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Curve Shortening Flow and Smooth Pro...

Hsu, Yu-Wen.

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Curve Shortening Flow and Smooth Projective Planes.

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Curve Shortening Flow and Smooth Projective Planes./
Author:	Hsu, Yu-Wen.
Description:	57 p.
Notes:	Source: Dissertation Abstracts International, Volume: 74-11(E), Section: B.
Contained By:	Dissertation Abstracts International74-11B(E).
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3571977
ISBN:	9781303297755

Curve Shortening Flow and Smooth Projective Planes.
Hsu, Yu-Wen.

Curve Shortening Flow and Smooth Projective Planes. - 57 p.

Source: Dissertation Abstracts International, Volume: 74-11(E), Section: B.

Thesis (Ph.D.)--Yale University, 2013.

In this dissertation, we study Gamma a family of curves on S 2 that defines a two-dimensional smooth projective plane. We use curve shortening flow (CSF) to prove that any two-dimensional smooth projective plane can be smoothly deformed through a family of smooth projective planes into one which is isomorphic to the real projective plane. In addition, as a consequence of our main result, we show that any two smooth embedded curves on RP 2 which intersect transversally at exactly one point converge to two different geodesics under the flow.

ISBN: 9781303297755Subjects--Topical Terms:

515831
Mathematics.

Curve Shortening Flow and Smooth Projective Planes.
LDR:02260nam a2200301 4500 001 1964735
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020 $a 9781303297755
035 $a (MiAaPQ)AAI3571977
035 $a AAI3571977
040 $a MiAaPQ $c MiAaPQ
100 1 $a Hsu, Yu-Wen. $3 2101236
245 1 0 $a Curve Shortening Flow and Smooth Projective Planes.
300 $a 57 p.
500 $a Source: Dissertation Abstracts International, Volume: 74-11(E), Section: B.
500 $a Adviser: Yair Minsky; Bruce Kleiner.
502 $a Thesis (Ph.D.)--Yale University, 2013.
520 $a In this dissertation, we study Gamma a family of curves on S 2 that defines a two-dimensional smooth projective plane. We use curve shortening flow (CSF) to prove that any two-dimensional smooth projective plane can be smoothly deformed through a family of smooth projective planes into one which is isomorphic to the real projective plane. In addition, as a consequence of our main result, we show that any two smooth embedded curves on RP 2 which intersect transversally at exactly one point converge to two different geodesics under the flow.
520 $a We first prove a regularity result: Gamma converges in the C infinity topology as t → infinity under CSF. We prove that if a smooth curve on S 2 is area-bisecting then under the flow its curvature function converges to zero uniformly and exponentially in the Cinfinity norm. This turns out to be a crucial estimate in proving the regularity result which leads to the proposition: CSF gives rise to a smooth homotopy when applied to an arbitrary two-dimensional smooth projective plane.
520 $a Then we show that the smooth projective structure is preserved under the smooth homotopy at any time. We study the solution to the linearized curve shortening equation and prove that it cannot vanish as t → infinity. This leads to our main result that the smooth homotopy is in fact a smooth homotopy of smooth projective planes.
590 $a School code: 0265.
650 4 $a Mathematics. $3 515831
650 4 $a Applied Mathematics. $3 1669109
690 $a 0405
690 $a 0364
710 2 $a Yale University. $b Mathematics. $3 2101237
773 0 $t Dissertation Abstracts International $g 74-11B(E).
790 $a 0265
791 $a Ph.D.
792 $a 2013
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3571977