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Some Problems in Four-dimensional Co...

Zhang, Siyi.

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Some Problems in Four-dimensional Conformal Geometry.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Some Problems in Four-dimensional Conformal Geometry./
作者:	Zhang, Siyi.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2019,
面頁冊數:	77 p.
附註:	Source: Dissertations Abstracts International, Volume: 81-02, Section: B.
Contained By:	Dissertations Abstracts International81-02B.
標題:	Mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=13882031
ISBN:	9781085638456

Some Problems in Four-dimensional Conformal Geometry.
Zhang, Siyi.

Some Problems in Four-dimensional Conformal Geometry. - Ann Arbor : ProQuest Dissertations & Theses, 2019 - 77 p.

Source: Dissertations Abstracts International, Volume: 81-02, Section: B.

Thesis (Ph.D.)--Princeton University, 2019.

This item must not be sold to any third party vendors.

In this thesis, we study some problems in four-dimensional conformal geometry. This thesis consists of two main parts: conformally invariant characterization of CP2 and conformally invariant gap theorems for Bach-flat metrics.In the first part, we extend the sphere theorem of [8] to give a conformally invariant characterization of (CP2, gFS). In particular, we introduce a conformal invariant β(M4,[g]) ≥ 0 defined on conformal four-manifolds satisfying a positivity condition; it follows from [8] that if 0 ≤ β(M4,[g]) 0 and 4 ≤ β(M4,[g]) 0 small enough, then M4 is diffeomorphic to CP2. The Ricci flow is used in a crucial way to pass from the bounds on β to pointwise curvature information. We also prove a lower bound for β(M) under some topological conditions.In the second part, we extend a conformal gap theorem for Bach-flat metrics with round sphere as model case established in [10] to prove conformally invariant gap theorems for Bach-flat 4-manifolds with (CP2, gFS) and (S2xS2,gp) as model cases. A Moser-type iteration argument plays an important role in the case of (CP2, gFS) and the convergence theory of Bach-flat metrics is of particular importance in the case of (S2xS2,gp).

ISBN: 9781085638456Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Analysis of PDEs

Some Problems in Four-dimensional Conformal Geometry.
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520 $a In this thesis, we study some problems in four-dimensional conformal geometry. This thesis consists of two main parts: conformally invariant characterization of CP2 and conformally invariant gap theorems for Bach-flat metrics.In the first part, we extend the sphere theorem of [8] to give a conformally invariant characterization of (CP2, gFS). In particular, we introduce a conformal invariant β(M4,[g]) ≥ 0 defined on conformal four-manifolds satisfying a positivity condition; it follows from [8] that if 0 ≤ β(M4,[g]) 0 and 4 ≤ β(M4,[g]) 0 small enough, then M4 is diffeomorphic to CP2. The Ricci flow is used in a crucial way to pass from the bounds on β to pointwise curvature information. We also prove a lower bound for β(M) under some topological conditions.In the second part, we extend a conformal gap theorem for Bach-flat metrics with round sphere as model case established in [10] to prove conformally invariant gap theorems for Bach-flat 4-manifolds with (CP2, gFS) and (S2xS2,gp) as model cases. A Moser-type iteration argument plays an important role in the case of (CP2, gFS) and the convergence theory of Bach-flat metrics is of particular importance in the case of (S2xS2,gp).
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