AN INTEGRAL FORMULA FOR COMPACT HYPERSURFACES IN SPACE FORMS AND ITS APPLICATIONS

Transcription

1 J. Aut. ath. Soc. 74 (2003), AN INTEGRAL FORULA FOR COPACT HYPERSURFACES IN SPACE FORS AND ITS APPLICATIONS LUIS J. ALÍAS (Received 5 December 2000; revied 8 arch 2002) Communicated by K. Wyocki Abtract In thi paper we etablih an integral formula for compact hyperurface in non-flat pace form, and apply it to derive ome intereting application. In particular, we obtain a characterization of geodeic phere in term of a relationhip between the calar curvature of the hyperurface and the ize of it Gau map image. We alo derive an inequality involving the average calar curvature of the hyperurface and the radiu of a geodeic ball in the ambient pace containing the hyperurface, characterizing the geodeic phere a thoe for which equality hold athematic ubject claification: primary 53C40, 53C42. Keyword and phrae: Ricci curvature, calar curvature, average calar curvature, Gau map image, geodeic phere.. Introduction In a recent paper, Dehmukh [] obtained an integral formula for compact hyperurface in Euclidean pace E n+, and applied it to derive ome intereting conequence on uch hyperurface, including a characterization of round phere a well a an inequality involving the average calar curvature and the extrinic radiu of the hyperurface (we alo refer the reader to [2, 3, 4] for ome other application of thi integral formulaobtained by the ame author). orerecently,vlacho [5] obtained an extenion of thi integral formula to the cae of compact hyperurface in the non-flat pace form, namely the phere S n+ and the hyperbolic pace H n+, and applied it to characterize geodeic phere in pace form. The author wa partially upported by Dirección General de Invetigación (CYT), Grant No. BF and Conejería de Educación y Univeridade (CAR) Programa Séneca, Grant No. PI-3/00854/FS/0. c 2003 Autralian athematical Society /03 $A2:00 + 0:00 239

2 240 Lui J. Alía [2] In thi paper we obtain another integral formula for compact hyperurface in the non-flat pace form (ee Propoition 2.), which i imilar to the previou one and allow u to derive new application. In particular, we characterize the geodeic phere in non-flat pace form a follow (ee Corollary 3.3 and Corollary 3.4): Let n be a compact, oriented hyperurface immered in S n+ uch that it Ricci curvature atifie Ric >.n + 2/.n /. Let u aume that it Gau map image N./ S n+ i contained in a tubular neighbourhood of radiu 0 <<³=2 around an equator of S n+. If it calar curvature S atifie S = in 2./, then i a geodeic phere of S n+. Let n be a compact, oriented hyperurface immered in H n+ uch that it Ricci curvature atifie Ric >.n + 2/.n /. Let u aume that the Gau map image N./ S n+ i contained in a tubular neighbourhood of (timelike) radiu >0around an equator of de Sitter pace S n+. If it calar curvature S atifie S = inh 2./, then i a geodeic phere of H n+. A another application of our integral formula, we alo extend Dehmukh inequality for the average calar curvature to the cae of compact hyperurface in non-flat pace form, characterizing the geodeic phere a thoe for which equality hold (ee Theorem 3.5 and Theorem 3.6): Let n be a compact, oriented hyperurface immered in S n+ uch that Ric.n + 2/.n /. If n i contained in a geodeic ball in S n+ of radiu 0 <%<³=2, then av.s/ in.%/ : oreover, the equality hold if and only if n i a geodeic phere of radiu %. Let n be a compact, oriented hyperurface immered in H n+ uch that Ric.n + 2/.n / and the calar curvature S i poitive on. If n i contained inageodeicballinh n+ of radiu %>0, then av.s/ inh.%/ : oreover, the equality hold if and only if n i a geodeic phere of radiu %. 2. The integral formula Throughout thi paper we will denote by n+.c/ the tandard model for an.n + /-dimenional, complete and imply connected pace with contant ectional curvature c, c = ;. Let E n+2, = 0;, denote the.n + 2/-dimenional real vector

