A REMARK ON COMPACT HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN SPACE FORMS. Key Words: constant mean curvature hypersurfaces, rigidity

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1 A REMARK ON COMPACT HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN SPACE FORMS GIOVANNI CATINO Abstract. I this ote we characterize compact hypersurfaces of dimesio 2 with costat mea curvature H immersed i space forms of costat curvature ad satisfyig a optimal itegral pichig coditio: they are either totally umbilical or, whe 3 ad H 0, they are locally cotaied i a rotatioal hypersurface. I dimesio two, the itegral pichig coditio reduces to a topological assumptio ad we recover the classical Hopf-Cher result. Key Words: costat mea curvature hypersurfaces, rigidity AMS subject classificatio: 53C40, 53C42, 53A0. Itroductio The study of costat mea curvature hypersurfaces i space forms of costat curvature is oe of the oldest subjects i differetial geometry. There are may iterestig results o this topic (see for example [2, 3, 6, 5, 8, 7, 8, 5, 2], ad may others. By costructig a holomorphic quadratic differetial, Hopf [2] showed that ay costat mea curvature twosphere i R 3 is totally umbilical. Cher [7] exteded Hopf s result to costat mea curvature two-spheres i three-dimesioal space forms. Compact immersed costat mea curvature tori i R 3 were first costructed by Wete [7]. To fix the otatio, let M, 2, be a compact hypersurface with costat mea curvature H immersed i a space form F + (c of costat curvature c. Deote by h the secod fudametal form of M ad by h its trace-free part. With this otatio, M is totally umbilical if ad oly if h vaishes. It is well kow [6, 8, 4] that if H = 0 ad h 2 c, c > 0, the M is either totally umbilical or a Clifford tori i S + (c, i.e. product of spheres S (r S 2 (r 2, + 2 =, of appropriate radii. This rigidity result was exteded by Alecar ad do Carmo [] to hypersurfaces with costat mea curvature. The aim of this ote is to show a characterizatio of compact hypersurfaces with costat mea curvature satisfyig a itegral pichig coditio o h. This improves the result i []. Moreover, i dimesio two, the itegral iequality reduces to a topological assumptio o the surface ad leads to a ew proof of Hopf-Cher Theorem. Our mai result reads as follows: Theorem.. Let M be a compact hypersurface with costat mea curvature immersed i a space form F + (c of costat curvature c. The h 2 M ( H2 h 2 2 H h + c ( 0

2 2 GIOVANNI CATINO ad equality occurs if ad oly if M is either totally umbilical or, whe 3 ad H 0, aroud every o-umbilical poit, it is locally cotaied i a rotatioal hypersurface of F + (c. Note that, if H = 0 ad c 0, the statemet is trivial. O the other had, there are Clifford tori i S + (c with h 2 c that are ot cotaied i a rotatioal hypersurface of S + (c. Hece, the secod part of the equality case i Theorem. caot be true if H = 0. I dimesio two, Gauss equatio ad Gauss-Boet theorem imply that the itegral iequality is equivalet to the o-positivity of the Euler characteristic of M 2 ad we recover Hopf-Cher result. Corollary.2 (Hopf Cher. Let M 2 be a compact surface with costat mea curvature immersed i a space form F 3 (c of costat curvature c. The, either M 2 is totally umbilical or χ(m 2 0. I particular, every compact costat mea curvature two-sphere immersed i a space form F 3 (c is totally umbilical. The proof of Theorem. relies o a improvemet of the Bocher method applied to Codazzi tesors with costat trace (sectio 2, which was observed by the the author i [6]. 2. Codazzi tesors with costat trace Let (M, g be a smooth Riemaia maifold of dimesio 3 ad cosider a Codazzi tesor T o M, i. e., a symmetric biliear form satisfyig the Codazzi equatio ( X T (Y, Z = ( Y T (X, Z, for every taget vectors X, Y, Z. For a overview o maifolds admittig a Codazzi tesor see [3, Chapter 6.C]. I all this sectio we will assume that T has costat trace. I particular, the trace-free tesor T = T tr(t g is agai a Codazzi tesor. I a local coordiate system, we have k Tij = j Tik. (2. Throughout the article, the Eistei covetio of summig over the repeated idices will be adopted. Takig the covariat derivative of the Codazzi equatio ad tracig we obtai T ij = k j Tik = j k Tik R ikjl Tkl + R jk Tik, where we have used the commutatio rules of covariat derivatives of symmetric two tesors. Here R ikjl ad R jk deote the compoets of the Riema ad Ricci tesor respectively. Now, sice T is trace-free, from (3. oe has k Tik = i Tkk = 0. Thus, ay trace-free Codazzi tesor T satisfies the followig elliptic system T ij = R ikjl Tkl + R jk Tik. (2.2 I particular, the followig Weitzeböck formula holds 2 T 2 = T 2 R ikjl Tij Tkl + R jk Tij Tik. (2.3 I this sectio we recall a vaishig theorem for Codazzi tesor with costat trace which was proved by the author i [6], followig the work of Gursky [] o coformal vector fields. As first observed by Bourguigo [4], trace-free Codazzi tesor satisfies the followig sharp iequality.

