Integral Formula of Minkowski Type and New Characterization of the Wulff Shape

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1 Acta athematca Snca, Engsh Seres Apr., 2008, Vo. 24, No. 4, pp Pubshed onne: Apr 5, 2008 DOI: 0.007/s Acta athematca Snca, Engsh Seres The Edtora Offce of AS & Sprnger-Verag 2008 Integra Formua of nkowsk Type and New Characterzaton of the Wuff Shape Y Jun HE Department of athematca Scences, Tsnghua Unversty, Beng 00084, P. R. Chna and Schoo of athematca Scences, Shanx Unversty, Tayuan , P. R. Chna E-ma: heyun@sxu.edu.cn Ha Zhong LI Department of athematca Scences, Tsnghua Unversty, Beng 00084, P. R. Chna E-ma: h@math.tsnghua.edu.cn Abstract Gven a postve functon F on S n whch satsfes a convexty condton, we ntroduce the r-th ansotropc mean curvature r for hypersurfaces n R n+ whch s a generazaton of the usua r-th mean curvature H r. We get ntegra formuas of nkowsk type for compact hypersurfaces n R n+. We gve some new characterzatons of the Wuff shape by the use of our ntegra formuas of nkowsk type, n case F whch reduces to some we-known resuts. Keywords Wuff shape, F -Wengarten operator, ansotropc prncpa curvature, r-th ansotropc mean curvature, ntegra formua of nkowsk type R(2000) Subect Cassfcaton 53C42, 53A30, 53B25 Introducton Let F : S n R + be a smooth functon whch satsfes the foowng convexty condton: (D 2 F + F ) x > 0, x S n, () where D 2 F denotes the ntrnsc Hessan of F on S n and denotes the dentty on T x S n, > 0means that the matrx s postve defnte. We consder the map φ: S n R n+, x F (x)x + (grad S nf) x, ts mage W F φ(s n ) s a smooth, convex hypersurface n R n+ caed the Wuff shape of F (see [ 5]). Now et X : R n+ be a smooth mmerson of a compact, orentabe hypersurface wthout boundary. Let ν : S n denotes ts Gauss map, that s, ν s a unt nner norma vector of. Let A F D 2 F +F, S F A F dν. S F s caed the F -Wengarten operator, and the egenvaues of S F are caed ansotropc prncpa curvatures. Let σ r be the eementary symmetrc functons of the ansotropc prncpa curvatures λ,λ 2,...,λ n : σ r λ λ r ( r n). < < r Receved arch 3, 2007, Accepted June 20, 2007 The frst author s supported partay by Tanyuan Fund for athematcs of NSFC (Grant No ) The second author s supported partay by Grant No of the NSFC and by Doctora Program Foundaton of the nstry of Educaton of Chna (2006)

2 698 He Y. J. and L H. Z. We set σ 0. Ther-ansotropc mean curvature r s defned by r σ r /Cn, r whch was ntroduced by Rey n [6]. In ths paper we frst gve the foowng ntegra formuas of nkowsk type for compact hypersurfaces n R n+. Theorem. Let X : R n+ be an n-dmensona compact hypersurface, F : S n R + be a smooth functon whch satsfes (). Then we have the foowng ntegra formuas of nkowsk type hod : (F r + r+ X, ν )da X 0, r 0,,...,n. (2) By the use of the above ntegra formuas of nkowsk type, we prove the foowng new characterzatons of the Wuff shape: Theorem.2 Let X : R n+ be an n-dmensona compact hypersurface, F : S n R + be a smooth functon whch satsfes (), and constand X, ν has fxed sgn. Then up to transatons and homothetes, X() s the Wuff shape. Theorem.3 Let X : R n+ be an n-dmensona compact hypersurface, F : S n R + be a smooth functon whch satsfes (). If constand r constfor some r, 2 r n, thenupto transatons and homothetes, X() s the Wuff shape. Theorem.4 Let X : R n+ be an n-dmensona compact convex hypersurface, F : S n R + be a smooth functon whch satsfes (). If r k constfor some k and r, wth0 k<r n, thenup to transatons and homothetes, X() s the Wuff shape. Theorem.5 Let X : R n+ be an n-dmensona compact hypersurface, F : S n R + be a smooth functon whch satsfes (). If k n constfor some k, wth0 k n, thenupto transatons and homothetes, X() s the Wuff shape. Choosng k 0 n Theorem.4, we get Coroary. Let X : R n+ be an n-dmensona compact convex hypersurface, F : S n R + be a smooth functon whch satsfes (), and for a fxed r wth r n, r const. Then up to transatons and homothetes, X() s the Wuff shape. Remark. When F, Wuff shape s ust the round sphere and r H r, formua (2) reduces to the cassca nkowsk ntegra formua (see [7] or [8]). Theorem.2 reduces to the cassca Theorem gven by Süss [4], Coroary. reduces to Theorem of Yano [9], Theorem.3 reduces to Theorem of Choe [0]. We aso note that n [], the authors proved the ntegra formua of nkowsk type for compact spaceke hypersurfaces n de Stter space. 2 Premnares Let {E,...,E n } s a oca orthogona frame on S n,ete E ν, where,...,n. Then {e,...,e n } s a oca orthogona frame of X : R n+. The structure equaton of S n s: dx θ E, de θ E θ x, dθ θ θ, dθ θ k θ k R k θ k θ θ θ, 2 k k (3) where θ + θ 0and R k δ k δ δ δ k. (4)