3 [3] An integral formula for compact hyperurface 24 pace R n+2 endowed with the metric tenor given by n+ v;w = v i w i +. / v n+2 w n+2 : i= Then n+./ = S n+ E n+2 i the.n + /-dimenional Euclidean phere, S n+ ={x E n+2 : x; x =}; and n+. / = H n+ E n+2 i the.n + /-dimenional hyperbolic pace, H n+ ={x E n+2 : x; x = ; x n+2 }: Let : n n+.c/ E n+2 be an immered, oriented, connected hyperurface in n+.c/, and let N be it globally defined unit normal field. We will denote by o, and the Levi-Civita connection of E n+2 the Gau and Weingarten formula for n in n+.c/ E n+2 by, n+.c/ and n, repectively. Then are given repectively () o X Y = X Y c X; Y = X Y + AX; Y N c X; Y ; and (2) A.X/ = o XN = X N; for all tangent vector field X; Y X./, wherea tand for the hape operator of n in n+.c/ aociated to N. For a fixed arbitrary vector a E n+2, let u conider the height function a; and the function a; N, which are defined on n. From () and (2) we know that their gradient are given by a; =a,and (3) a; N = A.a /; where (4) a = a a; N N c a; X./: By taking covariant derivative in (4) and uing () and (2), we obtain from o a = 0 that (5) X a = a; N A.X/ c a; X

4 242 Lui J. Alía [4] for X X./. Therefore, the Laplacian of a; i given by (6) a; = n Ei a ; E i =nh a; N nc a; ; i= where {E ;::: ;E n } i a local orthonormal frame on n and H = tr.a/=n i the mean curvature function of n,and (7) 2 a; 2 = a; a; + a; 2 = nh a; a; N nc a; 2 + a 2 : On the other hand, from (3) the Laplacian of a; N i given by n n a; N = Ei.Aa /; E i =. Ei A/a ; E i i= i= n A. Ei a /; E i : i= Uing now the Codazzi equation,. X A/.a / =. a A/.X/, jointly with (5) we conclude that where A 2 = tr.a 2 /,and a; N = tr. a A/ a; N A 2 + nch a; = n H; a a; N A 2 + nch a; ; (8) 2 a; N 2 = n a; N H; a a; N 2 A 2 +nch a; a; N + A.a / 2 : Now, let u recall that the Ricci curvature of n i given by Ric.X; Y / =.n /c X; Y +nh AX; Y AX; AY ; for X; Y X./. Thi allow u to write A.a / 2 =.n /c a 2 nh a; N ; a Ric.a ; a /; o that (8) become (9) 2 a; N 2 = n.h a; N /; a a; N 2 A 2 + nch a; a; N Ric.a ; a / +.n /c a 2 : Since.H a; N /; a =.H a; N /; a; = div.h a; N a; / H a; N a; ;

5 [5] An integral formula for compact hyperurface 243 we obtain from (9), uing (6), that 2 a; N 2 = ndiv.h a; N a; / Ric.a ; a / +.n /c a 2 +.n 2 H 2 A 2 / a; N 2 ch a; a; N : Finally, uing (7) and the fact that a; a = a 2 + a; N 2 + c a; 2 we conclude that 2 a; c.n / (0) N 2 + n div.h a; N a; / + a; 2 2 = Ric.a ; a /+.n +2/.n /c a 2 + S a; N 2 n.n /c a; a ; where S tand for the calar curvature of n, S = tr.ric/ = c + n 2 H 2 A 2 : Integrating now (0) on n, the divergence theorem implie our integral formula. PROPOSITION 2.. Let : n n+.c/ E n+2 be a compact, oriented hyperurface immered in n+.c/, c = ;, and let a E n+2 be a fixed arbitrary vector. Then, if Ric and S denote repectively the Ricci curvature and the calar curvature of n, it hold that ( () Ric.a ; a /.n + 2/.n /c a 2) dv =.S a; N 2 c a; a / dv; where dv i the n-dimenional volume element of n with repect to the induced metric and the choen orientation. 3. Application In thi ection we will derive ome conequence and application of our integral formula (). A a firt application, we obtain the following characterization of geodeic phere in n+.c/, which i inpired in the characterization of geodeic phere given by Dehmukh [, Theorem 3] in the Euclidean cae and by Vlacho [5, Theorem 2] in the pherical and hyperbolic cae. THEORE 3.. Let : n n+.c/ E n+2 be a compact, oriented hyperurface immered in n+.c/ uch that it Ricci curvature atifie Ric > c.n +2/.n /. If there exit a point a n+.c/ for which the calar curvature S atifie S a; N 2, then./ i a geodeic phere of n+.c/ centered at the point a.

