Pith Number

pith:YMVBYGV2

pith:2026:YMVBYGV2JBOR3WZZ2APFFVLHKD

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Discrete Einstein metrics on trees

Bobo Hua, Haoxuan Cheng, Shuliang Bai

Discrete Einstein metrics exist and are unique on trees under Lin-Lu-Yau curvature, but positive-curvature cases require caterpillar trees.

arxiv:2604.22449 v2 · 2026-04-24 · math.DG

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Claims

C1strongest claim

We establish the existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory. Notably, the existence of a positive-curvature Einstein metric implies the tree must be a caterpillar. Furthermore, these metrics exhibit radial monotonicity, with edge weights decreasing strictly away from the maximal edge.

C2weakest assumption

The curvature operator constructed from the Lin-Lu-Yau Ricci curvature on the tree must satisfy the positivity or irreducibility conditions required for Perron-Frobenius theory to guarantee a unique positive eigenvector; the abstract does not specify how this is verified or what happens if the tree has vertices of high degree that break positivity.

C3one line summary

Existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature is established via Perron-Frobenius theory, with positive curvature possible only on caterpillar trees and edge weights decreasing radially from the maximal edge.

Cited by

1 paper in Pith

A Classification of Positive-Curvature Discrete Einstein Metrics on Trees

Receipt and verification

First computed	2026-05-25T02:01:20.923816Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c32a1c1aba485d1ddb39d01e52d56750f69c01b94aea6ab7614cffa6e15f53e6

Aliases

arxiv: 2604.22449 · arxiv_version: 2604.22449v2 · doi: 10.48550/arxiv.2604.22449 · pith_short_12: YMVBYGV2JBOR · pith_short_16: YMVBYGV2JBOR3WZZ · pith_short_8: YMVBYGV2

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/YMVBYGV2JBOR3WZZ2APFFVLHKD \
  | 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: c32a1c1aba485d1ddb39d01e52d56750f69c01b94aea6ab7614cffa6e15f53e6

Canonical record JSON

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  "metadata": {
    "abstract_canon_sha256": "600f05a3b92f95ecede91eee7858f723fe844393c90cc3625f2ad2efa9cca63c",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "primary_cat": "math.DG",
    "submitted_at": "2026-04-24T11:07:15Z",
    "title_canon_sha256": "1ae7042af443ce9f98382cba2983eec062717b80b5f78e96471b2766d3d359b7"
  },
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  "source": {
    "id": "2604.22449",
    "kind": "arxiv",
    "version": 2
  }
}