局部對(duì)稱空間中線性Weingarten超曲面

張亞娟, 劉建成

(西北師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 蘭州 730070)

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局部對(duì)稱空間中線性Weingarten超曲面

張亞娟, 劉建成*

(西北師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 蘭州 730070)

研究了局部對(duì)稱空間中具有有界平均曲率的可定向完備線性Weingarten超曲面的剛性分類. 在對(duì)超曲面的全臍張量Φ的模長(zhǎng)進(jìn)行適當(dāng)限制下,應(yīng)用廣義極大值原理,得到了該類超曲面是全臍的或等距于一個(gè)具有2個(gè)不同主曲率的超曲面,且其中一個(gè)主曲率的重?cái)?shù)為1.

線性Weingarten超曲面; 局部對(duì)稱; δ-拼擠; 平均曲率

1 引言及主要結(jié)果

上述結(jié)論都是針對(duì)常數(shù)數(shù)量曲率超曲面而言的. 弱化這個(gè)條件的一種做法是考慮平均曲率H與標(biāo)準(zhǔn)數(shù)量曲率R滿足線性關(guān)系R=aH+b的超曲面,其中a,b,并把這種超曲面稱為線性Weingarten超曲面. 近幾年許多文獻(xiàn)對(duì)線性Weingarten超曲面進(jìn)行了相關(guān)研究,參見(jiàn)文獻(xiàn)[3-14],其中文獻(xiàn)[3]證明了單位球面Sn+1(1)中具有非負(fù)截曲率的線性Weingarten超曲面Mn,如果滿足(n-1)a2+4n(b-1)≥0,那么Mn要么是全臍的，要么等距于環(huán)面Sn-k(c1)×Sk(c2),其中1≤k≤n-1,c1,=1. 文獻(xiàn)[12]研究了雙曲空間Hn+1中的具有有界平均曲率的線性Weingarten超曲面Mn,通過(guò)對(duì)Mn的全臍張量Φ的模長(zhǎng)進(jìn)行恰當(dāng)?shù)南拗?得到了該類超曲面的剛性分類結(jié)果.

本文將更一般地研究外圍空間是局部對(duì)稱δ-拼擠黎曼流形中線性Weingarten超曲面的性質(zhì),得到了如下定理.

取Nn+1為單位球面Sn+1(1),則δ=c0=1,于是有

推論1 設(shè)Mn是Sn+1(1)中可定向完備超曲面. 假定Mn的平均曲率H有界,且標(biāo)準(zhǔn)數(shù)量曲率R與其平均曲率H滿足R=aH+b,a,b,a≤0,b>1. 如果

2 預(yù)備知識(shí)及引理

限制在超曲面Mn上時(shí),有

用hijk、hijkl表示hij的一階和二階協(xié)變微分,則

hijk-hikj=-K(n+1)ijk,

(2)

12△S=12 ∑ >i,j △h2ij=∑ i,j,k h2ijk+∑ i,j hij△hij=∑ i,j,k h2ijk+

(4)

(5)

等號(hào)成立當(dāng)且僅當(dāng)至少有n-1個(gè)μi相等.

證明 根據(jù)式(1)有

(6)

注意到Nn+1是一個(gè)局部對(duì)稱空間,對(duì)式(6)兩邊同時(shí)求協(xié)變微分,可得

因此

由Cauchy-Schwarz不等式得

(2n2H-n(n-1)a)2|H|2.

(7)

(2n2H-n(n-1)a)2-4n2S=

4n4H2+n2(n-1)2a2-4n3(n-1)aH-

4n4H2+n2(n-1)2a2-4n3(n-1)aH-

4n2(n(n-1)+n2H2-n(n-1)(aH+b))=

n2(n-1)2a2+4n3(n-1)(b-1)=

n2(n-1)((n-1)a2+4n(b-1))≥0.

(8)

結(jié)合式(7)、(8)可得

(9)

(10)

3 主要結(jié)果的證明

定理1的證明 由式(10)可得

(11)

△(n2H2-n(n-1)(aH+b))=

(12)

由式(4)、(11)、(12)可得

(13)

令μi=i-H,則. 由引理3可知H≠0,因此不妨設(shè)H>0. 由條件及引理1可得

(14)

(15)

nc0(n2H2-S)=-nc0|Φ|2.

(16)

所以由引理2及式(13)～(16)可得

2nδ|Φ|2-nc0|Φ|2=|Φ|2PH(|Φ|),

(17)

其中

因此

(18)

(nH)≥|Φ|2PH(|Φ|)≥0.