3 COMPACT HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN SPACE FORMS 3 Lemma 2.. Let T be a trace-free Codazzi tesor o a Riemaia maifold (M, g ad let Ω 0 = {p M : T (p 0}. The, o Ω 0, T T 2. Note that, if = 2, the equality holds. From the previous equatio, o Ω 0, we therefore have 2 T T 2 R ikjl Tij Tkl + R jk Tij Tik. (2.4 Moreover, to apply (2.4 o the whole M, we eed to measure the set M \ Ω 0. We have the followig result [6]. Lemma 2.2. Let T be a, o-trivial, trace-free Codazzi tesor o the Riemaia maifold (M, g ad let Ω 0 = {p M : T (p 0}. The Vol (M \ Ω 0 = 0. I particular (2.4 holds i a H -sese o M. Usig equatio (2.4, a itegratio by parts argumet implies the followig itegral iequality o trace-free Codazzi tesor [6]. Propositio 2.3. Let T be a, o-trivial, trace-free Codazzi tesor o a compact Riemaia maifold (M, g. For ε > 0, defie Ω ε = {p M : T (p ε}, ad The f ε = { T (p if p Ωε ε if p M \ Ω ε. +2 ( Rikjl Tij Tkl + R jk Tij Tik f ε 0. M 3. Proof of Theorem. ad Corollary.2 Let F + (c be a (+-dimesioal smooth Riemaia maifold with costat sectioal curvature c ad let M be a -dimesioal compact hypersurface immersed i F + (c. For ay p M we choose a local orthoormal frame {e,..., e, e + } i F + (c aroud p such that {e,..., e } are tagetial to M. Sice F + (c has costat sectioal curvature c, Codazzi ad Gauss equatios read (see for istace [9] k h ij j h ik = 0, (3. R ikjl = c (g ij g kl g il g jk + h ij h kl h il h jk, (3.2 where g deotes the iduced Riemaia metric o M, Rm its curvature tesor ad h the secod fudametal form of M. I particular, tracig Gauss equatio (3.2, we get R = ( c + H 2 h 2, (3.3 were R ad H deote the scalar curvature of g ad the mea curvature of M, respectively. Now, if M has costat mea curvature H, the by Codazzi equatio (3. the tesor h = h Hg is a trace-free Codazzi tesor. Thus, if h is ot idetically zero, amely if M is ot totally umbilical, the Propositio 2.3 applies ad we obtai the followig itegral iequality ( Rikjl hij hkl + R jk hij hik f ε 0, (3.4 M +2

