東華大學圖書館 |

語系: 繁體中文

說明(常見問題)

回圖書館首頁

手機版館藏查詢

登入

回首頁

切換: 標籤 | MARC模式 | ISBD

FindBook

Google Book

Amazon

博客來

Two Problems in Geometric Analysis.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Two Problems in Geometric Analysis./
作者:	Valentin De Jesus, Pedro.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2021,
面頁冊數:	97 p.
附註:	Source: Dissertations Abstracts International, Volume: 83-04, Section: B.
Contained By:	Dissertations Abstracts International83-04B.
標題:	Mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28645763
ISBN:	9798544291794

Two Problems in Geometric Analysis.
Valentin De Jesus, Pedro.

Two Problems in Geometric Analysis. - Ann Arbor : ProQuest Dissertations & Theses, 2021 - 97 p.

Source: Dissertations Abstracts International, Volume: 83-04, Section: B.

Thesis (Ph.D.)--The University of Iowa, 2021.

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

In this thesis, we prove two results concerning the global geometry of manifolds. The first part of this thesis studies asymptotically flat spaces (AF), which play a vital role in mathematical gravitational theory. For graph AF manifolds, we establish the Riemannian Positive Mass Theorem (RPMT) and the Riemannian Penrose Inequality (RPI) with respect to the Gauss-Bonnet-Chern (GBC) masses. Due to the coordinate-invariance required for any relativistic gravity model, the notion of global energy or mass of a gravitating system at a given time cannot be defined as the space integral of the energy density. The global mass is instead defined as an asymptotic boundary integral, making its positivity in theory far from obvious even with a positive energy density. The RPI is robust evidence for the Weak Cosmic Censorship Hypothesis and, hence, the causal structure in any relativistic model of gravity. In the second part of this thesis, we study a sharp form of the Isoperimetric inequality. Given a fixed quantitative distortion of the standard round sphere, we obtain a lower bound for the indeterminate optimal dimensional constant in the Bonnesen-type inequality in Fuglede's work for convex domains of dimension larger than three. To obtain precisely the lower bound, we use a variational approach to obtain the domains minimizing the isoperimetric quotient within a suitable family of domains. These Bonnesen-type inequalities allow us to use the isoperimetric quotient to bound the Gromov-Hausdorff distance between a domain's boundary and the round sphere of the same dimension. The three-dimensional case remains open.

ISBN: 9798544291794Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Geometric analysis

Two Problems in Geometric Analysis.
LDR:02701nmm a2200349 4500 001 2345160
005 20220531132453.5
008 241004s2021 ||||||||||||||||| ||eng d
020 $a 9798544291794
035 $a (MiAaPQ)AAI28645763
035 $a AAI28645763
040 $a MiAaPQ $c MiAaPQ
100 1 $a Valentin De Jesus, Pedro. $3 3684050
245 1 0 $a Two Problems in Geometric Analysis.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2021
300 $a 97 p.
500 $a Source: Dissertations Abstracts International, Volume: 83-04, Section: B.
500 $a Advisor: Fang, Hao.
502 $a Thesis (Ph.D.)--The University of Iowa, 2021.
506 $a This item must not be sold to any third party vendors.
520 $a In this thesis, we prove two results concerning the global geometry of manifolds. The first part of this thesis studies asymptotically flat spaces (AF), which play a vital role in mathematical gravitational theory. For graph AF manifolds, we establish the Riemannian Positive Mass Theorem (RPMT) and the Riemannian Penrose Inequality (RPI) with respect to the Gauss-Bonnet-Chern (GBC) masses. Due to the coordinate-invariance required for any relativistic gravity model, the notion of global energy or mass of a gravitating system at a given time cannot be defined as the space integral of the energy density. The global mass is instead defined as an asymptotic boundary integral, making its positivity in theory far from obvious even with a positive energy density. The RPI is robust evidence for the Weak Cosmic Censorship Hypothesis and, hence, the causal structure in any relativistic model of gravity. In the second part of this thesis, we study a sharp form of the Isoperimetric inequality. Given a fixed quantitative distortion of the standard round sphere, we obtain a lower bound for the indeterminate optimal dimensional constant in the Bonnesen-type inequality in Fuglede's work for convex domains of dimension larger than three. To obtain precisely the lower bound, we use a variational approach to obtain the domains minimizing the isoperimetric quotient within a suitable family of domains. These Bonnesen-type inequalities allow us to use the isoperimetric quotient to bound the Gromov-Hausdorff distance between a domain's boundary and the round sphere of the same dimension. The three-dimensional case remains open.
590 $a School code: 0096.
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Applied mathematics. $3 2122814
653 $a Geometric analysis
653 $a Asymptotically flat spaces
653 $a Isoperimetric inequality
690 $a 0405
690 $a 0642
690 $a 0364
710 2 $a The University of Iowa. $b Mathematics. $3 1683492
773 0 $t Dissertations Abstracts International $g 83-04B.
790 $a 0096
791 $a Ph.D.
792 $a 2021
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28645763