3 Integra Formua of nkowsk type, Wuff Shape 699 The structure equaton of X s (see [2], [3]): dx ω e, dν h ω e, de ω e + h ω ν, (5) dω dω k ω ω, ω k ω k R k θ k θ, 2 where ω + ω 0,R k + R k 0,andR k are the components of the Remannan curvature tensor of wth respect to the nduced metrc dx dx. From de d(e ν) ν de ν θ e ν θ ν,weget k { ω ν θ, ν θ h ω, (6) where ω + ω 0,h h. Let F : S n R + be a smooth functon. We denote the coeffcents of covarant dfferenta of F, grad S nf, D 2 F wth respect to {E },...,n by F,F,F k respectvey. From Rcc dentty and (4), we have F k F k m F m Rmk δ k F δ F k, (7) where F k denote the coeffcents of the covarant dfferenta of F on S n. So, f we denote the coeffcents of A F by A,thenwehavefrom(7) A k A k A k, (8) where A k denote the coeffcents of the covarant dfferenta of A F on S n. Let s k (A k ν)h k, S F A F dν. Then we have S F (e ) s e. We ca S F the F -Wengarten operator. From the postve defnteness of (A ) and the symmetry of (h ), we know the egenvaues of (s )arearea(nfact,becausea (A ) s postve defnte, there exsts a nonsnguar matrx C such that A C t C,wehaveS (s )AB has the same egenvaues wth the rea symmetrc matrx CBC T, whch foows from λi S λi AB λi C t CB λi CBC t, where B (h )). We ca them ansotropc prncpa curvatures, and denote them by λ,...,λ n. We have n nvarants, and the eementary symmetrc functon σ r of the ansotropc prncpa curvatures: σ r λ λ n ( r n). (9) < r For convenence, we set σ 0. Ther-ansotropc mean curvature r s defned by r σ r /Cn, r Cn r n! r!(n r)!. (0) Usng the characterstc poynoma of S F, σ r s defned by So, we have σ r r! det(ti S F ) n ( ) r σ r t n r. () r0,..., r ;,..., r δ r r s s r r, (2)

4 700 He Y. J. and L H. Z. where δ r r s the usua generazed Kronecker symbo,.e., δ r r equas + (resp. ) f r are dstnct and ( r ) s an even (resp. odd) permutaton of ( r ) and n other cases t equas zero. We defne (F ν), (F ν), (A ν) k by d(f ν) (F ν) ω, (3) d(f ν)+ (F ν)ω (F ν) ω, (4) d(a ν)+ (A k ν)ω k + k (A k ν)ω k k (A ν) k ω k. (5) By the use of (3), (5) and (6), we have by a drect cacuaton (F ν) h F ν, (F ν) h k F k ν, k (6) (A ν) k h k A ν. 3 Some Lemmas We ntroduce an mportant operator P r (aso see Rey [6]) by P r σ r I σ r S F + +( ) r SF r, r 0,,...,n. (7) We have the foowng emmas: Lemma 3. (S F A F ) t S F A F, (dν S F ) t dν S F, s k s k, h s k h ks, h k(p r ) h (P r ) k,wheres k are the components of the covarant dervarve of s. Proof Snce S F A F dν, anda F,dν are symmetrc operators, the frst two denttes are obvous. From the symmetry property (8) of A k, h h and Codazz equaton h k h k,wehave,bythe use of (6), ( ) s k A h (A ν) k h + A h k k (A m ν)h h km + A h k,m (A m ν) h mk + ( ) A h k A h k s k. (8) m h s k h A m h mk h km A m h h k s.,m,m By the use of the above formua and the defnton of P r, we get the ast dentty n Lemma 3.. Lemma 3.2 The matrx of P r s gven by : (P r ) δ r r! r s s r r. (9),..., r ;,..., r Proof We prove Lemma 3.2 nductvey. For r 0, t s easy to check that (9) s true. We can check drecty δ δ 2 δ q δ q δ 2 δ 2 2 δ q 2 δ q 2 δ q q... (20) δ q δ 2 q δ q q δ q q δ q δ 2 q δ q q δ q q

