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oai:www.ideals.illinois.edu:2142/19419 2023-07-10 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Miles, Joseph B. Chiappari, Stephen Anthony 2011-05-07T12:06:57Z 2011-05-07T12:06:57Z 10000-01-01 1990 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 AAI9114202 (UMI)AAI9114202 http://hdl.handle.net/2142/19419 eng Copyright 1990 Chiappari, Stephen Anthony Mathematics Proper holomorphic mappings of positive codimension in several complex variables text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.