Pith Number

pith:SPE5Z5VV

pith:2026:SPE5Z5VVUSYYMXJT5Y4KGQH6JT

not attested not anchored not stored refs resolved

Separable surfaces that are critical points of the Dirichlet energy

Rafael L\'opez

When the constant Λ is nonzero, the only separable surfaces satisfying the PDE φ_xx + φ_yy = Λ/2 are surfaces of revolution or graphs of the form z = f(x) + g(y).

arxiv:2605.13033 v1 · 2026-05-13 · math.DG

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\pithnumber{SPE5Z5VVUSYYMXJT5Y4KGQH6JT}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

if Λ≠0, we show that the surfaces must be either surfaces of revolution or of the type z=f(x)+g(y); furthermore, explicit parametrizations of these surfaces are obtained.

C2weakest assumption

the surfaces are the zero level sets of an implicit equation of the type f(x)+g(y)+h(z)=0 where f, g and h are smooth functions of one variable.

C3one line summary

Separable surfaces obeying the PDE φ_xx + φ_yy = Λ/2 are revolution surfaces or z = f(x) + g(y) when Λ is nonzero.

References

12 extracted · 12 resolved · 0 Pith anchors

[1] J. E. Brothers, F. Morgan, The isoperimetric theorem for general integrands, Michigan Math. J. 41 (1994), 419--431 1994

[2] T. Hasanis, R. L\'opez, A characteristic property of Delaunay surfaces. Proc. Amer. Math. Soc. 148 (2020), 5291--5298 2020

[3] D. Hoffman, H. Karcher, Complete embedded minimal surfaces of finite total curvature, Geometry, V, Encyclopaedia Math. Sci., vol. 90, Springer, Berlin, 1997, pp. 5--93 1997

[4] M. Koiso and B. Palmer, Geometry and stability of surfaces with constant anisotropic mean curvature, Indiana Univ. Math. J. 54 (2005), 1817--1852 2005

[5] M. Koiso and B. Palmer, Stability of anisotropic capillary surfaces between two parallel planes, Calc. Var. Partial Differ. Equ. 25, (2006), 275--298 2006

Receipt and verification

First computed	2026-05-18T03:08:59.676828Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

93c9dcf6b5a4b1865d33ee38a340fe4ceb8afe43198a2d555ea5c9d49c1fe863

Aliases

arxiv: 2605.13033 · arxiv_version: 2605.13033v1 · doi: 10.48550/arxiv.2605.13033 · pith_short_12: SPE5Z5VVUSYY · pith_short_16: SPE5Z5VVUSYYMXJT · pith_short_8: SPE5Z5VV

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/SPE5Z5VVUSYYMXJT5Y4KGQH6JT \
  | 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: 93c9dcf6b5a4b1865d33ee38a340fe4ceb8afe43198a2d555ea5c9d49c1fe863

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b1cb35f809e5533554829bbbe623cc98c69887c28342628e89d196814b910817",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.DG",
    "submitted_at": "2026-05-13T05:36:39Z",
    "title_canon_sha256": "790eeb796f48e0acd32e6f4b8b474b182638de473fffbd79897ac3e8e98924be"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13033",
    "kind": "arxiv",
    "version": 1
  }
}