6 244 Lui J. Alía [6] PROOF. It follow from the aumption on the Ricci curvature that Ric.a ; a /.n + 2/.n /c a 2 0 with equality if and only if a = a; vanihe everywhere. Therefore, uing our integral formula () we obtain that (2).S a; N 2 / dv 0; with equality if and only if a; i contant on n. On the other hand, from the aumption on the calar curvature we alo have.s a; N 2 / dv 0; which give the equality in (2) and implie that./ ={x n+.c/ : a; x =b} for a real contant b. If c =, then b = co.%/ with 0 <%<³and./ i a geodeic phere in S n+ centered at a with radiu %. Ifc =, then b = coh.%/ with %>0and./ i a geodeic phere in H n+ centered at a with radiu %. In the cae where c = (pherical hyperurface) the reult above can be paraphraed in a more geometric tatement a follow. Since c =, the Gau map N can be regarded a a map N : n S n+, and for every a S n+ the function a; N give a meaure of the pherical ditance between a and N. Actually, for every p we know that the pherical ditance between a and N.p/ i given by % a.p/ = arco. a; N.p/ /: For intance, when = S n.a;/ S n+ i a geodeic phere in S n+ centered at a with radiu, 0<<³, then it i eay to ee that % a = ³=2 i contant on. oreover, i a round n-phere of radiu in./ and it calar curvature i given by S = = in 2./ = = co 2.% a /. Now, Theorem 3. can be rewritten in term of the pherical ditance a follow. COROLLARY 3.2. Let : n S n+ E n+2 be a compact, oriented hyperurface immered in S n+ uch that Ric >.n + 2/.n /. For a given a S n+ and p, let u denote by % a.p/ the pherical ditance between a and the image of p under the Gau map N. If there exit a point a S n+ for which co 2.% a / ; S then./ i a geodeic phere of S n+ centered at the point a.

7 [7] An integral formula for compact hyperurface 245 A another conequence of thi, we have the following. COROLLARY 3.3. Let : n S n+ E n+2 be a compact, oriented hyperurface immered in S n+ uch that Ric >.n + 2/.n /. Let u aume that it Gau map image N./ S n+ i contained in a tubular neighbourhood of radiu 0 <<³=2 around an equator of S n+. If it calar curvature S atifie S in 2./ ; then./ i a geodeic phere of S n+ parallel to that equator. PROOF. Let u aume that the pherical image of n i contained in a tubular neighbourhood U.a;/ in S n+ of radiu 0 <<³=2 around the equator given by S n+ a. Oberve that U.a;/ ={x S n+ : in./ a; x in./}; o that a; N 2 in 2./ on n. At each point p n we have that 0 < S.p/ = in 2./ and, by the aumption on S, o that we may apply Theorem 3.. S.p/ a; N.p/ 2 S.p/ in 2./ ; On the other hand, when c = (hyperbolic hyperurface) the Gau map of n can be regarded a a map N : n S n+,wheres n+ denote the.n + /-dimenional unitary de Sitter pace, that i, S n+ ={x E n+2 : x; x =}: A i well known, for n 2 de Sitter pace i the tandard imply connected Lorentzian pace form with contant ectional curvature. In thi cae, if a H n+,the interection S n+ a define a round n-phere of radiu which i a totally geodeic pacelike hyperurface in S n+. We will refer to that phere a the equator of de Sitter pace determined by a. In thi term, when c = Theorem 3. can be rewritten a follow. COROLLARY 3.4. Let : n H n+ E n+2 be a compact, oriented hyperurface immered in H n+ uch that Ric >.n + 2/.n /. Let u aume that the Gau map image N./ S n+ i contained in a tubular neighbourhood of (timelike) radiu >0around an equator of S n+ determined by a H n+. If it calar curvature S atifie S inh 2./ ; then./ i a geodeic phere of H n+ centered at a.

