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The Inverse Mean Curvature Flow : = Singularities, Dynamical Stability, and Applications to Minimal Surfaces.

Record Type:	Electronic resources : Monograph/item
Title/Author:	The Inverse Mean Curvature Flow :/
Reminder of title:	Singularities, Dynamical Stability, and Applications to Minimal Surfaces.
Author:	Harvie, Brian Donald.
Description:	1 online resource (122 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 83-02, Section: A.
Contained By:	Dissertations Abstracts International83-02A.
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28541324click for full text (PQDT)
ISBN:	9798538101160

The Inverse Mean Curvature Flow : = Singularities, Dynamical Stability, and Applications to Minimal Surfaces.
Harvie, Brian Donald.

The Inverse Mean Curvature Flow :Singularities, Dynamical Stability, and Applications to Minimal Surfaces. - 1 online resource (122 pages)

Source: Dissertations Abstracts International, Volume: 83-02, Section: A.

Thesis (Ph.D.)--University of California, Davis, 2021.

Includes bibliographical references

This dissertation concerns the Inverse Mean Curvature Flow of closed hypersurfaces in Euclidean Space, and its relationship with minimal surfaces. Inverse Mean Curvature Flow is an extrinsic geometric flow which has become prominent in differential geometry because of its applications to geometric inequalities and general relativity, but deep questions persist about its analytic and geometric structure. The first four chapters of this dissertation focus on singularity formation in the flow, the flow behavior near singularities, and the dynamical stability of round spheres under mean-convex perturbations.On the topic of singularities, I establish the formation of a singularity for all embedded flow solutions which do not have spherical topology within a prescribed time interval. I later show that mean-convex, rotationally symmetric tori undergo a flow singularity wherein the flow surfaces converge to a limit surface without rescaling, contrasting sharply with the singularities of other extrinsic geometric flows. On the topic of long-time behavior, I show that all flow solutions whichexist and remain embedded for some minimal time depending only on initial data must exist for all time and asymptotically converge to round spheres at large times. In the fourth chapter, I utilize this characterization to establish dynamical stability of the round sphere under certain mean-convex, axially symmetric perturbations that are not necessarily star-shaped. In the last chapter, I relate questions of singularities and dynamical stability for the InverseMean Curvature Flow to the mathematics of soap films. Specifically, I show that certain families of solutions to Plateau's problem do not self-intersect and remain contained within a given region of Euclidean space. I accomplish this using a barrier method arising from global embedded solutions of Inverse Mean Curvature Flow. Conversely, I also use minimal disks to establish that a singularity likely forms in the flow of a specific mean-convex embedded sphere.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2023

Mode of access: World Wide Web

ISBN: 9798538101160Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Dynamical stabilityIndex Terms--Genre/Form:

542853
Electronic books.

The Inverse Mean Curvature Flow : = Singularities, Dynamical Stability, and Applications to Minimal Surfaces.
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504 $a Includes bibliographical references
520 $a This dissertation concerns the Inverse Mean Curvature Flow of closed hypersurfaces in Euclidean Space, and its relationship with minimal surfaces. Inverse Mean Curvature Flow is an extrinsic geometric flow which has become prominent in differential geometry because of its applications to geometric inequalities and general relativity, but deep questions persist about its analytic and geometric structure. The first four chapters of this dissertation focus on singularity formation in the flow, the flow behavior near singularities, and the dynamical stability of round spheres under mean-convex perturbations.On the topic of singularities, I establish the formation of a singularity for all embedded flow solutions which do not have spherical topology within a prescribed time interval. I later show that mean-convex, rotationally symmetric tori undergo a flow singularity wherein the flow surfaces converge to a limit surface without rescaling, contrasting sharply with the singularities of other extrinsic geometric flows. On the topic of long-time behavior, I show that all flow solutions whichexist and remain embedded for some minimal time depending only on initial data must exist for all time and asymptotically converge to round spheres at large times. In the fourth chapter, I utilize this characterization to establish dynamical stability of the round sphere under certain mean-convex, axially symmetric perturbations that are not necessarily star-shaped. In the last chapter, I relate questions of singularities and dynamical stability for the InverseMean Curvature Flow to the mathematics of soap films. Specifically, I show that certain families of solutions to Plateau's problem do not self-intersect and remain contained within a given region of Euclidean space. I accomplish this using a barrier method arising from global embedded solutions of Inverse Mean Curvature Flow. Conversely, I also use minimal disks to establish that a singularity likely forms in the flow of a specific mean-convex embedded sphere.
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650 4 $a Applied mathematics. $3 2122814
650 4 $a Partial differential equations. $3 2180177
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650 4 $a Spheres. $3 3684734
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650 4 $a Geometry. $3 517251
650 4 $a Symmetry. $3 536815
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653 $a Extrinsic curvature
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653 $a Minimal surfaces
653 $a Partial differential equations
653 $a Singularities
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