{"id":"ac5cd71c-c742-4b68-b669-2acb885684e4","arxiv_id":"2608.09878","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On gravitational instantons and several special-holonomy manifolds, nonzero-eigenvalue L^2 eigenforms exist if and only if L^2 eigenfunctions exist, and on ALE, A_n ALF, and asymptotically conical manifolds they never exist.","lead":"This paper proves that on four-dimensional gravitational instantons, the existence of nonzero-eigenvalue L^2 eigenforms is completely controlled by ordinary L^2 eigenfunctions; if eigenfunctions are absent, all eigenforms are absent. It also shows that ALE, A_n ALF, and asymptotically conical manifolds have no nonzero-eigenvalue L^2 eigenforms, which matters for string and M-theory compactifications because such modes would appear as massive particles.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.2's boundary-term decay is asserted from L² alone; the pointwise estimate (5.5) is unjustified, so Theorem 1.8(b) rests on an invalid proof step unless the standard flux-sequence repair is supplied.","rationale":"The reader's weakest_assumption identifies the same issue: Lemma 5.2's boundary-term vanishing is the linchpin of Theorem 1.8(b), and the proof as written relies on an unjustified pointwise decay estimate. This is the most load-bearing concern because both Corollary 1.13 and Corollary 1.15(a) ultimately pass through Theorem 1.8(b) to convert vanishing of L² eigenfunctions into vanishing of all L² eigenforms. The defect is repairable by the standard flux-sequence argument, which is why this should not change the verdict from CONDITIONAL to REJECT: the central claim is likely correct, but the submitted proof has a real rigorous gap. Other concerns, such as the sketched extension of [6] in Theorem 6.13 and the omitted ALG*/ALH* cases, are also present, but they are either less central or similarly repairable. Therefore the reader's CONDITIONAL verdict with moderate confidence remains appropriate.","tokens_in":27349,"tokens_out":17040,"duration_ms":172082,"concrete_test":"Re-derive Lemma 5.2 without Eq. (5.5): for an AT^mC gravitational instanton, use the coarea formula and Fubini to show ∫_{R0}^∞ ∫_{Σ_r} |α∧dα| dσ_r dr ≤ ∥α∥_{L²}∥dα∥_{L²} < ∞, hence liminf_{r→∞} ∫_{Σ_r} α∧dα = 0, so Stokes's theorem applies on the exhaustion {r ≤ r_j}. If this argument goes through (including ALG*/ALH* with their model geometries), the central theorem survives as a corrected proof; if not, Proposition 5.3 and Theorem 1.8(b) fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5, Lemma 5.2, Eq. (5.5): the proof asserts |α∧dα| = O(r^{-4+m}) because α,dα ∈ L². This does not follow from L² integrability and is false in general for L² forms. The lemma is used in Proposition 5.3 to show that the anti-self-dual part d_{-}α of an eigen-1-form is exact and L², hence zero; that is the only step forcing E¹(λ)=0 from E⁰(λ)=0, and Theorem 1.8(b) depends on it. The standard repair is to use integrability of α∧dα via Cauchy-Schwarz and the coarea formula to choose radii r_j with the spherical flux ∫_{Σ_{r_j}} α∧dα → 0; the boundary term then vanishes along a sequence. The paper does not give this argument, and for ALG*/ALH* it explicitly omits the model-space details. If the repair cannot be carried out, Proposition 5.3 and Theorem 1.8(b) are unproved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies L² eigenforms of the Hodge Laplacian on complete non-compact Ricci-flat manifolds with special holonomy, with the main focus on 4-dimensional gravitational instantons. It defines the eigenspaces E^k(λ) and proves Theorem 1.8: for λ ≠ 0 on any gravitational instanton, the absence of L² eigenfunctions implies the absence of L² eigenforms in every degree, while the presence of an eigenfunction produces eigenforms in all degrees. Combining this with known non-existence of eigenfunctions on ALE spaces (via Donnelly's exhaustion theorem) and with a new non-existence theorem on A_n ALF/multi-Taub–NUT spaces (via dimensional