{"id":"c5946018-2c89-442f-a23e-4243b8a711e1","arxiv_id":"2502.09433","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Free boundary hypersurfaces and marginally outer trapped surfaces satisfying scalar curvature and energy bounds force warped product or product splitting, with Einstein slices and sharp volume lower bounds.","lead":"These rigidity results show that compact hypersurfaces meeting a manifold's boundary at right angles, under energy and curvature bounds, force a rigid product or warped product geometry in a neighborhood. The theorems extend known splitting and volume-comparison results from closed manifolds to free boundary and trapped-surface settings used in mathematical relativity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem B's equality case rests on unproved Lemmas 4.3–4.4 (constant-θ+ foliation), deferred to [32]; this is a genuine verification gap, not an observed error, so the conditional verdict stands.","rationale":"We read the paper as a straightforward extension of Galloway–Jang and Barros–Cruz/Mendes to free boundary MOTS. The main computations in the visible proofs are coherent: the K=-εg reduction in Theorem A, the stability argument, the Yamabe-estimate chain in Proposition 4.1, and the equality analysis in Proposition 4.2 are internally consistent and contain no obvious algebraic error. The examples are consistent with the stated sharpness. The central weakness is the equality case of Theorem B, which borrows two substantial results (Lemmas 4.3 and 4.4) from the second author's prior work without proof. These lemmas are not peripheral: the constant-θ+ foliation is what converts the infinitesimal rigidity of Proposition 4.2 into a neighborhood product splitting. If the analogy with [32] fails — for instance because the free-boundary Robin conditions with drift X alter the simplicity argument, or because the weaker energy bound μ-|J|≥-c changes the linearized analysis — then the rigidity part of Theorem B would not follow. We found no evidence that the lemmas are false; the issue is that the paper does not supply enough detail to verify them. This matches the reader's weakest-assumption assessment, so we agree with the conditional verdict; no adjustment is proposed.","tokens_in":26427,"tokens_out":19850,"duration_ms":181513,"concrete_test":"Independently supply the proof of Lemma 4.4 following [32, Lemma 3.6]. Concretely, verify that the linearized map at t=0 from free-boundary normal variations (u with Bu=0) to (θ+'(u) - constant, boundary condition) is an isomorphism, using Lemma 4.3's simplicity of λ=0 for L and L*. If the proof in [32] uses self-adjointness (X=0) or the stronger DEC μ+J(N)≥0 rather than μ+J(N)≥-c, then the lemma as stated here is not established; in that case Theorem B's equality case needs an additional argument or a strengthened hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is the equality case of Theorem B. After the volume inequality is proved (Proposition 4.1), the rigidity conclusions (1)–(4) depend on Lemma 4.4, which asserts the existence of a neighborhood foliation by free boundary hypersurfaces Σ_t with constant null mean curvature θ+(t), metric g=φ²dt²+h_t and boundary condition ∂φ/∂ν_t=II∂M(N_t,N_t)φ. The proof is omitted ('entirely analogous' to Lemma 3.6 of [32]), and Lemma 4.3 (simplicity of λ=0 for the non-self-adjoint operator L and its adjoint, with the Robin boundary conditions B and B*) is likewise deferred. This is load-bearing because the subsequent argument defines f=θ+, η, ρ, ξ from this foliation, applies Lemma 2.9, and uses weak outerness to conclude θ+=0; without a constant-θ+ foliation, the equality case has no neighborhood in which to run the argument. The omission is flagged in the text itself, so it is a verification gap rather than a discovered contradiction. Theorem A's Lemma 3.1 (short-time existence for ε=1) is a secondary, more standard gap; the central concern remains the deferred foliation lemma.