Pith Number

pith:VMECTXNZ

pith:2026:VMECTXNZKLAOJZJJINSZB7WVC7

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Analytic local resolution of Medvedev's Morse index conjecture for the critical hyperbolic catenoid in $\mathbb{H}^3$

Alexander Pigazzini

The critical hyperbolic catenoid has Morse index 4 and nullity 2 for a slightly larger than 1/2.

arxiv:2605.13562 v1 · 2026-05-13 · math.DG · math.AP · math.SP

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4 Citations open

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Claims

C1strongest 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).

C2weakest 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.

C3one 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).

References

11 extracted · 11 resolved · 1 Pith anchors

[1] E.A. Coddington and N. Levinson,Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955 1955

[2] T. H. Colding, W. P. Minicozzi,A course in minimal surfaces, Graduate Studies in Mathematics, 121, AMS, 2011 2011

[3] Devyver,Index of the critical catenoid, Geom 2019

[4] Eastham,The Spectral Theory of Periodic Differential Equations, Scottish Acad 1973

[5] A. Fraser, R. Schoen,Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. Math. 203 (2015), no. 3, 823–890 2015

Receipt and verification

First computed	2026-05-18T02:44:23.501428Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

ab0829ddb952c0e4e529436590fed517dc0a2585e09f427b8ced41f9fcb2cb86

Aliases

arxiv: 2605.13562 · arxiv_version: 2605.13562v1 · doi: 10.48550/arxiv.2605.13562 · pith_short_12: VMECTXNZKLAO · pith_short_16: VMECTXNZKLAOJZJJ · pith_short_8: VMECTXNZ

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/VMECTXNZKLAOJZJJINSZB7WVC7 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: ab0829ddb952c0e4e529436590fed517dc0a2585e09f427b8ced41f9fcb2cb86

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "7ea10ba4e56a679034026d9983a950e13bbfb369cc326886df04a86fa7949bb0",
    "cross_cats_sorted": [
      "math.AP",
      "math.SP"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.DG",
    "submitted_at": "2026-05-13T14:03:12Z",
    "title_canon_sha256": "dd8e22edb5e553707221edc503030a7e8b7f2ba24fac627a95688018d1e931ad"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13562",
    "kind": "arxiv",
    "version": 1
  }
}