reduction and magnetic Schrödinger vanishing), the paper derives Corollaries 1.13 and 1.15: no nonzero-eigenvalue L² eigenforms on these families, while Atiyah–Hitchin and its double cover have infinitely many such eigenforms. The paper also proves partial analogues for Calabi–Yau 3-folds, G₂ manifolds and Spin(7) manifolds using holonomy decompositions, and it gives a physical discussion of massive Kaluza–Klein modes in string and M-theory.","tokens_in":27607,"tokens_out":10412,"duration_ms":95560,"significance":"If the main theorems hold, this is a clean and useful structural result: on gravitational instantons it reduces a form-spectrum question to the scalar Laplacian, and it gives the first rigorous non-existence statements for nonzero-eigenvalue L² eigenforms on ALE and A_n ALF spaces. The proof strategy is attractive: it uses the hyperkähler parallel self-dual 2-forms, the stability of the self-dual/anti-self-dual splitting, and exactness to force dα = d*α = 0 for an eigen-1-form once no eigenfunctions exist. The paper also gives genuinely useful partial extensions in higher special-holonomy settings and correctly identifies where the argument uses curvature decay. The reliance on external results (Donnelly [21], scattering calculus [17], and magnetic Schrödinger theory [6]) is appropriate; no quantity is fitted and the argument is a priori rather than circular. However, one boundary-integration step in the proof of the central theorem is not justified as written, and the ALG*/ALH* and line-bundle extensions are asserted rather than proved in full detail.","major_comments":[{"comment":"The assertion that |α∧dα| = O(r^{-4+m}) follows from α,dα ∈ L² is not valid. L² integrability gives only average decay over spheres, not pointwise decay, and for general L² forms the pointwise estimate is false. This step is load-bearing: Proposition 5.3 uses Lemma 5.2 to conclude dα = 0 for an eigen-1-form once E⁰(λ) = 0, and Theorem 1.8(b) and Corollary 1.15(a) depend on it. The standard repair is to avoid pointwise decay: use Cauchy–Schwarz and the coarea formula to choose radii r_j → ∞ with ∫_{Σ_{r_j}} |α∧dα| dσ → 0, so that the boundary term in (5.4) vanishes along a subsequence. The same repair is needed in Lemma 7.1, whose proof copies this argument. Please supply this sequence argument and state explicitly that the boundary term vanishes along a sequence of cutoffs rather than by pointwise decay.","section":"§5.2, Lemma 5.2, Eq. (5.5)"},{"comment":"Lemma 3.3 and the final paragraph of Lemma 5.2 dismiss the ALG* and ALH* cases with 'the argument is essentially the same' and 'we omit the details.' These cases are included in the statement of Theorem 1.8, so the proof is incomplete for a class of gravitational instantons in the theorem. In particular, one needs a radius function, the volume growth of the model spheres at infinity, and the analogue of (5.5)–(5.6) for these geometries; the paper does not provide them. Please either add these estimates or restrict Theorem 1.8 to the four AT^mC classes and state the ALG*/ALH* case as an open extension.","section":"§3.2 and §5.2, ALG*/ALH* cases"},{"comment":"The proof of Theorem 6.13 is an extension of [6] from Euclidean R³ to a Hermitian line bundle over an exterior domain with possibly nontrivial topology at infinity. The manuscript correctly identifies the needed replacement of the magnetic term by s ι_X dη, but the key step 'the proof of [6, Proposition 4.1] goes through' is asserted rather than demonstrated. It must be checked that the hypotheses of [6, Prop. 4.1] hold for H_s on D_{R_0}, including the treatment of the inner boundary and the nontrivial holonomy of L_s at infinity. The estimates (6.33), (6.36) and (6.37) are the right ingredients, but the derivation of (6.53)–(6.54) from them is not written out. Since Theorem 6.13 is the technical heart of Theorem 1.14, and hence of Corollary 1.15(a), this point needs to be closed before the result can be considered fully proved.","section":"§6.7, Theorem 