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes two rigidity theorems for compact free boundary hypersurfaces in initial data sets with boundary. Theorem A extends the Galloway--Jang scalar-curvature splitting theorem to the free boundary setting: under R_M ≥ -n(n+1)ε, H_∂M ≥ 0, H_Σ ≤ nε, a topological non-existence condition, and local weak outerness with respect to H_0 = nε, a neighborhood of Σ is isometric to a warped product [0,δ) × Σ with metric dt² + e^{2εt}h, where (Σ,h) is Ricci-flat with totally geodesic boundary. Theorem B extends the Barros--Cruz/Mendes volume estimate to free boundary stable MOTS: for n ≥ 3, if Σ is a compact free boundary stable, weakly outermost MOTS with σ_{1,0}(Σ,∂Σ) < 0, and μ - |J| ≥ -c, the boundary dominant energy condition, and n-convexity of K hold on M_+, then vol(Σ) ≥ (|σ_{1,0}|/(2c))^{n/2}; equality forces a product neighborhood dt² + h, an Einstein slice with scalar curvature -2c and totally geodesic boundary, K = a(t)dt², μ = -c, J = 0, and saturation of the boundary dominant energy condition. The proofs adapt the Galloway--Jang method for Theorem A and the Mendes method for Theorem B, with auxiliary propositions giving volume estimates and infinitesimal rigidity.","tokens_in":26620,"tokens_out":25879,"duration_ms":227099,"significance":"If the technical lemmas are supplied or properly verified, the results are significant and natural extensions of known rigidity statements to the free boundary and MOTS settings. The DEC/BDEC verification for K = -εg, the stability argument via the first eigenvalue, the volume estimate in Proposition 4.1, and the equality analysis in Proposition 4.2 are coherent and contain no evident algebraic error. The paper also provides concrete examples showing the necessity of some hypotheses. However, the equality case of Theorem B depends on two substantial lemmas whose proofs are omitted, and Theorem A depends on a short-time existence assertion that is not proved. These are verification gaps rather than demonstrated contradictions, but they are load-bearing and should be addressed before publication.","major_comments":[{"comment":"The equality case of Theorem B rests on Lemma 4.4, which asserts the existence of a neighborhood foliation by free boundary hypersurfaces Σ_t with constant null mean curvature θ_+(t), metric g = φ²dt² + h_t, and boundary condition ∂φ/∂ν_t = II_∂M(N_t,N_t)φ. Its proof is omitted as 'entirely analogous' to Lemma 3.6 of [32], and Lemma 4.3, the simplicity of λ = 0 for L and L* with the Robin boundary conditions B and B*, is likewise deferred. These lemmas are not cosmetic: the subsequent argument defines f = θ_+, η, ρ, and ξ from this foliation, applies Lemma 2.9, and uses weak outerness to conclude θ_+ ≡ 0. Without a constant-θ_+ foliation, the equality case has no neighborhood in which to run the analysis, so conclusions (1)--(4) of Theorem B do not follow. Since the present hypotheses only give μ - |J| ≥ -c rather than the full DEC used in parts of [32], the authors should either include complete proofs or formulate and verify a precise theorem from [32] that applies verbatim.","section":"Section 4, Lemmas 4.3 and 4.4"},{"comment":"Short-time existence of the free boundary null mean curvature flow (3.1) for ε = 1 is asserted without proof. For ε = 0 the flow is exactly Stahl's free boundary mean curvature flow, but for ε = 1 the equation includes an additional normal lower-order term, so it is not literally covered by the cited result. This existence statement is needed to produce the strict inequality H_Σ' < nε that rules out H_Σ ≢ nε in Theorem A. Please provide a proof or a precise reference covering the ε = 1 case.","section":"Section 3, Lemma 3.1, Eq. (3.1)"},{"comment":"The application of Lemma 2.9 requires the condition max{f,ρ} ≥ 0. The paper does not verify this condition explicitly. It presumably follows from ρ ≥ 0, which in turn follows from τ = tr K ≥ 0 under the n-convexity assumption on K, but this should be stated. This is a local but necessary check in the chain leading to θ_+ ≤ 0.","section":"Section 4, proof of Theorem B, application of Lemma 2.9"}],"minor_comments":[{"comment":"Theorem 2.3 is stated for a weakly outermost MOTS, whereas Theorem A only assumes local weak outerness. The proof implicitly restricts to a sufficiently small neighborhood U where Σ is weakly