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(1)The Integral of. the. Mean. Curvature. By. Masao. MAEDA*. (Received May 31, 1978). Let M be an m-dimensional compact Riemannian rpanifold immersed an n-dimensional Euclidean space E", n>m and H the mean curvature vector field on M of the immersion. Then,it holds an -lnterestlng. isometrically in inequality:. S.IHIMdv lll c.. and equality sign holds in the above when and only when M is an imbedded m-dimensional sphere in EM'i, an (m+1)-dimensional linear subspace of En, where dv is the volume element of M, c. the volume of the m-dimensional unit. sphere Sm(1) and IHI the Euclidean norm of H, see Chen's book [1;Theorem 3.2].. The purpose of this note is to consider whether we can derive an inequality such as above when the ambient space of M is the simply connected space form of constant negative sectional curvature -c, c>O. And we will give an inequality concerning the above problem when the dimension of M is two.. Let H"(-c) be the n-dimensional simply connected space form of constant negative sectional curvature -c represented on the set {x=(xi,...,x.)GE";. n lx12:2x?･<1} with the Riemannian metric g=(41c(1-lx12)2)2dx?･. Let i=1. i: M.Hn( - c) be an isometric immersion of a compact m-dimensional Riemannian. manifold M. Then from i, we have the immersion i: M->{xeEn; lxl<1}cEn. Let g=2dx?･ be the Euclidean metric of E". 7 and 7 denote the Riemannian connections of H"(-c) and E" respectively, Let {U;ui,...,u.} be a local ,. coordinate of M, and put ei:==0/Oui, i=1,..., m, gNij:==g(ei, ej) and gij:=g(ei, ej). Let (gNij) and (gij) be the inverse matrixes of (gNij) and (gij) respectively.. Then putting op:=41(c(1-lx12)2), we have gNij=opgij and giJ'=11op･g`j. Put. V:=logop. Let fV:TH"(-c)->TM and N:TE"-->TM be the orthogonal projections. Then lg7 =N,' 7W denotes the gradient vector field of ut on {xeEn;. lxl<1} with respect to the Euclidean metric g. Let fi and H be the mean curvature vector fields of the immersions i and i respectively. H is given as follows.. For a point p e M, 7},(M) denotes the tangent space of M at p and N. the normal " Department of Mathematics, Faculty of Education, Yokohama National University..

(2) 18 M. MAEDA. space of M at p in E". The second fundamental form s,: 7)(M)× 7},(M)->N,(M) on M at p is, by definition,. s,(X,Y)-=(7.Y)" fbr (X,Y)G7},(M)×7'},(M) where Y is an extension of Y as a local field. Since the above definition does not. depend on an extension Y of Y, we may write as s.(X, Y)=(7xY)N. Then H is defined as. H(p)=itraces, at pGM. fi is defined similarly. dvN and dv denote the volume elements of (M, i'"gN) and. (M, i"g) respectively. Then dfi= m. op2dv. From these preparation, we have. LEMMA1. H---ll7wNMd,=IAIydfi where l･l and 1･lg are the norms with respect to the metrics g and gN respectively.. PRooF. On U, using the fact [2;IA-l (vi)], mili= :lli gij(it.,ej)N. i,j=1 =EIigij(7.,ej+'It[ei(!p()ej+ej(ipf)ei-gij(J7ipf)])" == 2 g'Vij(J7 .,ej)N - z!- 2 g'Vijg,j J7 !PfN. == 'il)- 2g`j(7.,ej)" - -217s- 2gijgi,･J7 gef". =-Z:H- 2Mop 7WN. Thus opilt == H - -S- 7W".. So Hd-S-7W" =oplfil==q-2ili71g.. -m Thus, noticing dvN =2dv, q. we have. H--li-7get" Mdv=IAI17dv". q.e.d. For any vector field X:M.TE" along M, djvMX, the divergence of X on M, is defined as the trace of the mapping 7X: TM.TE" given by 7X(Y)=(7yX)', Ye TM, where (･)t denotes the tangent part of(･). Thus locally on U, divM-X =:`hi,IIi.ll=igij < 7.,X, ej > ,. where < , > denotes the Riemannian metric g. Then fbllowing lemma can be proved easily..

