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Lectures on Mean Curvature Flow s
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AMS/IP
Studies in Advanced Mathematics Volume 32
Lectures on Mean Curvature Flows
Xi-Ping Zhu
American Mathematical Society • International Press
https://doi.org/10.1090/amsip/032
Shing-Tung Yau , Genera l Edi to r
2000 Mathematics Subject Classification. P r imar y 53C44 ; Secondar y 35K55 , 52A20 , 53C20, 53C21 , 58J35 .
Library o f Congres s Cataloging-in-Publicatio n Dat a
Zhu, Xi-Ping . Lectures o n mea n curvatur e flows / Xi-Pin g Zhu .
p. cm . — (AMS/I P studie s i n advance d mathematics , ISS N 1089-328 8 ; v. 32 ) Includes bibliographica l reference s an d index . ISBN 0-8218-3311- 1 (alk . paper ) 1. Surface s o f constan t curvature . 2 . Flow s (Differentiabl e dynamica l systems ) I . Title .
II. Series .
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Contents 1 Th e curv e shortenin g flow fo r conve x curve s 1
1.1 Shrinkin g t o a Poin t 2 1.2 Asymptoti c Behavio r 6
2 Th e Shor t Tim e Existenc e an d Th e Evolutio n Equatio n o f Cur -vatures 1 5 2.1 Loca l Existenc e 1 7 2.2 Evolutio n o f Metri c an d Curvatur e 1 8 2.3 Pinchin g Estimat e 2 0
3 Contractio n o f Conve x Hypersurface s 2 5 3.1 Som e Fact s o n Conve x Hypersurfac e 2 5 3.2 MC F FO R CONVE X HYPERSURFACE S 2 9
4 Monotonicit y an d Self-Simila r Solution s 3 5 4.1 Typ e I Limits 3 5 4.2 Th e Classificatio n o f Self-simila r Solution s 3 9
5 Evolutio n o f Embedde d Curve s o r Surface s (I ) 4 7 5.1 Isoperimetri c Estimate s 4 8 5.2 Blow-u p Argumen t 5 0 5.3 Convexit y Theore m 5 2
6 Evolutio n o f Embedde d Curve s an d Surface s (II ) 5 5 6.1 Curve s wit h Finit e Tota l Absolut e Curvatur e 5 5 6.2 Lon g Tim e Existenc e fo r Complet e Curve s 6 0
7 Evolutio n o f Embedde d Curve s an d Surface s (III ) 6 7 7.1 Th e Evolutio n Equatio n o f Gradien t Functio n 6 8 7.2 Gradien t Estimate s 6 9 7.3 Curvatur e Estimate s 7 1 7.4 Lon g Tim e Existenc e fo r Entir e Graph s 7 5
8 Convexit y Estimate s fo r Mea n Conve x Surface s 7 7 8.1 Evolutio n Equation s 7 8 8.2 L p Estimate s 8 0 8.3 D e Giorg i Iteratio n Argumen t 8 4
v
vi CONTENTS
9 Li-Ya u Estimate s an d Typ e I I Singularitie s 8 9 9.1 Translatin g Solito n 9 0 9.2 Li-Ya u Typ e Inequalit y 9 1 9.3 Typ e I I Limit s 9 7
10 Th e Mea n Curvatur e Flo w i n Riemannia n Manifold s 10 1 10.1 Hypersurface s i n Riemannia n Manifold s 10 1 10.2 Evolutio n Equation s 10 4
11 Contractin g Conve x Hypersurface s i n Riemannia n Manifold s 10 9 11.1 Th e Pinchin g Estimate s 10 9 11.2 A Geometri c Lemm a 11 2 11.3 Huiske n Theore m 11 5
12 Definitio n o f Cente r o f Mas s fo r Isolate d Gravitatin g System s 12 3 12.1 Approximatel y Roun d Surface s 12 4 12.2 Existenc e o f Constan t Mea n Curvatur e Surface s 12 8 12.3 Cente r o f Gravit y 13 7
References 14 5
Index 149
Preface In thes e note s w e discus s i n som e detai l th e evolutio n o f a hypersurfac e whos e normal velocit y i s give n b y th e mea n curvature . I t wa s pose d a s eithe r a phe -nomenological mode l fo r theorie s o f sharp-interface s i n continuu m mechanic s and image processing, o r as the asymptotic behavio r o f certain systems in chem-ical reaction and mathematical biology . W e will concentrate on singularities an d asymptotic behavio r o f motions .