outermost and then applies Theorem 2.3 to U. This reduction should be made explicit.","section":"Section 3, proof of Theorem A"},{"comment":"The definition of 'locally weakly outermost with respect to H_0' is introduced only after Lemma 3.1, although it is used in the statement of Theorem A. Moving this definition to Section 2.1 or to the introduction would improve readability.","section":"Section 2.2, definition of weak outerness with respect to H_0"},{"comment":"The notation for the Yamabe quotient Q^{1,0}_h and the normalization by (∫ u^{2n/(n-2)}dv)^{(n-2)/n} is used heavily but could be defined more explicitly at the point of first use in Proposition 4.1 to avoid confusion with the normalized functional of Section 2.2.","section":"Section 4, Eq. (4.10) and Eq. (4.11)"},{"comment":"In Example 3.4 the notation S^n_+ and h_{S^n_+} should be defined, and in Example 3.5 the role of the interval [a,b] × T^{n-1} in satisfying all hypotheses except condition (4) deserves a few more words.","section":"Examples 3.4 and 3.5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on the second author's prior papers [30] and [32], and the two pivotal lemmas in the equality case of Theorem B are literally deferred to [32]. This is not improper, but it means the referees cannot verify the central rigidity conclusion without checking that [32, Lemma 3.6] applies under the weaker energy hypothesis μ - |J| ≥ -c. I recommend major revision rather than rejection because the core chain of inequalities and the equality analysis in Propositions 4.1 and 4.2 appear sound; the missing items are verifiable rather than contradictory."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, what to know: the paper proves two genuine extensions. Theorem A is the free boundary analogue of Galloway-Jang splitting: if Σ is compact, free boundary, H_Σ ≤ nε, admits no psc metric with minimal boundary, and is locally weakly outermost with respect to H0 = nε, then an outer neighborhood splits as dt² + e^{2εt}h with h Ricci-flat and totally geodesic boundary. Theorem B gives a volume lower bound vol(Σ) ≥ (|σ_{1,0}|/2c)^{n/2} for free boundary stable weakly outermost MOTS under BDEC and n-convexity, with product rigidity at equality. The framework is the second author's prior work on free boundary MOTS, and the paper leans on it honestly.\n\nThe core computations are sound. The DEC/BDEC verification for K = -εg is clean; the chain of inequalities in Proposition 4.1 and Theorem B is coherent; the use of Lemma 2.9 to force θ+ = 0 from the integral inequality is clever. The examples showing necessity of the hypotheses are useful and match the closed case.\n\nThe soft spots are the deferred lemmas. Lemma 4.4, imported from [32] with 'entirely analogous' proof, supplies the constant-θ+ foliation of an outer neighborhood with g = φ²dt² + h_t and the Robin boundary condition ∂φ/∂ν_t = II_{∂M}(N_t,N_t)φ. That foliation is load-bearing for the equality case of Theorem B: without it there is no neighborhood in which to define f, η, ρ, ξ and run the argument. Lemma 4.3 (simplicity of the zero eigenvalue for L and L*) is also deferred. The paper flags these omissions itself, so they are verification gaps rather than observed errors. The secondary gap, Lemma 3.1's short-time existence for ε = 1, is also asserted rather than proved, but that one is more standard.\n\nSelf-citation is heavy but legitimate: the cited theorems are independent published results with proofs, so there is no circularity. The novelty is modest—expected extensions along known lines—but the results are correctly stated and the reductions are nontrivial. Specialists in scalar curvature rigidity and MOTS will want this. The paper deserves a serious referee; the referee should ask for full proofs of Lemmas 4.3 and 4.4, or at least a detailed account of how [32] gives them.","headline":"Two credible free-boundary extensions of known rigidity theorems; Theorem B's equality case leans on omitted proofs, but the conditional verdict is fair.","tokens_in":27201,"tokens_out":2127,"would_cite":true,"duration_ms":18363,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","53C42","53C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two