5 Integra Formua of nkowsk type, Wuff Shape 70 Assume that (9) s true for r k, we ony need to show that t s aso true for r k +. For r k +, usng (2) and (20), we have RHS of (9) (k +)! (k +)! (k +)! δ k+ k+ s s k+ k+,..., k+ ;,..., k+ δ δ 2 δ k+ δ δ 2 δ 2 2 δ k+ 2 δ 2.. s s k+ k+ δ k+ δ 2 k+ δ k+ k+ δ k+ δ δ 2 δ k+ δ (δ δ k+ k+ δ k+ δ k k k+ + )s s k+ k+ σ k+ δ (k +)! σ k+ δ (P k ) k+ s k+ (P k+ ). δ k+ δ k k k+ s s k+ k+ + Lemma 3.3 For each r, we have () (P r) 0, () tr(p r S F )(r +)σ r+, () tr(p r )(n r)σ r. Proof () Notng (, r ) s skew-symmetrc n δ r r and (, r ) s symmetrc n s s r r (from Lemma 3.), we have (P r ) δ r (r )! r s s r r 0. () Usng (9) and (2), we have,, r ;,, r ; tr(p r S F ) (P r ) s r!,..., r ;,..., r ;, (r +)σ r+. () Usng () and the defnton of P r,wehave δ r r s s r r s tr(p r )tr(σ r I) tr(p r S F )nσ r rσ r (n r)σ r. Remark 3. When F, Lemma 3.3 s a we-known resut (for exampe, see Barbosa Coares [4]). Lemma 3.4 If λ λ 2 λ n const 0, then up to transatons and homothetes, X() s the Wuff shape. Proof Choose a oca orthogona frame e,e 2,...,e n such that A F s dagonazed: A F dag(μ,...,μ n ), (2) where μ > 0for,...,n by the convexty condton. Then we have S μ h. From (0) and (2), we get ( n ) 2 μ h 2 n(n ) μ μ (h h h 2 ) <

6 702 He Y. J. and L H. Z. { ( ) 2 (n ) μ n 2 h 2n } μ μ (h h h 2 ) (n ) < {(μ n 2 h μ h ) 2 +2nμ μ h 2 }, (n ) < so, μ h μ 2 h 22 μ n h nn and h 0when. Then, from [] or [3], [5], up to transatons and homothetes, X() sthewuffshape. 4 Proofs of Theorem. Theorem.5 Proof of Theorem. By the use of (5), we have X, ν h X, e, X, e δ + h X, ν, (22) so, from (6), Lemma 3. and (), (), () of Lemma 3.3, we have the foowng cacuaton dv{p r ( X, ν grad S nf Fgrad X 2 /2)} {(P r ) ( X, ν F F X, e )} { (P r ) } h k ( X, e k F + X, ν F k F k X, e ) Fδ Fh X, ν k k h k (P r ) X, e k F + k h k (P r ) X, e F k X, ν k (P r ) (F k + Fδ k )h k F (P r ) k h k (P r ) X, e k F + k h (P r ) k X, e k F X, ν k (P r ) A k h k F (P r ) X, ν (P r ) s F (P r ) X, ν tr(p r S F ) F tr(p r ) X, ν (r +)σ r+ F (n r)σ r (n r)cn(f r r + r+ X, ν ). Integratng the above formua over, we get (2) by the use of Stokes Theorem. Proof of Theorem.2 From (2), we have (F + X, ν )da X 0, (23) (F + 2 X, ν )da X 0. (24) By the assumpton const, we get from (23) and (24) X, ν ( 2 2 )da X 0. (25) On the other hand, 2 2 n 2 (n ) (λ λ ) 2 0. (26) <

