{"paper":{"title":"Geometric uncertainty principles for Schr\\\"odinger evolutions on negatively curved manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Schrödinger solutions with Gaussian decay at two times are identically zero on asymptotic hyperbolic manifolds.","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Changxing Miao, Ruihan Zhou, Yilin Song","submitted_at":"2026-05-17T03:01:00Z","abstract_excerpt":"In this paper, we study the Hardy type uncertainty principle for Schr\\\"odinger equations with $L^\\infty$ bounded potentials on certain Cartan-Hadamard manifolds endowed with an asymptotic hyperbolic metric in dimensions $n\\geq2$. The classical Hardy uncertainty principle in Euclidean space, as developed in the works of Escauriaza-Kenig-Ponce-Vega (JEMS, 2008; Duke Math. J., 2010), reveals a rigidity phenomenon for solution $u$ to Schr\\\"odinger equations: sufficiently strong Gaussian decay at two distinct times yields $u\\equiv0$. In this work, we show that a similar rigidity persists in the set"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that a similar rigidity persists in the setting of hyperbolic geometry, despite the absence of translation invariance and Fourier representation.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The manifolds are Cartan-Hadamard and endowed with an asymptotic hyperbolic metric, allowing the construction of a new weight function and mollifier via the exponential map and Jacobi fields that yield the required Carleman estimates and logarithmic convexity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The paper establishes a Hardy-type uncertainty principle showing Gaussian decay at two times implies the solution is identically zero for Schrödinger equations on Cartan-Hadamard manifolds with asymptotic hyperbolic metrics.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Schrödinger solutions with Gaussian decay at two times are identically zero on asymptotic hyperbolic manifolds.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b9dac9aeafac75423f0156b07a13c4820240b2c181ed0d31fb4f717b768e87ad"},"source":{"id":"2605.17233","kind":"arxiv","version":1},"verdict":{"id":"8e356fa8-652f-4c67-b698-e9bbdfa046ef","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T23:22:15.188357Z","strongest_claim":"We show that a similar rigidity persists in the setting of hyperbolic geometry, despite the absence of translation invariance and Fourier representation.","one_line_summary":"The paper establishes a Hardy-type uncertainty principle showing Gaussian decay at two times implies the solution is identically zero for Schrödinger equations on Cartan-Hadamard manifolds with asymptotic hyperbolic metrics.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The manifolds are Cartan-Hadamard and endowed with an asymptotic hyperbolic metric, allowing the construction of a new weight function and mollifier via the exponential map and Jacobi fields that yield the required Carleman estimates and logarithmic convexity.","pith_extraction_headline":"Schrödinger solutions with Gaussian decay at two times are identically zero on asymptotic hyperbolic manifolds."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17233/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:31:20.342786Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:31:18.190165Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T22:01:57.895168Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.801264Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"236d9aabd268cf67fa2ab01facd8649494a73c913abf978f2066579123af125c"},"references":{"count":24,"sample":[{"doi":"","year":1977,"title":"Almgren, Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents, inMinimal submanifolds and geodesics (Proc","work_id":"750a2234-c753-45fe-a941-ba1d13e53aee","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"Anderson, Hardy’s uncertainty principle on hyperbolic spaces, Bull","work_id":"aaeaba10-13e0-46b1-9eb8-2acedd83c7bc","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"Anderson,L p versions of Hardy type uncertainty principle on hyperbolic space, Proc","work_id":"12fa03e7-a569-44d1-af23-653a1160a2fd","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2009,"title":"J.-P. 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