Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-11T14:03:05.576034Z
Paper Citation Record · LEDGER
A sharp spectral splitting theorem
As of 12 August 2026, this Paper Citation Record lists 15 of 15 outbound references and 10 inbound Pith citation observations for arXiv:2412.12707.
A citation records a reference. It does not transfer a finding from one paper to another.
Coverage vector
One-hop event checks from named stored sources.
Source: scholarly_work_events, retraction_status_cache, observed 2026-08-12T06:34:41.77262+00:00
Pith citing papers itemized under the disclosed page cap.
Source: paper_references, paper_reference_links, observed 2026-08-07T14:43:24.738002Z
A source-named dated measurement, never combined with another source.
Source: pith, observed 2026-07-08T04:14:29.564261Z
Reference resolution
15 of 15 outbound references displayed
- verified exact2
- verified fuzzy5
- unresolved8
- parse uncertain0
- malformed identifier0
- metadata mismatch0
External citation measurements
No source-named external measurement is stored.
Outbound references
Observation 792423bd-97c3-4981-9819-b6456a045e7d · outbound
This paper cites Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds.
A sharp spectral splitting theorem Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds
Reference 1
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T14:03:05.974794Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 92169be1-39e2-45e9-b311-b17ce28215e2 · outbound
This paper cites Kato meets Bakry-\'Emery.
Reference 4
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T14:03:05.487526Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation e2e8a0f2-02b9-4dd7-b426-36c27ba607e9 · outbound
This paper cites The splitting theorem for manifolds of nonnegative Ricci curva- ture.
A sharp spectral splitting theorem The splitting theorem for manifolds of nonnegative Ricci curva- ture
Reference 5
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T14:03:05.953900Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 303d572a-10ef-4800-8552-e748638353e4 · outbound
This paper cites Stable minimal hypersurfaces in $\mathbf{R}^5$.
A sharp spectral splitting theorem Stable minimal hypersurfaces in $\mathbf{R}^5$
Reference 7
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T14:03:05.514077Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 4de74805-dc3c-4d1e-b848-bb7eb4963089 · outbound
This paper cites Homological $n$-systole in $(n+1)$-manifolds and bi-Ricci curvature.
A sharp spectral splitting theorem Homological $n$-systole in $(n+1)$-manifolds and bi-Ricci curvature
Reference 8
- Resolution
- verified exact
- local_arxiv, observed 2026-08-11T14:03:05.743967Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 6581ace8-36ad-407e-acfd-d241b3f760d5 · outbound
This paper cites On K\"ahler manifolds with non-negative mixed curvature.
A sharp spectral splitting theorem On K\"ahler manifolds with non-negative mixed curvature
Reference 9
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T14:03:05.533312Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 523b5b0f-d816-4fb9-9921-9bd954bda3e6 · outbound
This paper cites Ricci curvature, geodesics and some geom etric properties of Riemannian manifolds with boundary.
A sharp spectral splitting theorem Ricci curvature, geodesics and some geom etric properties of Riemannian manifolds with boundary
Reference 12
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T14:03:05.883951Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation eb63111f-eee3-497d-a60c-a632a9e04766 · outbound
This paper cites Stable minimal hypersurfaces in $\mathbb R^6$.
A sharp spectral splitting theorem Stable minimal hypersurfaces in $\mathbb R^6$
Reference 14
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T14:03:05.568773Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 758fa523-d8fc-4bce-addf-71845eda1ea0 · outbound
This paper cites Dimension constraints in some problems involving intermediate curvature.
A sharp spectral splitting theorem Dimension constraints in some problems involving intermediate curvature
Reference 15
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T14:03:05.576034Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 79ee6ac2-c75b-49ba-b71c-c3bd144d9d75 · outbound
This paper cites Generalized soap bubbles and th e topology of manifolds with positive scalar curvature.
A sharp spectral splitting theorem Generalized soap bubbles and th e topology of manifolds with positive scalar curvature
Reference 22
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T14:03:05.930437Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation a94866c2-ea82-4f9a-acdb-3dd644382574 · outbound
This paper cites an unresolved cited work.
A sharp spectral splitting theorem Unresolved cited work
Reference 132
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T14:03:05.906896Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 7262aeaf-e9fa-4a00-8d98-851a17d3b9d2 · outbound
This paper cites An introduction to geomet ric measure theory.
