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The Hyperbolic Positive Mass Theorem...

Zhang, Yiyue.

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The Hyperbolic Positive Mass Theorem and Volume Comparison Involving Scalar Curvature.

Record Type:	Electronic resources : Monograph/item
Title/Author:	The Hyperbolic Positive Mass Theorem and Volume Comparison Involving Scalar Curvature./
Author:	Zhang, Yiyue.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2021,
Description:	169 p.
Notes:	Source: Dissertations Abstracts International, Volume: 82-12, Section: B.
Contained By:	Dissertations Abstracts International82-12B.
Subject:	Mathematics. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28321619
ISBN:	9798738637476

The Hyperbolic Positive Mass Theorem and Volume Comparison Involving Scalar Curvature.
Zhang, Yiyue.

The Hyperbolic Positive Mass Theorem and Volume Comparison Involving Scalar Curvature. - Ann Arbor : ProQuest Dissertations & Theses, 2021 - 169 p.

Source: Dissertations Abstracts International, Volume: 82-12, Section: B.

Thesis (Ph.D.)--Duke University, 2021.

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

In this thesis, we study how the scalar curvature relates to the ADM mass and volume of the manifold.The positive mass theorem, first proven by Schoen and Yau in 1979, states that nonnegative local energy density implies nonnegative total mass. The harmonic level set technique pioneered by D. Stern [Ste19] has been used to prove a series of positive mass theorems, such as the Riemannian case [BKKS19], the spacetime case [HKK20] and the charged case. Using this novel technique, we prove the hyperbolic positive mass theorem in the spacetime setting, as well as some rigidity cases. A new interpretation of mass is introduced in this context. Then we solve the spacetime harmonic equation in the hyperbolic setting. We not only prove the positive mass theorem, but we also give a lower bound for the total mass without assuming the nonnegativity of the local energy density.Additionally, We prove a scalar curvature volume comparison theorem, assuming some boundedness for Ricci curvature. The proof relies on the perturbation of the scalar curvature [BM11].

ISBN: 9798738637476Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Hyperbolic

The Hyperbolic Positive Mass Theorem and Volume Comparison Involving Scalar Curvature.
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300 $a 169 p.
500 $a Source: Dissertations Abstracts International, Volume: 82-12, Section: B.
500 $a Advisor: Bray, Hubert.
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520 $a In this thesis, we study how the scalar curvature relates to the ADM mass and volume of the manifold.The positive mass theorem, first proven by Schoen and Yau in 1979, states that nonnegative local energy density implies nonnegative total mass. The harmonic level set technique pioneered by D. Stern [Ste19] has been used to prove a series of positive mass theorems, such as the Riemannian case [BKKS19], the spacetime case [HKK20] and the charged case. Using this novel technique, we prove the hyperbolic positive mass theorem in the spacetime setting, as well as some rigidity cases. A new interpretation of mass is introduced in this context. Then we solve the spacetime harmonic equation in the hyperbolic setting. We not only prove the positive mass theorem, but we also give a lower bound for the total mass without assuming the nonnegativity of the local energy density.Additionally, We prove a scalar curvature volume comparison theorem, assuming some boundedness for Ricci curvature. The proof relies on the perturbation of the scalar curvature [BM11].
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650 4 $a Theoretical mathematics. $3 3173530
653 $a Hyperbolic
653 $a Positive mass theorem
653 $a Scalar curvature
653 $a Arnowitt-Deser-Misner mass
653 $a Perturbation
653 $a Ricci curvature
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710 2 $a Duke University. $b Mathematics. $3 1036449
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