Annals of mathematics

Annals of Mathematics Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature Author(s): William Meeks III, Leon Simon, Shing-Tung Yau Source: The Annals of Mathematics, Second Series, Vol 116, No (Nov., 1982), pp 621-659 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/2007026 Accessed: 27/08/2011 13:35 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive We use information technology and tools to increase productivity and facilitate new forms of scholarship For more information about JSTOR, please contact support@jstor.org Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematics http://www.jstor.org 116 (1982), 621-659 AnnalsofMathematics, Embedded minimalsurfaces,exotic spheres, and manifoldswith positive Ricci curvature By WILLIAM MEEKS III, LEON SIMON and SHING-TUNGYAU Let N be a threedimensionalRiemannianmanifold.Let E be a closed embeddedsurfacein N Then it is a questionofbasic interestto see whetherone can deform : in its isotopyclass to some "canonical" embeddedsurface.From a natural"canonical"surfacewillbe the extremal the pointofviewofgeometry, surfaceof some functionaldefinedon the space of embedded surfaces.The simplestfunctionalis the area functional.The extremalsurfaceof the area functionalis called the minimalsurface.Such minimalsurfaceswere used extensively by Meeks-Yau[MY21in studyinggroupactionson threedimensional manifolds In [MY2],thetheoryofminimalsurfaceswas used to simplify and strengthen the classicalDehn's lemma,loop theoremand the spheretheorem.In the setting there,one minimizesarea among all immersedsurfacesand proves that the extremalobject is embedded In this paper, we minimizearea among all embeddedsurfacesisotopicto a fixedembeddedsurface.In thecategoryofthese surfaces,we prove a general existencetheorem(Theorem 1) A particular consequence of this theoremis that for irreduciblemanifoldsan embedded surfaceis isotopicto an embeddedincompressible surfacewith incompressible minimalarea We also provethatthereexistsan embeddedsphereof least area volumeis not a standardball, enclosinga fakecell,providedthe complementary and providedthereexistsno embeddedone-sidedRP2 we are By makinguse ofthelastresult,and a cuttingand pastingargument, able to settlea well-known problemin thetheoryofthreedimensionalmanifolds orientablethreedimensional We provethatthecoveringspace ofanyirreducible manifoldis irreducible.It is possibleto exploitour existencetheoremto study finitegroupactionson threedimensionalmanifoldsas in [MY2] In thesecondpartofthepaper,we applyourexistencetheoremto studythe withnon-negative Ricci curvatopologyof compactthreedimensionalmanifolds 003-486X/82/0116-3/0621/039$03.90/1 ? 1982 by PrincetonUniversity (MathematicsDepartment) 622 W MEEKS III, L SIMON, S T YAU exceptin thecase whenthemanifoldis covered ture.We classifythesemanifolds As a consequence,if one can prove the irreducible an homotopy sphere by existenceof a metricwith positiveRicci curvatureon any compact simply connectedthreedimensionalmanifold,thenthe Poincareconjectureis valid It shouldbe mentionedthatour existencetheoremwas used by Schoen-Yau[SY2] to provethatthe onlycompletenon-compactthreedimensionalmanifoldwith to R3 In thispaper,we also classifythe is diffeomorphic positiveRicci curvature whoseboundaryhas non-negatopologyof compactthreedimensionalmanifolds tivemean curvaturewithrespectto the outwardnormal In the above process, we study the topology of compact embedded to orientableminimalsurfacesin a threedimensionalmanifolddiffeomorphic X is with a metric with scalar curvaS2 which non-negative S' equipped s3#n11 ture.We findtheconditionforwhichtwocompactembeddedorientablesurfaces of the ambientspace If the are conjugateto each otherundera diffeomorphism to the threedimensionalsphere,then the minimal manifoldis diffeomorphic This generalizesa previoustheoremof Lawson is surface unique topologically [LH] where the metrichas positiveRicci curvatureand a theoremof Meeks Ricci curvature [MW2] wherethe metrichas non-negative In the last section,we studycompletemanifolds(non-compact)withpositive Ricci curvaturewhose boundaryhas non-negativemean curvaturewith respectto the outwardnormal.We