Pith Number

pith:Z4AIQH3U

pith:2026:Z4AIQH3UXZQWCORRXSHNQNEFNY

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Symmetry and Rigidity Results for the Mean Field Equation and Hawking Mass on ( \mathbb{S}^2 )

Amir Moradifam, Changfeng Gui

Solutions of the mean field equation on the sphere are symmetric for 1/3 ≤ α < 1/2, forcing rigidity of the Hawking mass for stable CMC spheres.

arxiv:2605.15448 v1 · 2026-05-14 · math.AP

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Claims

C1strongest claim

Symmetry results for solutions of the mean field equation (α/2) Δu + e^u - 1 = 0 on S^2 for 1/3 ≤ α < 1/2, applied to demonstrate rigidity of the Hawking mass for stable CMC spheres, unifying and extending previous results for surfaces that are not nearly spherical.

C2weakest assumption

The Sphere Covering Inequality together with topological arguments on S^2 suffice to establish the claimed symmetry for the full range 1/3 ≤ α < 1/2 (abstract, paragraph on proofs).

C3one line summary

Symmetry results for the mean field equation on S^2 for 1/3 ≤ α < 1/2 are established via the Sphere Covering Inequality and topology, then used to prove Hawking mass rigidity for stable CMC spheres.

References

24 extracted · 24 resolved · 0 Pith anchors

[1] Meilleures constantes dans le th´ eor` eme d’inclusion de Sobolev et un th´ eor` eme de Fredholm non lin´ eaire pour la transformation conforme de la courbure scalaire.J 1979

[2] Mass and 3-metrics of non-negative scalar curvature 2002

[3] A singular sphere covering inequality: uniqueness and symmetry of solutions to singular Liouville- type equations.Math 1922

[4] Localizing solutions of the Einstein constraint equations.Invent 2016

[5] Chang and Paul C 1988

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:00:59.189018Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

cf00881f74be61613a31bc8ed834856e1941e6fe78acdd824149fc860f7b80ad

Aliases

arxiv: 2605.15448 · arxiv_version: 2605.15448v1 · doi: 10.48550/arxiv.2605.15448 · pith_short_12: Z4AIQH3UXZQW · pith_short_16: Z4AIQH3UXZQWCORR · pith_short_8: Z4AIQH3U

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/Z4AIQH3UXZQWCORRXSHNQNEFNY \
  | 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: cf00881f74be61613a31bc8ed834856e1941e6fe78acdd824149fc860f7b80ad

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "2c7cc6771df2e9f4d0869e8bdd00be5694caa766272508808e2eb67348788af7",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-05-14T22:16:55Z",
    "title_canon_sha256": "626d0ba6857f40806de1939bb7e9f828e7dc0cb248d181ae70975e0ef3a2b65b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15448",
    "kind": "arxiv",
    "version": 1
  }
}