A sharp spectral splitting theorem An introduction to geomet ric measure theory
Reference 135
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T14:03:05.864508Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 9e3634db-f378-428a-91b4-bdcdcfffd627 · outbound
This paper cites Four Lectures on Scalar Curvature.
A sharp spectral splitting theorem Four Lectures on Scalar Curvature
Reference 2021
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T14:03:05.546643Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 56495f99-57ca-4587-8a81-ba9a14c239ae · outbound
This paper cites Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds.
A sharp spectral splitting theorem Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds
Reference 2023
- Resolution
- verified exact
- local_arxiv, observed 2026-08-11T14:03:05.843503Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 7ee726af-74aa-4c7f-9b2b-954a7c16f7ca · outbound
This paper cites New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry.
A sharp spectral splitting theorem New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry
Reference 2024
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T14:03:05.480118Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Pith citing papers
Observation cbb56e36-2031-4d6f-b15e-33340a6942d4 · inbound
Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces cites this paper.
Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces A sharp spectral splitting theorem
Reference 1
- Resolution
- verified exact
- arxiv_id, observed 2026-05-23T07:17:42.539108Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation ba8fd372-eefc-43a8-ae84-cfebda21d170 · inbound
Connected sum of manifolds with spectral Ricci lower bounds cites this paper.
Connected sum of manifolds with spectral Ricci lower bounds A sharp spectral splitting theorem
Reference 1
- Resolution
- unresolved
- no resolver link, observed 2026-08-07T14:43:24.738002Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation a5747f4b-1a88-4579-a4a0-75b12b867601 · inbound
Some rigidity theorems for spectral curvature bounds cites this paper.
Some rigidity theorems for spectral curvature bounds A sharp spectral splitting theorem
Reference 1
- Resolution
- unresolved
- no resolver link, observed 2026-07-13T11:27:57.743687Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 06982599-5410-408a-a5de-b97a87b3f5b2 · inbound
Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$ cites this paper.
Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$ A sharp spectral splitting theorem
Reference 3
- Resolution
- verified exact
- arxiv_id, observed 2026-05-10T11:40:18.998759Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 21e3c4e8-ead2-494b-85e3-929d16357038 · inbound
Intermediate curvature and splitting theorem cites this paper.
Intermediate curvature and splitting theorem A sharp spectral splitting theorem
Reference 1
- Resolution
- metadata mismatch
- arxiv_id, observed 2026-05-12T09:26:25.847589Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 35620d8e-e2bb-4c78-8ae1-fe922f8400dd · inbound
Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary cites this paper.
Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary A sharp spectral splitting theorem
Reference 1
- Resolution
- metadata mismatch
- arxiv_id, observed 2026-05-13T02:07:08.828873Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 055cc098-9e3f-41d7-8580-fb23d8eed370 · inbound
Spectral splitting theorem and ends of minimal hypersurfaces cites this paper.
Spectral splitting theorem and ends of minimal hypersurfaces A sharp spectral splitting theorem
Reference 1
- Resolution
- metadata mismatch
- arxiv_id, observed 2026-07-01T14:45:49.650292Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 2980bd33-1f8a-4641-b38a-2988c85f7997 · inbound
Band Width Estimates and Rigidity of Manifolds with Negative Curvature cites this paper.
Band Width Estimates and Rigidity of Manifolds with Negative Curvature A sharp spectral splitting theorem
Reference 2
- Resolution
- verified exact
- arxiv_id, observed 2026-07-04T19:10:04.848719Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 8d81f63d-8665-4fc9-82ec-30d0508431a9 · inbound
Band Width Estimates and Rigidity of Manifolds with Negative Curvature cites this paper.
Band Width Estimates and Rigidity of Manifolds with Negative Curvature A sharp spectral splitting theorem
Reference 2
- Resolution
- verified exact
- arxiv_id, observed 2026-07-03T22:59:02.691220Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 20728244-24cc-4dcf-ab67-9af4f78c114a · inbound
Bottom spectrum and parabolicity of 3-manifolds with scalar curvature lower bound cites this paper.
Bottom spectrum and parabolicity of 3-manifolds with scalar curvature lower bound A sharp spectral splitting theorem
Reference 1
- Resolution
- verified exact
- local_arxiv, observed 2026-07-08T04:14:29.567303Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.