provethatthe boundaryis connectedunless it is a Riemannianproductor is a handlebody.As in the paper of Frankel[FT], about the fundamental groupof the boundary thisgivessomeinformation of the extremalembedded that the regularity Finally,we shouldmention minimalsurfacein the main existencetheoremdepends on the theoryof [AS],wheretheydeal withminimalsurfacesin R3 It shouldalso Almgren-Simon Freedman-Hass-Scott wereable to improveone be mentionedthatveryrecently, if aspect of our theoremand prove that a compact incompressibleminimal surfaceminimizesarea in its homotopyclass and if it is homotopicto an embeddedsurface,thenit is embedded and statement ofmainresults Terminology B, willdenotethe closed3-ballof radiusp and center0 in R3, B = B1 S2 = aB D willdenotethe closedunitdisc withcenter0 in R2 N will denote a complete(not necessarilyorientable)Riemannian3-manifold If : C A is in a smooth surface,we let I2 denote the area (two measure)of E dimensionalHausdorff EMBEDDED MINIMAL 623 SURFACES N willalwaysbe supposedto have the following"homogeneousregularity" propertyforsomepo > 0: For each x0 E N thereis an open geodesicball Gp0(x0)withcenterx0 and radiusposuchthattheexponentialmap expxoprovidesa diffeomorphism 9pofBPo onto Gpo(X0),satisfying (1.1)~ ldyqg1, lid T y CBP,30 11c2, z c po(xo) We also requirethattherebe a constant[tindependentof x0 such that (1.2) supB | a k < C/P0, SupBpa -1 kaka fori, I, k, 1= 1,2, 3, wheregigdx2dxi is themetricrelativeto normalcoordinates forGPO(xO) Of courseit is trivialthatsucha poand sucha [t existin case N is compact geometry,we can prove that a By using comparisontheoremsin differential manifoldis homogeneousregularif and onlyif it is a completemanifoldwhose injectivityradius is bounded frombelow and whose sectional curvatureis bounded C will denote the collectionof all connected compact (not necessarily embeddedin N C1 surfaces-without-boundary orientable)smooth2-dimensional will denote the collectionof compact embedded surfacesl such that each componentof l is an elementof C Givenl E QJ,we let J(2)denotethe isotopyclass of l; thatis J(2)is the collectionof all l E C1 such that l is isotopicto Y via a smoothisotopyzp: of N onto N [0,1] X N N, whereqp ='N and each q9tis a diffeomorphism Here ptis definedby qgt(x)= (t, x), (t, x) E [0, 1] X N; we shall oftenwrite In case N is non-compact, we also requirethattherebe a fixed m {=Ptli't? compactK C N suchthat ptIN-K= 'N-K for each t E [0,1] I =J# Now suppose EC is given If infiE.) 0, then we may select a sequence {2k} C J(2) withlimI k I=inf E.) | I We call such a sequence a minimizingsequence for J(2) More generally,{2kl C is called a minimizing sequence if I Ek 1I infE (k) I I2 +Ek with ek -? as k o and if limsupk I Ek I + genus(Ek)) < 00 By a standardcompactnesstheoremfor Borel measures(applied to the measuresIlk given by yk(f) = fjkf, f e Co(N)), we know that there is a subsequence{2k } C {fk} and a BorelmeasuretLwith (1.3) IL(f) lim k/400 f, feCO(N) 624 III, L SIMON, W MEEKS S T YAU Our mainexistencetheoremis thengivenas follows: seTHEOREM Suppose N is compact,and {ok} C is a minimizing quence; let {fk'}, [t be as in (1.3) above, and supposelm I 2V > Thenthereare positiveintegersR, n, , nR and pairwisedisjointminimal surfacesE(), ,2) E C such that k'> n (') + n2 (2) + (in the sense that ,u(f) = iclt 2(R) fE C(N), ttas in (1.3)) 21njlffJ genus(V:')), then if g= Furthermore, (1.4) +n 1n7(g - 1) + n~gj? genus(Y-k,) i C- large k', where Qt { j: E) is one-sidedin N) and ? for all sufficiently forall j E Qt; hence all termsin { i: E(i) is two-sided in N) (Notice that g (1.4) are non-negative.) If each 2k' is two-sidedin N theneach E) satisfiesthestability (1.5) f( 1(iA 12+ Ric(v, v)) - I 12)? 