6.13, Steps 4–5"}],"minor_comments":[{"comment":"The word 'gravitional' should be 'gravitational'.","section":"§1.1, Remark 1.7"},{"comment":"In the displayed equations, the terms written as '2∫_M f_k ∧ α ∧ ∗dα' are missing the factor df_k; they should read '2∫_M f_k df_k ∧ α ∧ ∗dα'. The same typo appears in the estimate after (3.14).","section":"§3.2, Eqs. (3.12) and (3.15)"},{"comment":"'dual cover' should be 'double cover' in both occurrences.","section":"§6.5, Remark 6.5"},{"comment":"The notation 'E^0_λ(s)' is used where the paper elsewhere writes 'E^0_s(λ)'; please make the notation uniform.","section":"§6.4"},{"comment":"In '⟨v_k, i[H_s,D]]v_k⟩' there is a stray closing bracket; it should be '⟨v_k, i[H_s,D]v_k⟩'.","section":"§6.7, Step 5"},{"comment":"'analogus' should be 'analogous'.","section":"§7.3, Lemma 7.7"}],"recommendation":"major_revision","confidential_remarks":"The paper is squarely within the scope of the journal and the main results are valuable if the proof gaps are fixed. I see no circularity or fitting of parameters; the self-cited physics papers enter only as motivation. My principal concern is that the central theorem currently rests on an unjustified boundary estimate in Lemma 5.2, and that the ALG*/ALH* and line-bundle extensions are asserted rather than proved in detail. I believe these are repairable, which is why I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main results here are probably right, but one proof step in Section 5 is written incorrectly and needs a routine fix. The stress-test note is accurate: Lemma 5.2 claims the boundary term vanishes because |α∧dα| = O(r^{-4+m}) follows from α,dα ∈ L^2. That pointwise estimate does not follow from L^2 integrability and is false in general. The repair is standard, though: use Cauchy–Schwarz to get integrability of |α||dα|, then choose radii r_j along which the sphere integral of α∧dα tends to zero, and take limits after applying Stokes on annuli. I expect the authors can supply this without changing the architecture of the proof. So the gap is real but repairable, not fatal to the paper's central claims.\n\nWhat is genuinely new: Theorem 1.8(b), reducing the nonzero-eigenvalue L^2 eigenform problem to the scalar Laplacian on hyperkähler 4-manifolds, is a clean and useful observation. The Fourier/magnetic-Schrödinger argument in Section 6 proving vanishing of L^2 eigenfunctions on A_n ALF spaces is a substantial extension of [6], and the paper is honest that the line-bundle case requires additional work. Section 7's decompositions for SU(3), G2, and Spin(7) give partial results that look correct and will be handy for people working on spectra in higher dimensions. The scattering-calculus result Proposition 1.17 is less novel but fills a gap in the literature.\n\nSoft spots beyond Lemma 5.2: the ALG* and ALH* cases in Lemma 5.2 are explicitly omitted, with the paper saying the argument is essentially the same. That is probably true, but a referee should push for details. The proof of Theorem 6.13 is a high-level sketch following [6]; the radial-gauge patching for nontrivial line bundles is plausible, but the estimates deserve line-by-line checking. The phrase \"and nothing else\" in Section 8.1 overstates what a spectral theorem shows—absence of massive L^2 eigenforms does not by itself rule out all possible states—though the authors soften this later.\n\nOverall, the paper shows serious thinking, cites external tools appropriately, and the flaws are repairable. I would send it to a good journal without hesitation, asking for the Lemma 5.2 repair and more details on the omitted cases.","headline":"The paper's central spectral rigidity results are credible and likely correct, but Lemma 5.2 contains a real gap that needs a standard repair before the proof of Theorem 1.8(b) is sound.","tokens_in":28184,"tokens_out":3070,"would_cite":true,"duration_ms":28337,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58J50","53C26","53C29","53C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"On