Lectures on Mean Curvature Flow s

This page intentionally left blank

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 .

QA645 .Z48 200 2 516.3'62—dc21 200202797 7

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

Republication, systemati c copying , o r multipl e reproductio n o f any materia l i n thi s publicatio n is permitted onl y unde r licens e fro m th e America n Mathematica l Societ y an d Internationa l Press . Requests fo r suc h permissio n shoul d b e addresse d t o th e Acquisition s Department , America n Mathematical Society , 20 1 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s can als o b e mad e b y e-mai l t o reprint-permissionOams.org .

© 200 2 b y th e America n Mathematica l Societ y an d Internationa l Press . Al l right s reserved . The America n Mathematica l Societ y an d Internationa l Pres s retai n al l right s

except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

Visit th e AM S hom e pag e a t ht tp:/ /www.ams.org / Visit th e Internationa l Pres s hom e pag e a t URL : h t tp : / /www. in t lp ress .com /

10 9 8 7 6 5 4 3 2 1 0 7 0 6 0 5 0 4 0 3 0 2

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 .

This page intentionally left blank

This page intentionally left blank

References [I] U . Abresc h an d J . Langer , Th e normalize d curv e shortenin g flo w an d

homothetic solutions , J . Differentia l Geometry . 23(1986) , 175-196 .

[2] B . Andrews , Contractio n o f conve x hypersurface s i n Euclidea n spaces , Calc. Var . PDE , 2(1994) , 151-17 1

[3] S . B. Angenent, Th e zero set o f a solution o f a parabolic equation , J . Rein e Angrew. Math. , 390(1988) , 79-96 .

[4] S . B . Angenent , Shrinkin g doughnuts , Proc . Conf . Ellipti c Paraboli c Equations, Gre y nog, Wales , 1989 .

[5] L . Caffarelli , L . Nirenber g an d J . Spruck , O n a for m o f Bernstein' s theo -rem, Analyse mathematique e t applications , 55-66 , Gauthier-Villars, Mon -trouge, 1988 .

[6] K . S . Chou(K . Tso) , Deformin g a hypersurfac e b y it s Gauss-Kronec k curvature, Com m Pur e Appl . Math. , 38(1985) , 867-88 2

[7] K . S. Chou and X. P. Zhu, Shortening complete plane curves, J. Differentia l Geom, 50(1998) , 471-504 .

[8] D . DeTurck , Deformin g metric s i n directio n o f thei r Ricc i tensors , J . Differential Geom. , 18(1983) , 157-162 .

[9] K . Ecker an d G . Huisken, Mea n curvature evolution o f entire graphs , Ann . of Math. , 130(1989) , 453-471 .

[10] K . Ecke r an d G . Huisken , Interio r estimate s fo r hypersurface s movin g b y mean curvature , Invent . Math. , 105(1991) , 547-569 .

[II] M . E . Gag e an d R . S . Hamilton , Th e hea t equatio n shrinkin g conve x plane curves , J . Differentia l Geom. , 23(1986) , 69-96 .

[12] M . A . Grayson , Th e hea t equatio n shrink s embedde d plan e curve s t o round points , J . Differentia l Geom. , 26(1987) , 285-314 .

[13] R . S . Hamilton, Three-manifold s wit h positiv e Ricc i curvature , J . Differ -ential Geom. , 17(1982) , 255-306 .

[14] R . S . Hamilton, Four-manifold s wit h positive positive curvatur e operator , J. Differentia l Geom. , 24(1986) , 153-179 .

[15] R . S . Hamilton , Conve x hypersurface s wit h pinche d secon d fundamenta l form, Comm . Anal . Geom. , 2(1994) . 167-172 .

145

146 References

[16] R . S. Hamilton, Isoperimtri c estimate s fo r the curve shrinking flo w in the plan, Ann . o f Math. Stud. , 137(1995) , 201-222 .

[17] R . S . Hamilton , Harnac k estimate s fo r th e mea n curvatur e flow , J . Dif -ferential Geom. , 41(1995) , 215-226 .

[18] R . S . Hamilton , A compactnes s propert y fo r solutio n o f the Ricc i flow , Amer. J . Math. , 117(1995) , 545-572 .

[19] G . Huisken , Flo w b y mea n curvatur e o f conve x surface s int o sphere , J . Diff. Geom. , 20(1980) , 237-266 .

[20] G . Huisken , Contractin g conve x hypersurface s i n Riemannian manifold s by thei r mea n curvature , Invent . Math. , 84(1986) , 463-480 .

[21] G . Huisken , Asymptoti c behavio r fo r singularitie s o f the mean curvatur e flow, J . Diff . Geom. , 31(1990) , 285-299 .

[22] G . Huisken an d C. Sinestrari , Mea n curvatur e flo w singularitie s fo r mea n convex surface , Calc . Var . PDE , 8(1999), 1-14 .

[23] G . Huiske n an d C . Sinestrari , Convexit y estimate s fo r mea n curvatur e flow an d singularitie s o f mea n conve x surfaces , Act a Math. , 183(1999) . 47-70.

[24] G . Huisken an d S. T. Yau, Definition o f center o f mass fo r isolated phys -ical system s an d uniqu e foliation s b y stabl e sphere s wit h constan t mea n curvature, Invent . Math. , 124(1996) , 281-311 .

[25] N . V. Krylov , Nonlinea r Ellipti c an d Paraboli c Equation s o f the Secon d Order, D . Reidel Publishin g Company , 1987.

[26] O . A. Ladyzhenskaya , V . A. Solonnikov an d N. N. Uralceva, Linea r an d Quasilinear Equation s o f Parabolic Type , Amer . Math . Soc . Providence , RI, 1968.

[27] H . B. Lawson, Loca l rigidity theorem for minimal surfaces, Ann . of Math., 89(1969), 187-197 .

[28] P . Li and S. T. Yau, On the parabolic kerne l of the Schrodinger operator , Acta Math. , 156(1986) , 153-201 .

[29] G . M. Lieberman, Secon d Orde r Paraboli c Differentia l Equations , Worl d Scientific, Singapor e 1996.

[30] J . H . Michae l an d L . Simon , Sobole v an d mean-valu e inequalitie s o n generalized submanifold s o f R n, Comm . Pur e Appl . Math. , 26(1973) , 361-379.

[31] R . Schoen and S. T. Yau, On the proof o f the positive mass conjectur e i n general relativity , Comm . Math . Phys. , 65(1979) , 45-76 .

References 147

[32] R . Schoe n an d S . T . Yau , Proo f o f the positiv e mas s theore m II , Comm . Math. Phys. , 79(1981) , 231-260 .

[33] R . Scheon and S . T. Yau, Lectures on Differential Geometry , i n Conferenc e Proceedings an d Lectur e Note s i n Geometr y an d Topology , Volum e 1 , International Pres s Publications , 1994 .

[34] R . Schneider , Conve x Bodies : Th e Brum-Minkowsk i Theory , Cambridg e University Press , 1993 .

[35] L . Simon, Lecture s o n Geometri c Measur e Theory , Proceeding s o f Cente r for Mathematica l Analysis , Australia n Nationa l University , vol . 3 , 1983.

[36] J . Simons , Minima l varietie s i n Riemannia n manifolds , Ann . o f Math. , 88(1968), 62-105.

[37] B . White , Th e siz e o f th e singula r se t i n mea n curvatur e flo w o f mean -convex sets , J . Amer . Math . Soc , 13(2000) , 665-695.

This page intentionally left blank

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