{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/19419"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/19419","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Proper holomorphic mappings of positive codimension in several complex variables","abstract":"A holomorphic mapping f from a bounded domain $\\Omega$ in C$\\sp{\\rm n}$ to a bounded domain $\\Omega\\sp\\prime$ in C$\\sp{\\rm N}$ is proper if and only if (f(z$\\sb\\nu$)) tends to the boundary b$\\Omega\\sp\\prime$ for each sequence (z$\\sb\\nu$) that tends to b$\\Omega$. If the domains are balls B$\\sb{\\rm n}$ and B$\\sb{\\rm N}$, Forstneric has proved that if f is sufficiently smooth up to the sphere bB$\\sb{\\rm n}$, then it must be rational, and Cima and Suffridge have shown that it then extends to be holomorphic past bB$\\sb{\\rm n}$. We prove the more general result that if (i) $\\Omega$ lies on one side of a real analytic real hypersurface M in C$\\sp{\\rm n}$, (ii) F maps $\\Omega$ holomorphically into the ball B$\\sb{\\rm N}$, (iii) in some neighborhood of a point p of M, F is the quotient of a holomorphic mapping by a holomorphic function, and (iv) if for each point q of M sufficiently near p, (F(z$\\sb\\nu$)) tends to bB$\\sb{\\rm N}$ as (z$\\sb\\nu$) tends to q within $\\Omega$, then F extends to be holomorphic past M at p. We prove this extension result also for certain other target domains, e.g., generalized ellipsoids $\\rm\\{\\sum\\sb{j}\\ \\vert w\\sb{j}\\vert\\sp{2m\\sb j}< 1\\}$ in $\\rm C\\sp{N}.$","abstract_html":"A holomorphic mapping f from a bounded domain $\\Omega$ in C$\\sp{\\rm n}$ to a bounded domain $\\Omega\\sp\\prime$ in C$\\sp{\\rm N}$ is proper if and only if (f(z$\\sb\\nu$)) tends to the boundary b$\\Omega\\sp\\prime$ for each sequence (z$\\sb\\nu$) that tends to b$\\Omega$. If the domains are balls B$\\sb{\\rm n}$ and B$\\sb{\\rm N}$, Forstneric has proved that if f is sufficiently smooth up to the sphere bB$\\sb{\\rm n}$, then it must be rational, and Cima and Suffridge have shown that it then extends to be holomorphic past bB$\\sb{\\rm n}$. We prove the more general result that if (i) $\\Omega$ lies on one side of a real analytic real hypersurface M in C$\\sp{\\rm n}$, (ii) F maps $\\Omega$ holomorphically into the ball B$\\sb{\\rm N}$, (iii) in some neighborhood of a point p of M, F is the quotient of a holomorphic mapping by a holomorphic function, and (iv) if for each point q of M sufficiently near p, (F(z$\\sb\\nu$)) tends to bB$\\sb{\\rm N}$ as (z$\\sb\\nu$) tends to q within $\\Omega$, then F extends to be holomorphic past M at p. We prove this extension result also for certain other target domains, e.g., generalized ellipsoids <span class=\"etd-inline-math\">\\rm\\{\\sum\\sb{j} \\vert w\\sb{j}\\vert\\sp{2m\\sb j}&lt; 1\\}</span> in $\\rm C\\sp{N}.$","abstract_has_math":true,"creators":["Chiappari, Stephen Anthony"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Miles, Joseph B."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:06:57Z","date_published":"2011-05-07T12:06:57Z","updated_at":"2026-07-22T22:25:12Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1990 Chiappari, Stephen Anthony"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114202","(UMI)AAI9114202"],"render_values":[{"text":"AAI9114202","href":null,"code":true},{"text":"(UMI)AAI9114202","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/19419","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Miles, Joseph B."]},{"key":"dc:creator","label":"Author","values":["Chiappari, Stephen Anthony"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:06:57Z","10000-01-01","1990"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1990 Chiappari, Stephen Anthony"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114202","(UMI)AAI9114202","http://hdl.handle.net/2142/19419"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A holomorphic mapping f from a bounded domain $\\Omega$ in C$\\sp{\\rm n}$ to a bounded domain $\\Omega\\sp\\prime$ in C$\\sp{\\rm N}$ is proper if and only if (f(z$\\sb\\nu$)) tends to the boundary b$\\Omega\\sp\\prime$ for each sequence (z$\\sb\\nu$) that