any gravitational instanton, nonzero-eigenvalue L2 eigenforms exist exactly when L2 eigenfunctions do, and on ALE and A_n ALF spaces they do not.","keywords":["L2 eigenforms","Hodge Laplacian","gravitational instantons","special holonomy","hyperkähler manifolds","asymptotically conical manifolds","multi-Taub-NUT","magnetic Schrödinger operators"],"falsifier":"Exhibit a gravitational instanton and a nonzero $\\lambda$ with an $L^2$ eigenform of the Hodge Laplacian in some degree but no $L^2$ eigenfunction; this would falsify Theorem 1.8(b). More locally, find an $L^2$ 1-form $\\alpha$ with $d\\alpha\\in L^2$ on a gravitational instanton such that the flux of $\\alpha\\wedge d\\alpha$ through large spheres does not tend to zero, which would falsify Lemma 5.2 and the vanishing of $E^1(\\lambda)$ when $E^0(\\lambda)=0$.","tokens_in":2496,"feed_emoji":"🌀","tokens_out":3203,"duration_ms":90582,"temperature":0.7,"pith_summary":"This paper shows that on any gravitational instanton—a complete non-compact Ricci-flat 4-manifold with hyperkähler holonomy—the nonzero-eigenvalue spectrum of the Hodge Laplacian on forms is rigidly tied to the spectrum on functions: a nonzero-eigenvalue L2 eigenform exists in any degree exactly when a nonzero-eigenvalue L2 eigenfunction exists. Because asymptotically conical manifolds admit no L2 eigenfunctions, all ALE gravitational instantons therefore have no nonzero-eigenvalue L2 eigenforms at all. The same vanishing is proved for A_n ALF (multi-Taub–NUT) instantons, where the scalar eigenvalue problem is reduced by a circle-action Fourier decomposition to a family of magnetic Schrödinger operators. Partial analogues of the rigidity are proved for Calabi–Yau 3-folds, G2 manifolds, and Spin(7) manifolds. Physically, the vanishing theorems say that string and M-theory compactifications on ALE and A_n ALF spaces produce no massive Kaluza–Klein states from L2 forms.","feed_headline":"On gravitational instantons, L2 eigenforms reduce to eigenfunctions","feed_subtitle":"When no L^2 eigenfunction exists, no nonzero-eigenvalue L^2 eigenform exists either—so on ALE and A_n ALF spaces there are none.","key_machinery":"The mechanism carrying Theorem 1.8 is the hyperkähler structure: on a gravitational instanton the self-dual 2-form bundle is trivialised by three parallel Kähler forms $\\omega_1,\\omega_2,\\omega_3$, so every self-dual 2-form is $\\sum f_i\\omega_i$ and the Hodge Laplacian acts componentwise, giving $E^2_+(\\lambda) \\cong E^0(\\lambda)^3$. The 1-form and anti-self-dual cases are killed by Lemma 5.2, which uses Stokes theorem with the boundary term at infinity vanishing because $|\\alpha\\wedge d\\alpha|=O(r^{-4+m})$ on an $AT^mC$ end; this forces $\\|d\\alpha\\|_{L^2}^2=0$. For $A_n$ ALF instantons, the key machinery is the Gibbons–Hawking circle action: Fourier decomposition in the fibre direction reduces the scalar Laplacian eigenvalue equation to a family of magnetic Schrödinger operators $P_s^*P_s + W_{s,\\lambda}$ on $\\mathbb{R}^3\\setminus\\{x_1,\\dots,x_n\\}$, and the positive-energy modes are ruled out by an extension of the magnetic virial vanishing argument, adapted to a nontrivial line bundle over the sphere at infinity.","core_discovery":"The central discovery is that for $\\lambda \\neq 0$ the spaces $E^k(\\lambda)$ of $L^2$ eigenforms of the Hodge Laplacian on a gravitational instanton are all simultaneously trivial or all simultaneously nontrivial, and both alternatives are controlled by functions. The argument uses the hyperkähler structure: the self-dual 2-form bundle on a gravitational instanton is flat and trivial, spanned by three parallel Kähler forms $\\omega_1,\\omega_2,\\omega_3$, so the self-dual eigenform equation decomposes componentwise into three copies of the eigenfunction equation. The remaining form degrees are killed by a Stokes-boundary lemma: if $\\alpha,d\\alpha \\in L^2$, the boundary term at infinity vanishes and forces $d\\alpha = 0$ whenever $d\\alpha$ is purely