rigidity theorems: near a free boundary hypersurface, the geometry is forced into a warped or Einstein product splitting.","keywords":["free boundary hypersurfaces","marginally outer trapped surfaces","initial data sets","rigidity theorems","splitting theorems","Yamabe invariant","dominant energy condition","scalar curvature"],"falsifier":"One concrete way to disprove Theorem B would be to construct an explicit initial data set with boundary containing a free-boundary stable MOTS that satisfies all hypotheses and has volume below $(|\\sigma_{1,0}(\\Sigma,\\partial\\Sigma)|/(2c))^{n/2}$; even a numerical example would settle the sharpness of the bound. A second test is to run the free boundary null mean curvature flow (3.1) for $\\epsilon=1$ on a nontrivial compact free-boundary hypersurface and check whether the asserted short-time existence and $\\theta_+<0$ conclusion persist.","tokens_in":26145,"feed_emoji":"📐","tokens_out":10747,"duration_ms":89966,"temperature":0.7,"pith_summary":"This paper proves two rigidity theorems for compact free boundary hypersurfaces, submanifolds that meet the ambient boundary orthogonally, in Riemannian manifolds and initial data sets with boundary. The first extends a known splitting result: under a scalar curvature lower bound, a mean curvature bound on $\\Sigma$, no positive-scalar-curvature metric with minimal boundary, and a weak outermost condition, an outer neighborhood must be the warped product $[0,\\delta)\\times\\Sigma$ with metric $dt^2+e^{2\\epsilon t}h$, where $h$ is Ricci-flat and the boundary is totally geodesic. The second gives a sharp volume lower bound for stable, weakly outermost free boundary MOTS, marginally outer trapped surfaces, of dimension $n\\ge3$ with negative Yamabe invariant, under the dominant energy condition, the boundary dominant energy condition, and $n$-convexity of $K$. Equality forces the neighborhood to be a product $dt^2+h$ with an Einstein slice of scalar curvature $-2c$ and totally geodesic boundary, together with $K=a(t)\\,dt^2$, $\\mu=-c$, $J=0$, and saturated boundary energy. The upshot is that boundary conditions do not weaken the rigidity familiar from closed horizons and minimal hypersurfaces; sharp splitting and volume comparisons persist in the free boundary setting.","feed_headline":"Two rigidity theorems force rigid geometry near free-boundary surfaces","feed_subtitle":"Warped or Einstein product splittings follow when scalar curvature and energy inequalities saturate.","key_machinery":"The machinery centers on free boundary MOTS and their stability operator. For a two-sided free boundary hypersurface with null mean curvature $\\theta_+$, the first variation under a normal deformation $\\varphi N$ is $\\partial_t\\theta_+=L\\varphi+(-\\tfrac12\\theta_+^2+\\theta_+\\tau)\\varphi$, where $L$ is the stability operator with a Robin boundary condition coming from the free boundary condition. Lemma 2.1, imported from the second author's earlier work, is the engine that converts a positive solution of $Lu\\ge0$ with boundary inequality into Ricci-flatness and a totally geodesic boundary. Lemma 3.1 produces a nearby free boundary hypersurface with strictly negative null mean curvature by a free boundary null mean curvature flow, which is what upgrades 'weakly outermost' to an actual MOTS condition. Proposition 4.1 feeds the stability inequality into the Yamabe invariant $\\sigma_{1,0}(\\Sigma,\\partial\\Sigma)$, and Lemma 4.4, also imported, supplies the foliation by constant-null-mean-curvature leaves that carries the equality case of Theorem B.","core_discovery":"The paper's central claims are Theorem A and Theorem B. Theorem A states that if $(M,g)$ is a Riemannian manifold with boundary, $\\Sigma$ is a compact free boundary hypersurface, $R_M\\ge-n(n+1)\\epsilon$, $H_{\\partial M}\\ge0$, $H_\\Sigma\\le n\\epsilon$, $\\Sigma$ admits no metric of positive scalar curvature with minimal boundary, and $\\Sigma$ is locally weakly outermost with respect to $H_0=n\\epsilon$, then an outer neighborhood of $\\Sigma$ is isometric to $[0,\\delta)\\times\\Sigma$ with metric $dt^2+e^{2\\epsilon t}h$, where $h=g|_\\Sigma$ is Ricci-flat with totally geodesic boundary. Theorem B states that for $n\\ge3$, a compact free boundary stable, weakly outermost MOTS with $\\sigma_{1,0}(\\Sigma,\\partial\\Sigma)<0$ in an initial data set with boundary, satisfying $\\mu-|J|\\ge-c$, the boundary dominant energy condition, and $n$-convexity of $K$ on the outer region $M_+$, has $\\operatorname{vol}(\\Sigma)\\ge(|\\sigma_{1,0}|/(2c))^{n/2}$; equality forces the existence of an outer neighborhood with product metric $dt^2+h$, $\\Sigma$ Einstein with scalar curvature $-2c$ and totally geodesic boundary, $K=a(t)\\,dt^2$, $\\mu=-c$, $J=0$, and the boundary dominant energy condition saturated along $V\\cap\\partial M$. The authors also prove Proposition 4.1, which supplies these volume estimates at the level of stable MOTS before the outermost assumption is used.","pith_inferences":["A natural next test, not undertaken here, is whether Proposition 4.1(2) admits the same equality rigidity as Theorem B; one would expect a product splitting with $\\Sigma$ having a boundary of constant mean curvature and the boundary energy condition saturated.","The theorem suggests a free-boundary version of Penrose-type inequalities: for initial data satisfying dominant energy, the area of a free-boundary apparent horizon should be controlled by the boundary contribution, with equality only in the rigid, time-symmetric, radial-$K$ configuration described by Theorem B.","The unproved imported foliation in Lemma 4.4 is the point most worth checking; if the constant-null-mean-curvature foliation or the simplicity of the zero eigenvalue fails for some initial data, the equality conclusion of Theorem B may still hold but requires a different argument."],"forward_implications":["Corollary 3.2 makes the rigidity global on the outer end: if $(M,g)$ is complete, $(M_+,g|_{M_+})$ is isometric to $([0,\\infty)\\times\\Sigma, dt^2+e^{2\\epsilon t}h)$.","The examples of Section 3 show that omitting condition (3) of Theorem A destroys the conclusion for $\\epsilon=0$ and $\\epsilon=1$, and omitting condition (4) destroys it for $\\epsilon=1$.","Theorem B yields a sharp volume bound for free-boundary apparent horizons: stable, weakly outermost MOTS cannot be arbitrarily small when the energy density dominates the current and $K$ is $n$-convex.","In the equality case of Theorem B the initial data is rigid in an outer neighborhood: the metric is a product with an Einstein slice, the extrinsic curvature is purely radial, and all energy inequalities are saturated.","Proposition 4.1 also proves an analogous boundary volume estimate: under $\\mu+J(N)\\ge0$ and a boundary energy gap $\\bar c$, one gets $\\operatorname{vol}(\\partial\\Sigma)\\ge(|\\sigma_{0,1}(\\Sigma,\\partial\\Sigma)|/(2\\bar c))^{(n-1)}$."],"supporting_citations":[{"why":"Supplies the closed-hypersurface splitting theorem and the overall strategy via $K=-\\epsilon g$ that Theorem A extends to free boundary hypersurfaces.","marker":"[25]"},{"why":"Supplies the free boundary Yamabe volume comparison, Theorem 7, whose MOTS analogue is Theorem B.","marker":"[8]"},{"why":"Supplies Lemmas 2.1, 2.2, Theorem 2.3, and the foliation arguments whose analogues are imported as Lemmas 4.3 and 4.4.","marker":"[32]"},{"why":"Supplies the closed MOTS volume bound and equality rigidity, Theorems 8 and 9, that Theorem B generalizes to the boundary setting.","marker":"[30]"},{"why":"Introduces the stability notion and Robin boundary condition for free boundary MOTS used throughout Sections 2 and 4.","marker":"[1]"},{"why":"Provides the variation formula (2.1) for null mean curvature and the definition of the stability operator $L$.","marker":"[5]"},{"why":"Contains the Lemma 5.2 result used to show $\\Sigma$ is a MOTS in the initial data $(M,g,-\\epsilon g)$, which Lemma 3.1 adapts to the free boundary case.","marker":"[6]"},{"why":"Supplies short-time existence for the free boundary mean curvature flow used in Lemma 3.1 for $\\epsilon=0$.","marker":"[40]"},{"why":"Used in Proposition 2.8 to conclude that a Yamabe metric attaining negative $\\sigma_{1,0}$ is Einstein with totally geodesic