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Problems on the geometric function theory in several complex variables and complex geometry
Descriptive
TypeOfResource
Text
TitleInfo
(ID = T-1)
Title
Problems on the geometric function theory in several complex variables and complex geometry
Identifier
ETD_2788
Identifier
(type = hdl)
http://hdl.rutgers.edu/1782.1/rucore10001600001.ETD.000056872
Language
LanguageTerm
(authority = ISO639-2);
(type = code)
eng
Genre
(authority = marcgt)
theses
Subject
(ID = SBJ-1);
(authority = RUETD)
Topic
Mathematics
Subject
(ID = SBJ-2);
(authority = ETD-LCSH)
Topic
Isometrics (Mathematics)
Subject
(ID = SBJ-3);
(authority = ETD-LCSH)
Topic
Geodesy
Subject
(ID = SBJ-4);
(authority = ETD-LCSH)
Topic
Geometric function theory
Abstract
(type = abstract)
The thesis consists of two parts. In the first part, we study the rigidity for the local holomorphic isometric embeddings. On the one hand, we prove the total geodesy for the local holomorphic conformal embedding from the unit ball of complex dimension at least 2 to the product of unit balls and hence the rigidity for the local holomorphic isometry is the natural corollary. Before obtaining the total geodesy, the algebraic extension theorem is derived following the idea in cite{MN} by considering the sphere bundle of the source and target domains. When conformal factors are not constant, we twist the sphere bundle to gain the pseudoconvexity. Then the algebraicity follows from the algebraicity theorem of Huang in the CR geometry. Different from the argument in the earlier works, the total geodesy of each factor does not directly follow from the properness because the codimension is arbitrary. By analyzing the real analytic subvariety carefully, we conclude that the factor is either a proper holomorphic rational map or a constant map. Lastly the total geodesy follows from a linearity criterion of Huang. On the other hand, we also derive the total geodesy for the local holomorphic isometries from the projective space to the product of projective spaces. In the second part, we give a proof for the convergence of a modified K"{a}hler-Ricci flow. The flow is defined by Zhang on K"{a}hler manifolds while the K"{a}hler class along the evolution is varying. When the limit cohomology class is semi-positive, big and integer, the convergence of the flow is conjectured by Zhang and we confirm it by using the monotonicity of some energy functional. When the limit class is K"{a}hler, the convergence is proven by Zhang and we give an alternative proof by also using the energy functional. As a corollary, the convergence provides the solution to the degenerate Monge-Amp`{e}re equation on the Calabi-Yau manifold. Meanwhile we take the opportunity to describe the K"{a}hler-Ricci flow on singular varieties.
PhysicalDescription
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electronic resource
Extent
vii, 49 p.
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application/pdf
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text/xml
Note
(type = degree)
Ph.D.
Note
(type = bibliography)
Includes bibliographical references
Note
(type = vita)
Includes vita
Note
(type = statement of responsibility)
by Yuan Yuan
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Yuan
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Yuan
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author
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Yuan Yuan
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Huang
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Xiaojun
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chair
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Advisory Committee
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Xiaojun Huang
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NamePart
(type = family)
Song
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(type = given)
Jian
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internal member
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Advisory Committee
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Jian Song
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(type = personal)
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(type = family)
Jacobowitz
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(type = given)
Howard
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internal member
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Advisory Committee
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Howard Jacobowitz
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NamePart
(type = family)
Berhanu
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(type = given)
Shiferaw
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internal member
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Advisory Committee
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Shiferaw Berhanu
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NamePart
Rutgers University
Role
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degree grantor
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Graduate School - New Brunswick
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school
OriginInfo
DateCreated
(qualifier = exact)
2010
DateOther
(qualifier = exact);
(type = degree)
2010-10
Place
PlaceTerm
(type = code)
xx
RelatedItem
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TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Identifier
(type = RULIB)
ETD
RelatedItem
(type = host)
TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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NjNbRU
Identifier
(type = doi)
doi:10.7282/T3DR2V7N
Genre
(authority = ExL-Esploro)
ETD doctoral
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Rights
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(ID = rulibRdec0006)
The author owns the copyright to this work.
Copyright
Status
Copyright protected
Availability
Status
Open
Reason
Permission or license
RightsHolder
(ID = PRH-1);
(type = personal)
Name
FamilyName
Yuan
GivenName
Yuan
Role
Copyright Holder
RightsEvent
(ID = RE-1);
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Type
Permission or license
DateTime
2010-07-24 21:18:23
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Name
Yuan Yuan
Affiliation
Rutgers University. Graduate School - New Brunswick
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Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
RightsEvent
(ID = RE-2);
(AUTHORITY = rulib)
Type
Embargo
DateTime
2010-10-31
Detail
Access to this PDF has been restricted at the author's request. It will be publicly available after May 2nd, 2011.
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ETD
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application/pdf
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application/x-tar
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317440
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