self-dual or anti-self-dual. As a consequence, once $E^0(\\lambda)=0$, every $L^2$ eigenform with eigenvalue $\\lambda$ vanishes in every degree. Combining this rigidity with known non-existence of $L^2$ eigenfunctions on asymptotically conical manifolds yields vanishing of all nonzero-eigenvalue $L^2$ eigenforms on ALE instantons; for $A_n$ ALF instantons the same conclusion is reached by proving $E^0(\\lambda)=0$ via the circle action and a magnetic Schrödinger vanishing theorem.","pith_inferences":["If Theorem 1.8(b) holds for all gravitational instantons as stated, the hyperkähler rigidity should extend unchanged to ALG, ALH, ALG* and ALH* types; what is missing there is only the input $E^0(\\lambda)=0$, which the paper's Fourier method does not currently reach.","The stark contrast between $A_n$ ALF spaces (no massive eigenforms) and Atiyah–Hitchin (infinitely many) suggests the sign of the Gibbons–Hawking mass at infinity is the controlling feature; this could be tested by studying $D_n$ ALF instantons with negative-mass asymptotics.","In the physical picture, the vanishing results imply that only the zero-mode sector—ADE harmonic 2-forms together with brane states—contributes to the low-energy effective theory on ALE spaces; if any non-AC special-holonomy manifold admitted massive $L^2$ eigenforms, those would be genuine massive Kaluza–Klein particles rather than geometric artifacts.","The AC vanishing result (Proposition 1.17) is independent of holonomy, so the paper suggests that asymptotic conicality, not special holonomy per se, is what suppresses massive $L^2$ eigenforms; a direct test would be to look for nonzero-eigenvalue $L^2$ eigenforms on asymptotically cylindrical special-holonomy manifolds, where the paper predicts they should exist."],"forward_implications":["On every ALE gravitational instanton, the only $L^2$ eigenforms of the Hodge Laplacian are harmonic forms: there are no nonzero-eigenvalue $L^2$ eigenforms in any degree.","On every $A_n$ ALF/multi-Taub–NUT gravitational instanton, the same vanishing holds: $E^k(\\lambda)=0$ for all $\\lambda\\neq 0$ and all $k$.","On the Atiyah–Hitchin manifold and its double cover, there are infinitely many nonzero eigenvalues $\\lambda$ for which $L^2$ eigenforms exist in every degree $k=0,1,2,3,4$.","On any Kähler, $G_2$, or $Spin(7)$ manifold, a nonzero-eigenvalue $L^2$ eigenfunction forces the existence of nonzero-eigenvalue $L^2$ eigenforms in many form degrees, by wedging with parallel forms.","The vanishing results rule out massive Kaluza–Klein states in string and M-theory compactifications on ALE and $A_n$ ALF gravitational instantons, leaving only zero-mode physics coupled to the ADE gauge sector."],"supporting_citations":[{"why":"Supplies the vanishing theorem for positive eigenvalues of magnetic Schrödinger operators, which is adapted to rule out positive-energy Fourier modes on $A_n$ ALF spaces.","marker":"[6]"},{"why":"Provides the scattering-calculus eigenvalue vanishing result applied to prove non-existence of $L^2$ eigenforms on asymptotically conical manifolds.","marker":"[17]"},{"why":"Gives the exhaustion-function criterion used to prove that asymptotically conical manifolds admit no $L^2$ eigenfunctions.","marker":"[21]"},{"why":"Proves existence of $L^2$ eigenfunctions on the Atiyah–Hitchin manifold, which feeds directly into Corollary 1.15(b).","marker":"[8]"},{"why":"Provides the known eigenfunctions on Atiyah–Hitchin used to conclude the existence of infinitely many nonzero-eigenvalue eigenforms.","marker":"[27]"},{"why":"Establishes the Gibbons–Hawking ansatz that constructs the multi-Taub–NUT ($A_n$ ALF) gravitational instantons studied in Theorem 1.14.","marker":"[26]"},{"why":"Gives the cut-off and integration-by-parts method used to show that an $L^2$ eigenform has $d\\alpha,d^*\\alpha\\in L^2$ and hence nonnegative eigenvalue.","marker":"[34]"},{"why":"Provides the recent classification of gravitational instantons, ensuring the proof of Lemma 5.2 covers all