Let M n b e a n n-dimensiona l close d manifold . A hypersurfac e o f R n + 1 i s a map X : Mn — > R n + 1 . W e ca n le t eac h poin t X(-) mov e i n th e inne r norma l direction n wit h velocit y t o b e th e mea n curvatur e H. Thi s give s th e mea n curvature flow
OX — =Hn, o n M x [ 0 , T ) .
Consider th e specia l case : n = 1 . W e usuall y cal l one-dimensiona l mea n curvature flo w a s the curv e shortenin g flow. Writ e th e positio n vecto r a s
X =
By Prene t formula ,
Hn = kn =
ds2
\ ds 2 )
where s is arclength parameter. S o the curve shortening flow becomes the syste m
dx
dy_
dt
d2x
d2s
cPy
d2s
which say s tha t th e coordinat e function s evolv e b y hea t equatio n i n th e intrinsi c geometry o f th e curve , whic h i s itsel f changing .
In general , th e mea n curvatur e flo w ca n b e writ te n a s
ax dt
= &g(t)X
Vll
V l l l PREFACE
where g(t) i s th e induce d metri c o f th e evolvin g hypersurfac e X(-,t) : M —» R n + 1 .
This i s a quasilinea r paraboli c equatio n becaus e th e Laplacia n operato r i s taken i n th e induce d metric . Th e mea n curvatur e H o f X(- , t) satisfie s
— = A g(t)H + \A\ 2H,
where A i s th e secon d fundamenta l for m o f th e hypersurfac e X(-,t). Thi s i s a nonlinea r hea t equatio n wit h superlinea r growth . I t i s clea r tha t th e mea n curvature flow mus t generall y blo w u p i n finite time . However , th e geometri c nature o f the flow enable s on e t o obtai n mor e precis e result s fo r th e lon g tim e behavior. I n Chapte r 1 , as the tes t case , we consider th e one-dimensiona l mea n curvature flow (i.e., the curve shortening flow) for conve x closed curves . W e will prove the Gage-Hamilton theorem which states that th e flow preserves convexity and shrink s t o a poin t i n finite time ; furthermore , i f we dilat e th e flow so tha t its enclose d are a i s alway s equa l t o 7r , the normalize d flow converge s t o a uni t circle. I n Chapte r 2 , we discuss basi c result s suc h a s loca l existence , evolution s of metric an d curvature , an d pinchin g estimates . Th e high-dimensiona l versio n of the abov e Gage-Hamilto n theorem , i.e. , Huisken' s theorem , fo r conve x close d hypersurfaces wil l be proven i n Chapte r 3 and Chapte r 4 . I n Chapte r 4 we also give th e Huisken' s classificatio n o f Typ e I singularitie s fo r th e mea n curvatur e flow. Fro m Chapte r 5 t o 7 , w e stud y th e mea n curvatur e flow fo r nonconve x embedded curve s o r surfaces . I n Chapte r 5 w e prov e th e Grayso n theorem . This is , th e curv e shortenin g flow startin g a t an y close d embedde d curv e be -comes conve x befor e i t develop s a singularity . Further , th e curv e shortenin g flow for complet e noncompac t embedde d curve s i s studie d i n Chapte r 6 . W e give a ver y genera l lon g tim e existenc e theore m o f Cho u an d th e autho r whic h states tha t i f the initia l curv e divids the plan e int o two domains o f infinite area , then th e solutio n exist s fo r al l times . I n Chapte r 7 , w e presen t th e lon g tim e existence o f Ecker an d Huiske n fo r entir e graphs . I n Chapte r 8 and 9 we stud y the formatio n o f Typ e I I singularitie s fo r th e evolutio n o f mea n conve x hyper -surfaces. W e presen t th e convexit y estimate s o f Huiske n an d Sinestrar i whic h implies tha t an y Typ e I I limi t mus t b e a conve x eterna l solutio n o f th e mea n curvature flow. B y combinin g th e matri x Li-Ya u inequalit y o f Hamilton , w e conclude tha t th e limitin g flow is a translatin g soliton . Fro m Chapte r 1 0 to 1 2 we study th e mea n curvatur e flow in Riemannia n manifolds . I n Chapte r 1 0 we give severa l preliminar y result s suc h a s loca l existenc e an d th e evolutio n equa -tions o f extrinsic curvatures . W e have know n i n the previou s chapter s tha t th e mean curvatur e flow contract s a conve x compac t hypersurfac e i n a n Euclidea n space smoothl y t o a singl e poin t i n finite tim e an d th e shap e o f th e hypersur -faces become s spherica l a t th e en d o f the contraction . Th e purpos e o f Chapte r 11 is to show that thi s contraction property i s still holding in the general case, if the initia l hypersurfac e i s convex enoug h t o overcom e th e obstruction s impose d by th e geometr y o f th e ambien t Riemannia n manifolds . Finall y i n Chapte r 1 2 as an applicatio n t o the theory o f the mea n curvatur e flow, we present th e wor k
PREFACE IX
of Huiske n an d Ya u o n th e definitio n o f cente r o f mas s fo r isolate d gravitatin g systems.
