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Some parabolic and elliptic problems in complex Riemannian geometry

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Title
Some parabolic and elliptic problems in complex Riemannian geometry
Name (type = personal)
NamePart (type = family)
Guo
NamePart (type = given)
Bin
NamePart (type = date)
1985-
DisplayForm
Bin Guo
Role
RoleTerm (authority = RULIB)
author
Name (type = personal)
NamePart (type = family)
Song
NamePart (type = given)
Jian
DisplayForm
Jian Song
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
chair
Name (type = personal)
NamePart (type = family)
Han
NamePart (type = given)
Zheng-Chao
DisplayForm
Zheng-Chao Han
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
internal member
Name (type = personal)
NamePart (type = family)
Chanillo
NamePart (type = given)
Sagun
DisplayForm
Sagun Chanillo
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
internal member
Name (type = personal)
NamePart (type = family)
Wang
NamePart (type = given)
Xiaowei
DisplayForm
Xiaowei Wang
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
outside member
Name (type = corporate)
NamePart
Rutgers University
Role
RoleTerm (authority = RULIB)
degree grantor
Name (type = corporate)
NamePart
Graduate School - New Brunswick
Role
RoleTerm (authority = RULIB)
school
TypeOfResource
Text
Genre (authority = marcgt)
theses
OriginInfo
DateCreated (qualifier = exact)
2015
DateOther (qualifier = exact); (type = degree)
2015-05
CopyrightDate (encoding = w3cdtf); (qualifier = exact)
2015
Place
PlaceTerm (type = code)
xx
Language
LanguageTerm (authority = ISO639-2b); (type = code)
eng
Abstract (type = abstract)
This dissertation consists of three parts, the first one is on the blow-up behavior of K"ahler Ricci flow on $cp^n$ blown-up at one point, and the second one on the convergence of K"ahler Ricci flow on minimal projective manifolds of general type, and the last one is on the existence of canonical conical K"ahler metrics on toric manifolds. In the first part, we consider the Ricci flow on $cp^n$ blown-up at one point starting with any rotationally symmetric K"ahler metric. We show that if the total volume does not go to zero at the singular time, then any parabolic blow-up limit of the Ricci flow along the exceptional divisor is a non-compact complete shrinking K"ahler Ricci soliton with rotational symmetry on $mathbb C^n$ blown-up at one point, hence the FIK soliton constructed in cite{FIK}. In the second part, we consider the K"ahler Ricci flow on a smooth minimal model of general type, following the ideas of Song (cite{S1,S2}), we show that if the Ricci curvature is uniformly bounded below along the K"ahler-Ricci flow, then the diameter is uniformly bounded. As a corollary we show that under the Ricci curvature lower bound assumption, the Gromov-Hausdorff limit of the flow is homeomorphic to the canonical model of the manifold. Moreover, we will give a purely analytic proof of a recent result of Tosatti-Zhang (cite{TZ}) that if the canonical line bundle $K_X$ is big and nef, but not ample, then the Ricci flow is of Type IIb. In the last part, we give criterion for the existence of toric conical K"ahler-Einstein and K"ahler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci lower bound and Bakry-Emery-Ricci lower bound. It is shown that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with conical K"ahler-Einstein metrics in the Gromov-Hausdorff topology.
Subject (authority = RUETD)
Topic
Mathematics
Subject (authority = ETD-LCSH)
Topic
Riemann surfaces
Subject (authority = ETD-LCSH)
Topic
Geometry, Differential
Subject (authority = ETD-LCSH)
Topic
Kählerian structures
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
Identifier
ETD_6323
PhysicalDescription
Form (authority = gmd)
electronic resource
InternetMediaType
application/pdf
InternetMediaType
text/xml
Extent
1 online resource (viii, 129 p.)
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Note (type = statement of responsibility)
by Bin Guo
RelatedItem (type = host)
TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
PhysicalLocation (authority = marcorg); (displayLabel = Rutgers, The State University of New Jersey)
NjNbRU
Identifier (type = doi)
doi:10.7282/T31Z4676
Genre (authority = ExL-Esploro)
ETD doctoral
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The author owns the copyright to this work.
RightsHolder (type = personal)
Name
FamilyName
Guo
GivenName
Bin
Role
Copyright Holder
RightsEvent
Type
Permission or license
DateTime (encoding = w3cdtf); (qualifier = exact); (point = start)
2015-04-12 22:05:03
AssociatedEntity
Name
Bin Guo
Role
Copyright holder
Affiliation
Rutgers University. Graduate School - New Brunswick
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License
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Author Agreement License
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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.
Copyright
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Copyright protected
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Open
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Permission or license
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ContentModel
ETD
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windows xp
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