{"paper":{"title":"Analytic local resolution of Medvedev's Morse index conjecture for the critical hyperbolic catenoid in $\\mathbb{H}^3$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The critical hyperbolic catenoid has Morse index 4 and nullity 2 for a slightly larger than 1/2.","cross_cats":["math.AP","math.SP"],"primary_cat":"math.DG","authors_text":"Alexander Pigazzini","submitted_at":"2026-05-13T14:03:12Z","abstract_excerpt":"Let $\\Sigma_a\\subset B^3(r(a))\\subset\\mathbb{H}^3$ ($a>1/2$) be the critical hyperbolic catenoid of the Mori family, a free boundary minimal surface in the geodesic ball. The Medvedev conjecture [8] states ind$(\\Sigma_a)=4$ for all $a>1/2$. We study its strong form: ind$(\\Sigma_a)=4$ and nul$(\\Sigma_a)=2$. The nullity condition nul$(\\Sigma_a)=2$ combines the mode-$|k|=1$ result $\\text{nul}_R(\\Sigma_a)|_{|k|=1}=2$ of [10, Cor. 4.4] with vanishing kernel in modes $|k|=0,|k|\\ge2$; the latter, not in [10], is established here for $a\\in(1/2,1/2+\\delta_0)$. The main result is the analytic local reso"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"There exists δ0 > 0 such that ind(Σ_a) = 4 and nul(Σ_a) = 2 for all a ∈ (1/2, 1/2 + δ0). This follows from the expansion H(a) = sinh r(a)/K(a) = σ* cosh σ* + C0 (a - 1/2) + O((a-1/2)^2) with C0 > 0, combined with reductions (E), (F), (G).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The reduction of φ_a > 0 to H'(a) > 0 under the assumption sinh r(a) > 2K(a) (condition G), whose analytic closure on (1/2,1] is claimed via strict concavity of a transcendental function; if this inequality fails or the concavity argument has a gap, the local positivity may not hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Local analytic resolution of the strong Medvedev conjecture: the critical hyperbolic catenoid has Morse index 4 and nullity 2 for a in (1/2, 1/2 + δ0).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The critical hyperbolic catenoid has Morse index 4 and nullity 2 for a slightly larger than 1/2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8b0bf1158e67ec27dbb7782057614cf7d7c737be4739f141c2407e73ea401afd"},"source":{"id":"2605.13562","kind":"arxiv","version":1},"verdict":{"id":"84968c77-9112-44c8-b1b0-6cdc8e00013c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:55:46.818283Z","strongest_claim":"There exists δ0 > 0 such that ind(Σ_a) = 4 and nul(Σ_a) = 2 for all a ∈ (1/2, 1/2 + δ0). This follows from the expansion H(a) = sinh r(a)/K(a) = σ* cosh σ* + C0 (a - 1/2) + O((a-1/2)^2) with C0 > 0, combined with reductions (E), (F), (G).","one_line_summary":"Local analytic resolution of the strong Medvedev conjecture: the critical hyperbolic catenoid has Morse index 4 and nullity 2 for a in (1/2, 1/2 + δ0).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The reduction of φ_a > 0 to H'(a) > 0 under the assumption sinh r(a) > 2K(a) (condition G), whose analytic closure on (1/2,1] is claimed via strict concavity of a transcendental function; if this inequality fails or the concavity argument has a gap, the local positivity may not hold.","pith_extraction_headline":"The critical hyperbolic catenoid has Morse index 4 and nullity 2 for a slightly larger than 1/2."},"references":{"count":11,"sample":[{"doi":"","year":1955,"title":"E.A. Coddington and N. Levinson,Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955","work_id":"19869b66-ec41-4851-a6c0-26e37e3488b5","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2011,"title":"T. H. Colding, W. P. Minicozzi,A course in minimal surfaces, Graduate Studies in Mathematics, 121, AMS, 2011","work_id":"fcb9d2fc-2898-4f34-a811-4c0959bbea3d","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"Devyver,Index of the critical catenoid, Geom","work_id":"9b6d8474-acb3-45e8-954d-fd70b7dea3e1","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1973,"title":"Eastham,The Spectral Theory of Periodic Differential Equations, Scottish Acad","work_id":"5dfb9610-e328-4b20-bf81-cc707481b61b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"A. Fraser, R. Schoen,Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. Math. 203 (2015), no. 3, 823–890","work_id":"e7f25912-4ec5-4beb-bbb5-446762c1db49","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":11,"snapshot_sha256":"643568d4fbbf632746a52421a0b66a6fef2d00a97bfa358b2598e774d9cc9fd7","internal_anchors":1},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}