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For surfaces Sigma with genus gamma and k boundary components we obtain the upper bound sigma_1L(\\partial \\Sigma) \\leq 2(2gamma+k)\\pi. We attempt to find the best constant in this inequality for annular surfaces (gamma=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogona"},"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":"0912.5392","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-12-29T23:58:21Z","cross_cats_sorted":[],"title_canon_sha256":"76e406468f35ab4ade2a255583591aea7366ace560f479e5956bf95195c5f031","abstract_canon_sha256":"ac533a0362df229c03c982b45b585c45aac7797f30d2ef21e830d28ecfe40dcb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:10.134038Z","signature_b64":"qWMBoLxbY1BjvaJ1AsNNhnhAMZ3/HdFvjRrH5OWm5seNl4hHwzSZnRbav8HLelLyR7Xja/vzwBaWW6L85wdnCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0f827e31dad93df849a4f92dabc9c35f10ccd2b80fd4942ade7c4e93e09cb52","last_reissued_at":"2026-05-18T04:34:10.133388Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:10.133388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The first Steklov eigenvalue, conformal geometry, and minimal surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Ailana Fraser, Richard Schoen","submitted_at":"2009-12-29T23:58:21Z","abstract_excerpt":"We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue sigma_1 of the Dirichlet-to-Neumann map (Steklov eigenvalue). 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