0, t' (: E wherev is any unit normalfor E) (V(i) need not necessarilybe two-sided,so that here v is not assumed to be continuous.)In any case, even when 2k' is one-sided,(1.5) holdsforany E) whichis two-sidedin N conditiondescribed In case N merelysatisfiesthe homogeneousregularity I> must be replaced by the hypothesisthat above, the hypothesislimI liminfI k' n K I> for some compact K C N; then the above conclusions and which Sk' whichtendto infinity continueto hold In fact thereare currents can be writtenas finite sums of embeddedclosed surfaceswith uniformly bounded diameterand with area boundedfrombelow (by a fixed positive constant)so that 2k' - Sk' n12() + n22(2) +* +nk 2(k) and lim I Sk/| = lim I Y-k'1|-(nl I 7:(1) I + ***+nk I 2(k) 1) Furthermore (1.4) and (1.5) hold, wherein (1.4) we can add the corresponding sum associatedto thegenusof thesurfacesin Sk, In particular,if N satisfiesthe additionalconditionthatfor each c > thereexistsa compactset of N so that a geodesic ball of radius c in the EMBEDDED MINIMAL SURFACES 625 complement of thiscompactset is a subsetof someopen domaindiffeomorphic to theball, thenwe may takeSk/= (1.6) Remarks.1 We shall give a more precisestatementconcerningthe relation(up to isotopy)ofthe E), 5(R), n , nR and the sequence {fk'} at a laterstage.(See Remark(3.27).) thenall theEi) suchthat We shallalso showthatifeach 2k' is two-sided, also two-sided However Remark is odd are (see (3.27)), E) maybe one-sided n, in case n, is even to each 2k we have a varifoldV(2k) (see [AW, ? 3.5]); Corresponding since I2k I is bounded, a subsequence v(E4) of V(2k0) will convergeto a varifoldV suchthat11V 11= p (p as in (1.3)) In view oftheconstancy stationary theorem([AW]) the contentof the theoremis then V = n'v(2(1)) + *+n"v(E(R)) (1.7) withn, L:() as described and in factstable,is readilyseen as follows.If The factthatV is stationary, = {Tt}O tei is any smoothisotopyas above,then 8k (Ek -(* 0) M((,#V(Yk))= M(V(Tt(Yk))) >.IkI for each t E [0,1] and each k = 1,2, , by the assumptionthat {2k} minimizing sequence.Thus,takinglimitsas k -* o, we get (1.8) is a M((pt#V)' M(V) for everysuch isotopy.Notice that this is in fact a strongerconditionthan because here p is any isotopyas describedabove stability, in N we have thatthereare constants'q E (0, 1) and Since V is stationary c > (dependingonlyon t) suchthatc1rq< and (1 - Cp/pO)p-2 11V II(Gp(y)) is increasingin p forp ? 'qpo In particularit followsthat (1.9) o-2 11V II(G (y)) ? C2p-2 11V II(G (y)) forany < a < p ' po,withc2 dependingonlyon 110 For a proofof this(whichuses (1.1), (1.2)), see the example[SS, ? 5]; the ofstandardmonotonicity modification proofis a straightforward arguments (See e.g [AW],[MS].) Preliminary lemmas LEMMA Let po be as in (1.1) and (1.2) Thereis a number8 E (0, 1) (independentof N, po) such thatif l C(, satisfies 12 n Gpo(Xo)I< 82p2 (2.1) 626 W MEEKS III, L SIMON, S T YAU for each x0 e N, then thereexists a unique compact K2 C N with aKs (2.2) = and X E N vol(K, n GP(xo))' 8?2pg, This K2 also satisfies (2.3) vol(K) Also, if S2, I 13/2, ?c then K2Y c = c() B (2.4) Remark.Evidently,if the hypothesesof the above lemmahold and if as in Section1, thenN -h has : is isotopicto E via an isotopyT = { tp0','1 two componentsU,V suchthat U is diffeomorphic to Ke forN Proof We firstnotethatby (1.1), (1.2) we can finda triangulation XfC0 K K ? of JC0has diam such that each 3-cell po and such that there is a {(x, diffeomorphism X2, x3) PK ofKPO_ {x E N: dist(x,N) < po} ontosPO, S < {x E R3: dist(x,s) E R3 PO}, spo po}, with TK(K) = s iand - ? (2.5) SUPY&K IldyPK C1, supZ S Ild p>IK1?c1, cl1 cl0y For smallenoughwe can perturbNOYslightlyto give a new triangulation X{ such that L does not intersectthe 1-skeletonof X,* and such that foreach 3-cell K of XfC thereis a diffeomorphism {PK of K ontos with (2.5)' supYEKIdY4KII' clsupzEsI1dz4K i1 ? cl, cl = cl(y) thendependson l, but we have (2.5)' withcl dependingonly (Of courseXfC on It.) We notethatthisS(Ccan be selectedso that(by virtueof (2.1)) thereis at and aK0 transversally least one 3-cellK0 E Y{ suchthatl intersects c2 =C length(E n MKO) c2p, of YC,l mustbe containedin a Since l does not intersectthe 1-skeleton of the dual triangulation SfC'.ThereforeY is containedin a regularneighborhood handlebodyand it is thenstandardthatthereis a compactK2, containedin this handlebody,whichis boundedby l; thus (2.6) (2.7) MKY= , K nlE= 0, It is also standardthat this K2 is then of YuC whereE denotesthe 1-skeleton S2 to B in case diffeomorphic *To see this,let a be any edge of s, let D(a) be the disc normalto a withradiuspo/2 and centreat the mid-pointof a, and