tends to b$\\Omega$. If the domains are balls B$\\sb{\\rm n}$ and B$\\sb{\\rm N}$, Forstneric has proved that if f is sufficiently smooth up to the sphere bB$\\sb{\\rm n}$, then it must be rational, and Cima and Suffridge have shown that it then extends to be holomorphic past bB$\\sb{\\rm n}$. We prove the more general result that if (i) $\\Omega$ lies on one side of a real analytic real hypersurface M in C$\\sp{\\rm n}$, (ii) F maps $\\Omega$ holomorphically into the ball B$\\sb{\\rm N}$, (iii) in some neighborhood of a point p of M, F is the quotient of a holomorphic mapping by a holomorphic function, and (iv) if for each point q of M sufficiently near p, (F(z$\\sb\\nu$)) tends to bB$\\sb{\\rm N}$ as (z$\\sb\\nu$) tends to q within $\\Omega$, then F extends to be holomorphic past M at p. We prove this extension result also for certain other target domains, e.g., generalized ellipsoids $\\rm\\{\\sum\\sb{j}\\ \\vert w\\sb{j}\\vert\\sp{2m\\sb j}< 1\\}$ in $\\rm C\\sp{N}.$","We also investigate some properties of a certain variety associated to a proper holomorphic mapping and compute it for several mappings that are of importance to the study of proper mappings invariant under fixed point free finite unitary groups and to the classification of polynomial proper mappings between balls.","Made available in DSpace on 2011-05-07T12:06:57Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114202.pdf: 2400429 bytes, checksum: 123a4bd097d95783af20a065fe200698 (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:36:50Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:14:59-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Proper holomorphic mappings of positive codimension in several complex variables"]}]}],"canonical_facts":{"dc:contributor":["Miles, Joseph B."],"dc:creator":["Chiappari, Stephen Anthony"],"dc:date":["2011-05-07T12:06:57Z","10000-01-01","1990"],"dc:description":["A holomorphic mapping f from a bounded domain $\\Omega$ in C$\\sp{\\rm n}$ to a bounded domain $\\Omega\\sp\\prime$ in C$\\sp{\\rm N}$ is proper if and only if (f(z$\\sb\\nu$)) tends to the boundary b$\\Omega\\sp\\prime$ for each sequence (z$\\sb\\nu$) that tends to b$\\Omega$. If the domains are balls B$\\sb{\\rm n}$ and B$\\sb{\\rm N}$, Forstneric has proved that if f is sufficiently smooth up to the sphere bB$\\sb{\\rm n}$, then it must be rational, and Cima and Suffridge have shown that it then extends to be holomorphic past bB$\\sb{\\rm n}$. We prove the more general result that if (i) $\\Omega$ lies on one side of a real analytic real hypersurface M in C$\\sp{\\rm n}$, (ii) F maps $\\Omega$ holomorphically into the ball B$\\sb{\\rm N}$, (iii) in some neighborhood of a point p of M, F is the quotient of a holomorphic mapping by a holomorphic function, and (iv) if for each point q of M sufficiently near p, (F(z$\\sb\\nu$)) tends to bB$\\sb{\\rm N}$ as (z$\\sb\\nu$) tends to q within $\\Omega$, then F extends to be holomorphic past M at p. We prove this extension result also for certain other target domains, e.g., generalized ellipsoids $\\rm\\{\\sum\\sb{j}\\ \\vert w\\sb{j}\\vert\\sp{2m\\sb j}< 1\\}$ in $\\rm C\\sp{N}.$","We also investigate some properties of a certain variety associated to a proper holomorphic mapping and compute it for several mappings that are of importance to the study of proper mappings invariant under fixed point free finite unitary groups and to the classification of polynomial proper mappings between balls.","Made available in DSpace on 2011-05-07T12:06:57Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114202.pdf: 2400429 bytes, checksum: 123a4bd097d95783af20a065fe200698 (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:36:50Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:14:59-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9114202","(UMI)AAI9114202","http://hdl.handle.net/2142/19419"],"dc:language":["eng"],"dc:rights":["Copyright 1990 Chiappari, Stephen Anthony"],"dc:subject":["Mathematics"],"dc:title":["Proper holomorphic mappings of positive codimension in several complex variables"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:12Z"}