possible asymptotic types.","marker":"[48]"}],"fun_headline_variants":["Gravitational instantons: no nonzero L2 eigenforms","L2 eigenforms reduce to eigenfunctions on instantons","All L2 eigenforms vanish on ALE and ALF instantons","Eigenforms on grav instantons: only if eigenfunctions exist"],"cache_read_input_tokens":30208,"weakest_assumption_plain":"The load-bearing premise is that on a gravitational instanton, the boundary term at infinity in the Stokes identity for $\\int d(\\alpha\\wedge d\\alpha)$ vanishes whenever $\\alpha,d\\alpha\\in L^2$; the stated pointwise decay does not follow from $L^2$-integrability alone, so if the flux through large spheres fails to tend to zero, the proof of Theorem 1.8(b) breaks down.","fun_headline_variants_meta":{"raw":{"variants":["Gravitational instantons: no nonzero L2 eigenforms","L2 eigenforms reduce to eigenfunctions on instantons","All L2 eigenforms vanish on ALE and ALF instantons","Eigenforms on grav instantons: only if eigenfunctions exist"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000865,"raw_usage":{"total_tokens":3717,"prompt_tokens":880,"completion_tokens":2837,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":496,"completion_tokens_details":{"reasoning_tokens":2764}},"tokens_in":496,"tokens_out":2837,"duration_ms":20826,"temperature":1.0,"reasoning_tokens":2764,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:24:18.296499+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a gravitational instanton and a nonzero $\\lambda$ with an $L^2$ eigenform of the Hodge Laplacian in some degree but no $L^2$ eigenfunction; this would falsify Theorem 1.8(b). More locally, find an $L^2$ 1-form $\\alpha$ with $d\\alpha\\in L^2$ on a gravitational instanton such that the flux of $\\alpha\\wedge d\\alpha$ through large spheres does not tend to zero, which would falsify Lemma 5.2 and the vanishing of $E^1(\\lambda)$ when $E^0(\\lambda)=0$.","supporting_citations":[{"cited_title":"Ab- sence of positive eigenvalues of magnetic Schr¨ odinger operators.Calc","cited_arxiv_id":null,"evidence_quote":"Supplies the vanishing theorem for positive eigenvalues of magnetic Schrödinger operators, which is adapted to rule out positive-energy Fourier modes on $A_n$ ALF spaces."},{"cited_title":"Chru´ sciel, Luc Nguyen, Paul Tod, and Andr´ as Vasy","cited_arxiv_id":null,"evidence_quote":"Provides the scattering-calculus eigenvalue vanishing result applied to prove non-existence of $L^2$ eigenforms on asymptotically conical manifolds."},{"cited_title":"Exhaustion functions and the spectrum of Riemannian manifolds.Indiana Univ","cited_arxiv_id":null,"evidence_quote":"Gives the exhaustion-function criterion used to prove that asymptotically conical manifolds admit no $L^2$ eigenfunctions."},{"cited_title":"Schroers, and Kim Smedley-Williams","cited_arxiv_id":null,"evidence_quote":"Proves existence of $L^2$ eigenfunctions on the Atiyah–Hitchin manifold, which feeds directly into Corollary 1.15(b)."},{"cited_title":"Gibbons and Nicholas S","cited_arxiv_id":null,"evidence_quote":"Provides the known eigenfunctions on Atiyah–Hitchin used to conclude the existence of infinitely many nonzero-eigenvalue eigenforms."},{"cited_title":"Gibbons and Stephen W","cited_arxiv_id":null,"evidence_quote":"Establishes the Gibbons–Hawking ansatz that constructs the multi-Taub–NUT ($A_n$ ALF) gravitational instantons studied in Theorem 1.14."},{"cited_title":"Noncompact Riemannian manifolds with purely continuous spectrum.Michigan Math","cited_arxiv_id":null,"evidence_quote":"Gives the cut-off and integration-by-parts method used to show that an $L^2$ eigenform has $d\\alpha,d^*\\alpha\\in L^2$ and hence nonnegative eigenvalue."},{"cited_title":"Collapsing geometry of hyperk¨ ahler 4- manifolds and applications.Acta Math., 232(2):325–424, 2024","cited_arxiv_id":null,"evidence_quote":"Provides the recent classification of gravitational instantons, ensuring the proof of Lemma 5.2 covers all possible asymptotic types."}],"review_version":2}