foreach ( E D(a), let at be the segmentparallelto a with n mid-point ( and lengthequal to length(a) + po By (2.1) and (2.5) we musthave at n PK(I KPO)= exceptfora set of ( E D(a) of area < c82p2 For smallenoughit willthenevidently be possibleto choose YCso thateach edge of Yuis containedin one of the curvesin the family {q4j(a ):K E No, ( E D(a), a an edge of s} EMBEDDED 627 MINIMAL SURFACES It thusremainsonlyto prove(2.2) and (2.3) Firsttake K0 as in (2.6) By virtueof(2.5)', (2.6), and (2.7), we musthave (by the isoperimetric inequalityin R2 that Kmn MOK 1' c3 = C382P2, Then by (2.1), (2.5)' and the isoperimetric inequalityin R3 we have vol(K, n K0) (2.8) ? c4 C483p3, On the otherhand,forany 3-cellsK, K' E Yusuch thatK, K' sharea common two-dimensional face, we have by (2.1) and the Poincareinequalityin R3, that min{vol((K U K') n Km),vol((K U K') - K))} c563p Thus if is smallenoughwe musthave (since vol(K), vol(K') ? c6p3 by (2.5)), either max{vol(K n Km),vol(K' n Ki)) ? c6 3p0 or min{vol(K n Km),vol(K' n K,)))} ?2 p3 wherec6= c6(M).Since thisis trueforany K, K' ?E Yusharinga commonface, we thenhave from(2.8) that(for8 smallenough) vol(K n K2) ? C783PO3 (2.9) forevery3-cellK in the triangulation W3.By (2.5) and the Poincareinequality, this implies, again forsmall enough S, that (2.10) forall K E u; vol(K n Km) < c8IY n K 13/2 (2.2) and (2.3) now evidentlyfollowfrom(2.9), (2.10), provided8 is sufficiently small LEMMA Suppose MI, ,MR are diffeomorphicto D, suppose M, -_ 8M, c A - aA, aMCc aA, j = 1, ,R, whereA C N is diffeomorphic to B and supposethataMi n aM, = and thateitherMi n M, = or Mi intersects M, transversally forall i 7#j Then thereexistpairwise disjointM1, ,MR with M -_ aMi C A - aA, aM,= aM, and I M, - 2/2 It is thusclearthat,givenany sequence - (3-3) Yk [...]... anycomponentofM3 above in the statementof Theorem1 of [AS]: We wouldalso pointout a misprint the equality(N -M) n a U= 0 in (iii) should read (No-M) n U = 0 (so of aM pointsinto U at pointsof aM n a8U) thatthe inwardpointingco-normal Theorem1 of [AS] carriesoverdirectlyto Withthesemodifiedhypotheses, The is unchangedexcept that (4.6) of essentially proof the presentsetting Lemma 3 is used in place of inequality(3.1)... isotopyclass of a surfaceparallelto a boundary smallarea componentcontainssurfacesof arbitrarily Proofof Proposition 1 FirstnotethatifN is a handlebody,thenthearea of anycompactsurfacecan be shrunkdownin a standardway to a one-dimensional complexin sucha way thatthe area of the surfaceapproacheszero withfurther and further shrinkings Hence we shallonlybe concernedwiththe proofof the converseof the above... 2, be a component of X -L and minimal surface in X Let N,, j define I(Nj) to be the rank of the image of the induced map i*: ?T2(NA) H2(Ni,Z2) If A' is anotherembeddedorientableminimalsurfaceof the same of X takingL to A' if and onlyif genusin X, thenthereexistsa diffeomorphism thenumberof components of X - l is thesame and if thecorresponding indices I(NI) are thesame THEOREM Proofof Theorem7 The resultis... findthatthearea of Integrating than thearea ofS and theyare equal ifand onlyifAt = 0 in that greater is not S, region.Since S has leastarea in itsisotopyclass,thearea of S, is notless thanthe of S and a/at definesa parallel area of S and hence At= 0 in a neighborhood vector there.Therefore,a neighborhoodof S is isometricto a Riemannian productS X (-c, e) Now let N be theuniversalcoverofN and S be a componentof... one of the two componentsof WN.If the assertionwere to fail,thenby intersection therewould exist an embedded arc y: [0, 1] -Y theorywith Z2-coefficients such thaty(O) e E and y(l) lies on the othercomponentof aY1 and y is disjoint fromthe spheresS1, ,SI Let W and 6D= {D1, DR} be as in the proofof Assertion1 There,each componentof aW mustbe a two-sphere The sum of the boundaryspheresSj*,S , Sn *of W... and S2 are disjoint Proof First suppose that S1 and S2 intersect transversely.In this case Si n S2 consists of a collection of Jordan curves each of which is the boundary of exactly one disc on S1 and another disc on S2' Let 'D be the I= EMBEDDED 647 MINIMAL SURFACES collectionofall such discs.Choose a D E 63Dsuch that IDII