7 Integra Formua of nkowsk type, Wuff Shape 703 Thus, f X, ν has fxed sgn, then 2 2 0,so λ λ 2 λ n. Thus, from Lemma 3.4, up to transatons and homothetes, X() sthewuffshape. Proof of Theorem.3 We have the fact that f s compact and r > 0 then r (r )/r r, 2 r n (27) wth equaty hodng f and ony f λ λ 2 λ n on (cf. [0], [6]). Indeed (27) hods f r const, snce s compact, there exsts a pont p 0 on such that a prncpa curvatures are postve at p 0, so a ansotropc prncpa curvatures are postve at p 0. Appyng (27) nductvey, one sees that f r const, then r r, (28) here agan equaty hods f and ony f λ λ 2 λ n. Integratng F r (r )/r F r over, usng (2) and r const,weget F da X F r da X r X, ν da X. (29) (r )/r r On the other hand, our assumpton const (thus > 0) and (23) mpes X, ν da X FdA X. (30) Puttng (30) nto (29), we get r r. (3) Therefore equaty hods n (28) and λ λ 2 λ n on. Thus, from Lemma 3.4, up to transatons and homothetes, X() sthewuffshape. Proof of Theorem.4 From (2), we have (F k + k+ X, ν )da X 0, (32) (F r + r+ X, ν )da X 0. (33) From the assumptons r k const, r k (32) (33) mpes X, ν ( r+ r k+ )da X 0. (34) k From the convexty of, a the prncpa curvatures of are postve, so a the ansotropc prncpa curvature are postve, we have > 0, 0 n on. From (see [7]) k k+2 2 k+,..., r r+ 2 r, (35) where equaty hods n one of (35) f and ony f λ λ 2 λ n,wecancheck k r+ k+ r, that s, r+ r k+ 0. (36) k On the other hand, we can choose the poston of orgn O such that X, ν has fxed sgn. Thus, from (34) and (36), k r+ k+ r, so λ λ 2 λ n. Thus, from Lemma 3.4, up to transatons and homothetes, X() sthewuffshape.

8 704 He Y. J. and L H. Z. Proof of Theorem.5 From the proof of Theorem.3, we know that there exsts a pont p 0 such that the ansotropc prncpa curvature λ (p 0 ) > 0, n. From k constant, we have k n k n (p 0 ) > 0. Thus n 0on, by the contnuty of λ,wehaveλ > 0, n, on. Therefore, a prncpa curvatures of are postve on, and s convex. Theorem.5 foows from Theorem.4. References [] Carenz, U.: The Wuff-shape mnmzes an ansotropcwmore functona, Interfaces and Free Boundares, 6, (2004) [2] Koso,., Pamer, B.: Geometry and stabty of surfaces wth constant ansotropc mean curvature. Indana Unv. ath. J., 54, (2005) [3] Pamer, B.: Stabty of the Wuff shape. Proc. Amer. ath. Soc., 26, (998) [4] Süss, W.: Zur reatven Dfferentageometre. V., Tôhoku ath. J., 30, (929) [5] Tayor, J.: Crystane varatona probems. Bu. Amer. ath. Soc., 84, (978) [6] Rey, R.: The reatve dfferenta geometry of nonparametrc hypersurfaces. Duke ath. J., 43, (976) [7] Hsung, C. C.: Some ntegra formuas for cosed hypersurfaces. ath. Scand., 2, (954) [8] Smon, U.: nkowsksche ntegraformen und hre Anwendungen n der Dfferentageometre n Grossen. ath. Ann., 73, (967) [9] Yano, K.: Integra formuas n Remannan geometry, arce, Dekker, N. Y, 970 [0] Choe, J.: Suffcent condtons for constant mean curvature surfaces to be round. ath. Ann., 323, (2002) [] L, H., Chen, W. H.: Integra formuas for compact spaceke hypersurfaces n de Stter space and ther appcatons to Goddard s conecture. Acta athematca Snca, New Seres, 4, (998) [2] L, H.: Hypersurfaces wth constant scaar curvature n space forms. ath. Ann., 305, (996) [3] L, H.: Goba rgdty theorems of hypersurfaces. Ark. ath., 35, (997) [4] Barbosa, J. L.., Coares, A. G.: Stabty of hypersurfaces wth constant r-mean curvature. Ann. Goba Ana. Geom., 5, (997) [5] Wnkmann, S.: A note on the stabty of the Wuff shape. Arch. ath., (2006) [6] onte, S., Ros, A.: Compact hypersurfaces: The Aexandrov theorem for hgher order mean curvatures, n Lawson, B. and Tenenbat, K. (eds), Dfferenta Geometry, Ptman onographs, Vo. 52, Longman, Essex, 99, pp [7] Hardy, G. H., Lttewood, J. E., Poya, G.: Inequates, Cambrdge Unv. Press, London, 934 n