8 246 Lui J. Alía [8] The proof i imilar to that of Corollary 3.3 by oberving that the tubular neighbourhood U.a;/in S n+ of (timelike) radiu around the equator S n+ a i given by U.a;/={x S n+ : inh./ a; x inh./}: On the other hand, in [, Theorem, Theorem 2], Dehmukh gave an intereting inequality involving the extrinic radiu of a non-negatively Ricci curved compact hyperurface in Euclidean pace n and it average calar curvature av.s/, whichi defined by the Eintein functional av.s/ = S dv: vol./ In thi paper, and a another application of our integral formula (), we extend Dehmukh inequality to the cae of compact hyperurfacein the other pace form, alo characterizing when the equality hold, a follow. THEORE 3.5. Let : n S n+ E n+2 be a compact, oriented hyperurface immered in S n+ uch that Ric.n + 2/.n /. If the image./ S n+ i contained in a geodeic ball in S n+ of radiu 0 <%<³=2, then av.s/ in.%/ : oreover, the equality hold if and only if n i a geodeic phere of radiu %. THEORE 3.6. Let : n H n+ E n+2 be a compact, oriented hyperurface immered in H n+ uch that Ric.n + 2/.n / and the calar curvature S i poitive on. If the image./ H n+ i contained in a geodeic ball in H n+ of radiu %>0, then av.s/ inh.%/ : oreover, the equality hold if and only if n i a geodeic phere of radiu %. COROLLARY 3.7. Let : n H n+ E n+2 be a poitively Ricci curved, compact, oriented hyperurface immered in H n+. If the image./ H n+ i contained in a geodeic ball in H n+ of radiu %>0, then av.s/ inh.%/ : oreover, the equality hold if and only if n i a geodeic phere of radiu %.

9 [9] An integral formula for compact hyperurface 247 PROOF (Proof of Theorem 3.5). Let u aume that./ i contained in a geodeic ball B.a;%/in S n+ of radiu 0 <%<³=2 centered at a point a S n+. Recall that o that a; 2 co 2.%/, and B.a;%/={x S n+ : 0 < co.%/ a; x }; a; N 2 = a; 2 a 2 co 2.%/ = in 2.%/ on, with equality if and only if a = 0and a; =co.%/ i contant on. Since S > 0on, then S in 2.%/ S a; N 2 ; o that from () we obtain in 2.%/ S dv vol./ =.S in 2.%/ / dv.s a; N 2 / dv =.Ric.a ; a /.n + 2/.n / a 2 / dv 0: That i, av.s/ = S dv vol./ in 2.%/ : oreover, equality hold if and only if a; =co.%/ i contant on, which mean that i a geodeic phere of radiu % centered at a. The proof of Theorem 3.6 i imilar to that of Theorem 3.5 by oberving that the geodeic ball B.a;%/in H n+ of radiu % centered at a point a i given by B.a;%/ ={x H n+ : a; x coh.%/}; and uing now that a; N 2 = a; 2 a 2. Reference [] S. Dehmukh, An integral formula for compact hyperurface in a Euclidean pace and it application, Glagow ath. J. 34 (992), [2], Hyperurface of non-negative Ricci curvature in a Euclidean pace, J. Geom. 45 (992),

10 248 Lui J. Alía [0] [3], Iometric immerion of a compact Riemannian manifold into a Euclidean pace, Bull. Autral. ath. Soc. 46 (992), [4], Compact hyperurface in a Euclidean pace, Quart. J. ath. Oxford Ser. (2) 49 (998), [5] T. Vlacho, An integral formula for hyperurface in pace form, Glagow ath. J. 37 (995), Departamento de atemática Univeridad de urcia E-3000 Epinardo, urcia Spain ljalia@um.e