary initial hypersurface with positive support function.","Any even smooth strictly convex initial capillary hypersurface can be continuously deformed through strictly convex capillary hypersurfaces into a solution, so existence is obtained constructively rather than only by elliptic methods.","The uniform $C^k$ estimates and the monotone functional give a compactness mechanism: subsequential limits are automatically smooth strictly convex solutions.","The paper leaves the non-even cases $p=1$ and $p=n+1$ as open problems, which it identifies as targets for further flow-based work."],"supporting_citations":[{"why":"Supplies Lemma 2.6, the compactness lemma that produces the uniform $C^0$ bound for the hardest range $-n-1<p<0$ in Lemma 4.1.","marker":"[HI25b]"},{"why":"Supplies Lemma 4.8, which converts the $C^0$ bound into the uniform gradient bound used in Lemma 4.2.","marker":"[HIS25]"},{"why":"Provides the capillary support function identities and volume derivative formula (Prop. 2.9) underpinning the monotonicity computations.","marker":"[MWWX25]"},{"why":"Together with [Lie96], gives the parabolic regularity theory used to extend uniform estimates to long-time convergence.","marker":"[Don88]"},{"why":"Second-order parabolic PDE theory used with [Don88] for the regularity and asymptotic behavior of the flows.","marker":"[Lie96]"},{"why":"Solved the capillary Minkowski problem for $\\theta\\le\\pi/2$ via the continuity method, providing the baseline existence result that this flow approach extends.","marker":"[MWW25a]"},{"why":"Gave prior elliptic existence for capillary $L_p$ Minkowski solutions in even and higher-$p$ ranges, which the flow construction generalizes.","marker":"[MWW25b]"}],"fun_headline_variants":["Capillary L_p Minkowski solved by flow for all p>-n-1","Flow approach settles capillary L_p Minkowski for p>-n-1","Capillary curvature flow yields L_p Minkowski for all p>-n-1","Flow proof extends L_p Minkowski to capillary for p>-n-1","Anisotropic capillary flow resolves L_p Minkowski for p>-n-1"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For the hardest range $-n-1<p<0$, the proof that the support function stays uniformly bounded above and below relies on a compactness lemma in a companion paper by the same authors; if that lemma is false or does not cover the full range, the long-time convergence proof for negative $p$ collapses.","fun_headline_variants_meta":{"raw":{"variants":["Capillary L_p Minkowski solved by flow for all p>-n-1","Flow approach settles capillary L_p Minkowski for p>-n-1","Capillary curvature flow yields L_p Minkowski for all p>-n-1","Flow proof extends L_p Minkowski to capillary for p>-n-1","Anisotropic capillary flow resolves L_p Minkowski for p>-n-1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001126,"raw_usage":{"total_tokens":4641,"prompt_tokens":863,"completion_tokens":3778,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":3670}},"tokens_in":479,"tokens_out":3778,"duration_ms":24635,"temperature":1.0,"reasoning_tokens":3670,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:20:01.304280+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test is to search for a sequence of even, smooth, strictly convex capillary hypersurfaces $\\Sigma_i$ with fixed volume and a fixed even positive $f$ for which the normalized quantity $V(\\Xi_i)(\\int_{S^n}\\tilde f\\,\\hat h_i^p)^{-(n+1)/p}$ stays bounded below as in the paper's inequality (4.1) while the support functions $h_i$ fail to admit uniform positive lower and upper bounds; such a sequence would disprove the external compactness lemma and invalidate Lemma 4.1 for $p<0$. A numerical check for $p=-n$ on ellipsoidal cap initial data would test whether $h$ drifts toward zero or infinity under the normalized flow.","supporting_citations":[],"review_version":1}