boundary, a key equality-case step of Theorem B.","marker":"[16]"}],"fun_headline_variants":["Free-boundary rigidity forces product geometry near surfaces","MOTS bounds yield volume and rigidity for free-boundary surfaces","Splitting theorems extend to free-boundary initial data sets","Rigid free-boundary surfaces: product metrics and volume bounds","No positive scalar curvature? Then free-boundary splits as product"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on an imported, unproved lemma saying that near the surface one can slice space into free-boundary layers each with the same constant null expansion; if that slicing, or the short-time existence of the auxiliary flow used for case $\\epsilon=1$, fails, the equality conclusions are unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Free-boundary rigidity forces product geometry near surfaces","MOTS bounds yield volume and rigidity for free-boundary surfaces","Splitting theorems extend to free-boundary initial data sets","Rigid free-boundary surfaces: product metrics and volume bounds","No positive scalar curvature? Then free-boundary splits as product"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000228,"raw_usage":{"total_tokens":1542,"prompt_tokens":1079,"completion_tokens":463,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":695,"completion_tokens_details":{"reasoning_tokens":379}},"tokens_in":695,"tokens_out":463,"duration_ms":4555,"temperature":1.0,"reasoning_tokens":379,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T21:30:48.681636+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete way to disprove Theorem B would be to construct an explicit initial data set with boundary containing a free-boundary stable MOTS that satisfies all hypotheses and has volume below $(|\\sigma_{1,0}(\\Sigma,\\partial\\Sigma)|/(2c))^{n/2}$; even a numerical example would settle the sharpness of the bound. A second test is to run the free boundary null mean curvature flow (3.1) for $\\epsilon=1$ on a nontrivial compact free-boundary hypersurface and check whether the asserted short-time existence and $\\theta_+<0$ conclusion persist.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the closed-hypersurface splitting theorem and the overall strategy via $K=-\\epsilon g$ that Theorem A extends to free boundary hypersurfaces."},{"cited_title":"Barros and C","cited_arxiv_id":null,"evidence_quote":"Supplies the free boundary Yamabe volume comparison, Theorem 7, whose MOTS analogue is Theorem B."},{"cited_title":"A, Theory Methods 220 (2022), 15 (English), Id/No 112841","cited_arxiv_id":null,"evidence_quote":"Supplies Lemmas 2.1, 2.2, Theorem 2.3, and the foliation arguments whose analogues are imported as Lemmas 4.3 and 4.4."},{"cited_title":"Mendes, Rigidity of marginally outer trapped (hyper)surfaces with negative σ-constant, Trans","cited_arxiv_id":null,"evidence_quote":"Supplies the closed MOTS volume bound and equality rigidity, Theorems 8 and 9, that Theorem B generalizes to the boundary setting."},{"cited_title":"Alaee, M","cited_arxiv_id":null,"evidence_quote":"Introduces the stability notion and Robin boundary condition for free boundary MOTS used throughout Sections 2 and 4."},{"cited_title":"Andersson, M","cited_arxiv_id":null,"evidence_quote":"Provides the variation formula (2.1) for null mean curvature and the definition of the stability operator $L$."},{"cited_title":"Andersson and J","cited_arxiv_id":null,"evidence_quote":"Contains the Lemma 5.2 result used to show $\\Sigma$ is a MOTS in the initial data $(M,g,-\\epsilon g)$, which Lemma 3.1 adapts to the free boundary case."},{"cited_title":"Stahl, Regularity estimates for solutions to the mean curvature flo w with a Neumann boundary condition, Calc","cited_arxiv_id":null,"evidence_quote":"Supplies short-time existence for the free boundary mean curvature flow used in Lemma 3.1 for $\\epsilon=0$."},{"cited_title":"Cruz and A","cited_arxiv_id":null,"evidence_quote":"Used in Proposition 2.8 to conclude that a Yamabe metric attaining negative $\\sigma_{1,0}$ is Einstein with totally geodesic boundary, a key equality-case step of Theorem B."}],"review_version":1}