Pith Number

pith:J4ZIQ7LZ

pith:2026:J4ZIQ7LZC3PNQOSWXLMROBFCNY

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A comparison theorem with applications to sharp geometric inequalities for submanifolds

Chengyang Yi, Shengliang Pan

An explicit formula for the Jacobian determinant of the normal exponential map on a submanifold leads to a new comparison theorem and sharp geometric inequalities.

arxiv:2605.06074 v2 · 2026-05-07 · math.DG

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Claims

C1strongest claim

We derive an explicit expression for the Jacobian determinant of the normal exponential map on a submanifold, establishing a relationship with its ambient counterpart. This formula leads to a new comparison theorem which is closely related to the comparison theorem of Heintze-Karcher and the estimate of Brendle. As applications, we obtain a Fenchel-Borsuk-Chern-Lashof-type inequality and a Willmore-Chen-type inequality on closed submanifolds in complete noncompact manifolds with nonnegative curvature and Euclidean volume growth.

C2weakest assumption

The ambient manifold is complete and noncompact with nonnegative sectional curvature and Euclidean volume growth; the submanifolds are closed (and the normal exponential map is well-defined up to the cut locus).

C3one line summary

An explicit Jacobian formula for the normal exponential map produces a comparison theorem that implies Fenchel-Borsuk-Chern-Lashof-type and Willmore-Chen-type inequalities for closed submanifolds in complete noncompact manifolds with nonnegative curvature and Euclidean volume growth.

Cited by

1 paper in Pith

Fenchel-Willmore-Chen inequality under lower bounds on weighted intermediate Ricci curvature

Receipt and verification

First computed	2026-07-17T01:20:50.705819Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

4f32887d7916ded83a56bad91704a26e02bbda41762bee8cdcd4e2defb68f64c

Aliases

arxiv: 2605.06074 · arxiv_version: 2605.06074v2 · doi: 10.48550/arxiv.2605.06074 · pith_short_12: J4ZIQ7LZC3PN · pith_short_16: J4ZIQ7LZC3PNQOSW · pith_short_8: J4ZIQ7LZ

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/J4ZIQ7LZC3PNQOSWXLMROBFCNY \
  | 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: 4f32887d7916ded83a56bad91704a26e02bbda41762bee8cdcd4e2defb68f64c

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "ac07c4d84679c9c566edd613b890d6a58960fa370329f82eaa5500b1bcda01b8",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.DG",
    "submitted_at": "2026-05-07T11:59:15Z",
    "title_canon_sha256": "e6b74c235316a6aa3df676bf7fe1f09e9891b099b62210a947b4d0db6aa87bb7"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.06074",
    "kind": "arxiv",
    "version": 2
  }
}