(3) The lntegral of the Mean Curvature 19 LEMMA2, (1) divMXt=divMX+<X,H>, and for any Ci-function f on M,. (2) div.fX=fdiv.X+<Xt,7f>. For the proof, see [3].. From (1) of Lemma 2 and the divergence theorem for the tangent vector field. X, we have. -jM < X, H>dv= thSM divMx dv. '. Now in the following, we will calculate the integral of the left hand side in Lemma 1, when the dimension ofM is two. So it suffices to calculate the following three terms. j. H- -li-7wN 2dv=:j.<a H> dv -j. < H, 7W> dv +tj. <7utN, 7W> dv. Since W == log op == log-2- -2log(1 -- 1xl2), we see. 7 IPf = 41(1 - ixl2)x. So from (2) of Lemma 2,. divM Vut ==divM i-fxl2 x= i.-lxl2 divMx +<xt, V(i-lxl2 )>･ Since divM x'= 2, see [3; p. 720] and. 7(i-lx12 )= (i- Fx12)2 x, we have diVM7ut=: 1-¥x12 + (1--. rel2)2 <X', X'>,. '. Hence. --j.<IL VW> dv ==S.l1.lxI, + (1- fx1,), <xt, xt>ldv. From this and together with. t'<7WN, 7WN>= (1-fx12)2 <x", x">,we have. -j. <a 7WN> dv + tS.<7WN, 7wN> dv :jMI1-lx12 + (1.-- lx12)2 <xt, xt>+ (1-lx12)2 <xN, xN>ldv.. =SMI1-ixl2 + (1- f.I2)2 lx121dv.

(4) M. MAEDA. 20. =S.L(1-fi9x1252-dV==C･S.opdv. ==. ' cjMdfi =c･volume (M).. Thus we have j.IAt1Z-dfi==e･Volute(M)+S.1Hl2dv. As is stated in the introduction of this note it holds. '. j. IHI2dv )- e2 (=: 4n) 4. and equality sign holds when and only when M is imbedded as a 2-dimensional sphere S2 in En.. For a point pEHn(-c) and r>O, let B,(p): =={qGH"(-c);d(p, q)==r} where d is the distance functidn on H"(-c) induced from the Riemannian metric. g. And for a 3-dimensional totally geodesic submanifold H3(-c) of H"(-c) which contains point p, we will call the set B,(p) n H3(- c) as a geodesic 2-sphere (with radius r centered at p, if necessary).. Thus summarizing the above and changing the notation used above, we have. THEoREM. Let M be a compact 2-dimensional Riemannian manifold immersed isometrically in the n-dimensional simply connected space form H"(-c) of constant non-positive sectional curvature -c. Then for the mean curvature vector field H on M of this immersion, it holds. s 1H12dvM).c･Volume(M)+4n M ' when and only when M is imbedded. and equality sign holds as a geodesic 2-sphere in H"(-c), where IHI is the norm of H with respect to the Riemannian metric of Hn(- c) and dvM is the volume element of M. Putting Z=max {IH(p)1; p E M}, as a corollary of this theorem, we have i. CoRoLLARy. LetMbeamanifoldasintheabovetheorem. Then A> VZ[. ). In particular, there exist no compact minimal surfaces in H"(- c).. CoRoLLARy. Let M be a manifold as in the above theorem. Then Volume(M) l 4zl(A2 - c) and equality sign holds when and only when M is imbedded as a geodesic 2-sphere. in H"(-c) with radius r= j: N/6(12is2) ds where ct is the'positive solution of the equation: ct2+(X-. -4)oc--1==O･.

(5) The Integral of the Mean Curvature. 21. References. [1l B.Y.CHEN, GeometryofSubmanifolds, MarcelDekkerInc.NewYork1973. [2] D.GRoMoLL, W,Klingenberg and W.MEyER, Riemannsche Geometrie im Grossen, Springer-Verlag, 1968.. [3] D.HoFFMAN and J.SpRucK, Sobolev and Isoperimetric Inequalities for Riemannian Submanifolds, Comm. Pure and Appl. Math. XXVII (1974) 715-727.. t. -. '. -.

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