This monograp h i s a n outgrowt h o f my lecture s give n a t Th e Institut e o f Mathematical Science s o f Th e Chines e Universit y o f Hon g Kon g i n 2000 . I would lik e to than k Professo r S . T . Ya u an d Professo r Z . P . Xin invitin g m e t o deliver th e lecture s i n a ver y stimulatin g environment . I a m als o indebte d t o Professor K . S . Cho u an d my forme r studen t B . L . Che n fo r man y discussion s on geometri c flows ove r a perio d o f tim e o f a t leas t si x years . Thi s wor k wa s partially supporte d b y Th e IM S o f Th e Chines e Universit y o f Hon g Kon g an d the Foundatio n fo r Outstandin g Youn g Scholar s o f China .
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145
146 References
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Index Abresh-Langer curves , 14 , 39
Approximately roun d surfaces , 12 5
Asymptotic directions , 5 7
Asymptotically flat , 56 , 12 3
Blaschke Selectio n Theorem , 4
Center o f
gravity, 13 8
mass, 13 7
Codazzi equation , 10 2
Containment principle , 3 , 53, 104
Convex
curve, 1
hypersurface, 2 3
Curve shortenin g flow , vi i
Entropy, 1 0
Expanding self-simila r equation , 7 6
Gage-Hamilton Theorem , viii , 8
Gauss
equation, 15 , 102
map, 25 , 11 4
Gauss-Weingarten relations , 16 , 102
Gradient
function, 6 8
translating soliton , 9 1
Grayson Theorem , viii , 52
Huisken Theorem , viii , 44 , 11 5
Isoperimetric estimates , 48 , 58
Li-Yau inequality , 9 2
Local existence , 3 , 17 , 10 4
Locally conve x hypersurface , 10 9
Long time existence, viii , 58, 60, 68, 75
Maximal solution , 3 5
Mean conve x surfaces , 7 8
Mean curvatur e flow, vi i
Metric, 15 , 28, 78, 10 2
Monotonicity formula , 3 7
Normal angle , 2
Pinching estimate , 2 2
Positive mas s theorem , 12 3
Principal
curvature, 2 6
radii, 2 6
Quasi-conformal, 11 4
Radius
inner, 2 7
outer, 2 7
Schwarzschild metric , 12 4
Second fundamenta l form , 15 , 29 , 77 , 102
Self-similar
solution, 3 9
shrinking tori , 4 4
Simons identities , 10 2
Singularity, 3 6
Spacelike timeslice , 12 3
Stable, 13 8
Strictly stable , 13 8
Strong uniqueness , 6 5
Sturmian compariso n (o r oscillation ) theorem, 6 2
Support function , 2 , 2 5
Total
absolute curvature , 50 , 55
mass (o r ADM-mass) , 12 3
Translating soliton , 9 0
Traceless secon d fundamenta l form, 12 5
Type
I, 36 , 11 7
149
150 Index
II, 36 , 11 7
Width, 11 , 27