4 4 GIOVANNI CATINO where f ε = { h (p if p Ωε ε if p M \ Ω ε. ad Ω ε = {p M : h (p ε}. Usig Gauss equatio (3.2, a simple calculatio shows R ikjl hij hkl + R jk hij hik = H2 h 2 h 4 H h ij hik hjk + c h 2. Moreover, sice h is trace-free, we have the sharp Okumura iequality (for a proof, see for istace [, Lemma 2.6] hij hik hjk 2 ( h 3 (3.5 ad, if 3, equality occurs at some poit p M if ad oly if h ca be diagoalized at p with ( -eigevalues equal to λ ad oe eigevalue equals to ( λ, for some λ R. Hece, we obtai R ikjl hij hkl + R jk hij hik h 2( H2 h 2 2 H h + c, ( ad from (3.4, we get ( h 2 M H2 h 2 2 H h + c h +2 ( f +2 ε +2 ( Rikjl hij hkl + R jk hij hik f ε 0. M Takig the limit as ε 0, sice h +2 ( h 2 M H2 h 2 2 H h + c ( f +2 ε a.e. o M by Lemma 2.2, we coclude 0 (3.6 ad, if 3 ad H 0, equality occurs if ad oly if, at every poit, either h is ull or it has a eigevalue of multiplicity ( ad aother of multiplicity. Now we ca coclude the proof of Theorem. ad Corollary.2. If = 2, the we have showed that either M 2 is totally umbilic, or the followig itegral pichig iequality holds ( 2 H2 h 2 + 2c 0. Sice h 2 = h 2 2 H2, we have M 2 ad from Gauss equatio (3.3, we obtai M 2 ( H 2 h 2 + 2c 0 M 2 R 0. Corollary.2 ow simply follows from Gauss-Boet theorem. If 3, we have that iequality (3.6 holds ad equality occurs if ad oly if either h is ull ad M is totally umbilical or, whe H 0, aroud every o-umbilical poit, h splits with a eigevalue of multiplicity ( ad aother of multiplicity. Notice that, from Lemma 2.2, the ope set of o-umbilical poits is dese i M. Theorem. ow follows

5 COMPACT HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN SPACE FORMS 5 from [0, Theorem 4.2], where the authors showed that every hypersurfaces i a space form with this property is cotaied i a rotatioal hypersurface of F + (c. Ackowledgmets. The author is members of the Gruppo Nazioale per l Aalisi Matematica, la Probabilità e le loro Applicazioi (GNAMPA of the Istituto Nazioale di Alta Matematica (INdAM ad is supported by the GNAMPA project Equazioi di evoluzioe geometriche e strutture di tipo Eistei. Refereces. H. Alecar ad M. do Carmo, Hypersurfaces with costat mea curvature i spheres, Proc. Amer. Math. Soc. 20 (994, B. Adrews ad H. Li, Embedded costat mea curvature tori i the three-sphere, arxiv preprit server to appear o J. Diff. Geom., A. L. Besse, Eistei maifolds, Spriger Verlag, Berli, J.-P. Bourguigo, The magic of Weitzeböck formulas, Variatioal Methods Paris (998, Progess i Noliear Differetial Equatios ad Applicatios IV, Birkauser, 990, pp S. Bredle, Embedded miimal tori i s 3 ad the Lawso cojecture, Acta Mathematica 2 (203, G. Catio, O coformally flat maifolds with costat positive scalar curvature, arxiv preprit server S.S. Cher, O surfaces of costat mea curvature i a three-dimesioal space of costat curvature, Lecture otes i Math. 007 (983, S.S. Cher, M. do Carmo, ad S. Kobayashi, Miimal submaifolds of a sphere with secod fudametal form of costat legth, Fuctioal Aalysis ad Related Fields (970, M. do Carmo, Riemaia geometry, Mathematics: Theory & Applicatios, vol. 290, Birkhäuser, M. do Carmo ad M. Dajczer, Rotatio hypersurfaces i spaces of costat curvature, Tras. Amer. Math. Soc. 277 (983, M. J. Gursky, Coformal vector fields o four-maifolds with egative scalar curvature, Math. Z. 232 (999, o. 2, H. Hopf, ber Fläche mit eier Relatio zwische de Hauptkru mmuge, Math. Nachr. 4 (95, , Differetial geometry i the large, Lecture Notes i Mathematics 000 ( B. Lawso, Local rigidity theorems for miimal hypersurfaces, A. of Math. 89 (969, K. Nomizu ad B. Smyth, A formula of simos type ad hypersurfaces with costat mea curvature, J. Diff. Geom. 3 (969, J. Simos, Miimal varieties i Riemaia maifolds, A. of Math. (2 88 (968, H.C. Wete, Couterexample to a cojecture of H. Hopf, Pacific. J. Math. 2 (986, S.T. Yau, Submaifolds with costat mea curvature. I, II, Amer. J. Math. 96 (974, (Giovai Catio Dipartimeto di Matematica, Politecico di Milao, Piazza Leoardo da Vici 32, 2033 Milao, Italy address: giovai.catio@polimi.it