(19)

另一方面,由引理4可知,存在點(diǎn)列{pk}?Mn,使得

(20)

由式(1)可得

(21)

注意到定理1的條件a≤0,結(jié)合式(20)、(21)可得

(22)

結(jié)合式(19)、(20)、(22)可得

sup|Φ|2Psup H(sup|Φ|)≥0.于是有sup|Φ|2Psup H(sup|Φ|)=0,繼而可知Φ0,Mn是全臍的,或. 如果存在點(diǎn)pMn,使得,則a≤0時(shí)H在p點(diǎn)也取得最大值. 由引理3可知,算子是橢圓算子,結(jié)合式(19),由Hopf極大值原理可以得到H在Mn上為常數(shù). 因此在Mn上|Φ|于是式(5)等號(hào)成立,因此Mn等距于具有2個(gè)不同主曲率的超曲面,且其中1個(gè)主曲率的重?cái)?shù)為1. 定理1證畢.

0=∫M(nH)*1≥∫M|Φ|2PH(|Φ|)*1.

(23)

[1]CHENGSY,YAUST.Hypersurfaceswithconstantscalarcurvature[J].MathematicscheAnnalen,1977,225(3):195-204.[2]SHUSC.Completehypersurfaceswithconstantscalarcurvatureinahyperbolicspace[J].BalkanJournalofGeometryandItsApplications,2007,12(2):107-115.

[3]LIHZ,SUHYJ,WEIGX.LinearWeingartenhypersurfacesinaunitsphere[J].BulletinoftheKoreanMathematicalSociety,2009,46(2): 321-329.

[4]SHUSC.LinearWeingartenhypersurfacesinarealspaceform[J].GlasgowMathematicalJournal,2010,52(3):635-648.

[5]LPEZR.RotationallinearWeingartensurfacesofhyperbolictype[J].IsraelJournalofMathematics,2008,167(1):283-301.

[6]BARROSA,SILVAJ,SOUSAP.RotationallinearWeingartensurfacesintotheEuclideansphere[J].IsraelJournalofMathematics,2012,192(2):819-830.

[7]YANGD.LinearWeingartenspacelikehypersurfacesinlocallysymmetricLorentzspace[J].BulletinoftheKoreanMathematicalSociety,2012,49(2):271-284.

[8]DELIMAHF,DELIMAJR.CompletelinearWeingartenspacelikehypersurfacesimmersedinalocallysymmetricLorentzspace[J].ResultsinMathematics,2013,63(3/4):865-876.

[9]AQUINOCP,DELIMAHF.OnthegeometryoflinearWeingartenhypersurfacesinthehyperbolicspace[J].MonatsheftefürMathematik,2013,171(3/4):259-268.

[10]AQUINOCP,DELIMAHF,VELSQUEZMAL.AnewcharacterizationofcompletelinearWeingartenhypersurfacesinrealspaceforms[J].PacificJournalofMathematics,2013,261(1):33-43.

[11]AQUINOCP,DELIMAHF,VELSQUEZMAL.Gene-ralizedmaximumprinciplesandthecharacterizationoflinearWeingartenhypersurfacesinspaceforms[J].MichiganMathematicalJournal,2014,63(1):27-40.[12]AQUINOCP,DELIMAHF,VELSQUEZMAL.LinearWeingartenhypersurfaceswithboundedmeancurvatureinthehyperbolicspace[J].GlasgowMathematicalJournal,2015,57(3):653-663.

[13]CHAOXL,WANGPJ.LinearWeingartenhypersurfacesinlocallysymmetricmanifolds[J].BalkanJournalofGeometryandItsApplications,2014,19(2):50-59.

[14]CHAOXL.LinearWeingartenhypersurfacesinaunitsphere[J].BulletinoftheIranianMathematicalSociety,2015,41(2):353-362.

[15]OKUMURAM.Hypersurfacesandapinchingproblemonthesecondfundamentaltensor[J].AmericanJournalofMathematics,1974,96(1):210.

【中文責(zé)編：莊曉瓊 英文責(zé)編：肖菁】

Linear Weingarten Hypersurfaces in Locally Symmetric Manifolds

ZHANG Yajuan， LIU Jiancheng*

(College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China)

The rigidity classifications for complete orientable linear Weingarten hypersurfaces with bounded mean curvature in locally symmetric manifold are studied. By supposing a suitable restriction on the norm of the totally umbilical tensorΦand using the generalized maximum principle, it is proved that such a hypersurface must be either totally umbilical or isometric to the hypersurface with two distinct principal curvatures, one of which is simple.

linear Weingarten hypersurfaces; locally symmetric;δ-pinching; mean curvature

2016-01-21 《華南師范大學(xué)學(xué)報(bào)(自然科學(xué)版)》網(wǎng)址：http://journal.scnu.edu.cn/n

國(guó)家自然科學(xué)基金項(xiàng)目(11261051)

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*通訊作者:劉建成,教授,Email:liujc@nwnu.edu.cn.