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We prove that for all sufficiently small $\\lambda>0$\n  there exists a family of \"bubbling\" conformal metrics $g_\\lambda=e^{u_\\lambda}g$ such that their Gauss curvature is given by the sign-changing function $K_{g_\\lambda}=-f+\\lambda^2$. Moreover, the family $u_\\lambda$ satisfies $$u_\\lambda(p_j) = -4\\log\\lambda -2\\log \\lef"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1407.1912","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-08T00:23:01Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"289430cf276d27f6d18f4aa2f5ef896a69794912bc791f064f9aafce58fb9ced","abstract_canon_sha256":"81917a989b349d46b222ebf9b84f71ec2183312df779b2c04abfc2af8c9b92d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:52.905862Z","signature_b64":"NyVx7Y5HKzYQ45JeO1PhpjPVgVyuiSICFAI0WlzWJ8JzCQ2VjRxdloVBBo5UxMaTRmcrKXOFzicb8kc91sqQBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6336996d3dee9048de9f3dcb9f68715a4911b1dd3dd54b51f3c0105663f0f4dd","last_reissued_at":"2026-05-18T00:44:52.905220Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:52.905220Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large Conformal metrics with prescribed sign-changing Gauss curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Carlos Rom\\'an, Manuel del Pino","submitted_at":"2014-07-08T00:23:01Z","abstract_excerpt":"Let $(M,g)$ be a two dimensional compact Riemannian manifold of genus $g(M)>1$. Let $f$ be a smooth function on $M$ such that $$f \\ge 0, \\quad f\\not\\equiv 0, \\quad \\min_M f = 0. $$ Let $p_1,\\ldots,p_n$ be any set of points at which $f(p_i)=0$ and $D^2f(p_i)$ is non-singular. We prove that for all sufficiently small $\\lambda>0$\n  there exists a family of \"bubbling\" conformal metrics $g_\\lambda=e^{u_\\lambda}g$ such that their Gauss curvature is given by the sign-changing function $K_{g_\\lambda}=-f+\\lambda^2$. Moreover, the family $u_\\lambda$ satisfies $$u_\\lambda(p_j) = -4\\log\\lambda -2\\log \\lef"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.1912","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1407.1912","created_at":"2026-05-18T00:44:52.905326+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.1912v1","created_at":"2026-05-18T00:44:52.905326+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.1912","created_at":"2026-05-18T00:44:52.905326+00:00"},{"alias_kind":"pith_short_12","alias_value":"MM3JS3J552IE","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_16","alias_value":"MM3JS3J552IERXU7","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_8","alias_value":"MM3JS3J5","created_at":"2026-05-18T12:28:38.356838+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MM3JS3J552IERXU7HXFZ62DRLJ","json":"https://pith.science/pith/MM3JS3J552IERXU7HXFZ62DRLJ.json","graph_json":"https://pith.science/api/pith-number/MM3JS3J552IERXU7HXFZ62DRLJ/graph.json","events_json":"https://pith.science/api/pith-number/MM3JS3J552IERXU7HXFZ62DRLJ/events.json","paper":"https://pith.science/paper/MM3JS3J5"},"agent_actions":{"view_html":"https://pith.science/pith/MM3JS3J552IERXU7HXFZ62DRLJ","download_json":"https://pith.science/pith/MM3JS3J552IERXU7HXFZ62DRLJ.json","view_paper":"https://pith.science/paper/MM3JS3J5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.1912&json=true","fetch_graph":"https://pith.science/api/pith-number/MM3JS3J552IERXU7HXFZ62DRLJ/graph.json","fetch_events":"https://pith.science/api/pith-number/MM3JS3J552IERXU7HXFZ62DRLJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MM3JS3J552IERXU7HXFZ62DRLJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MM3JS3J552IERXU7HXFZ62DRLJ/action/storage_attestation","attest_author":"https://pith.science/pith/MM3JS3J552IERXU7HXFZ62DRLJ/action/author_attestation","sign_citation":"https://pith.science/pith/MM3JS3J552IERXU7HXFZ62DRLJ/action/citation_signature","submit_replication":"https://pith.science/pith/MM3JS3J552IERXU7HXFZ62DRLJ/action/replication_record"}},"created_at":"2026-05-18T00:44:52.905326+00:00","updated_at":"2026-05-18T00:44:52.905326+00:00"}