The length of the second fundamental form, a tangency principle and applications

ISSN 0001-3765 www.scielo.br/aabc

The length of the second fundamental form, a tangency principle and applications

FRANCISCO X. FONTENELE1 and SÉRGIO L. SILVA2

1_{Universidade Federal Fluminense, Instituto de Matemática, Departamento de Geometria} 24020-140 Niterói, RJ, Brasil

2_{Universidade Estadual do Rio de Janeiro-UERJ, Departamento de Estruturas Matemáticas-IME} 20550-013 RJ, Brasil

Manuscript received on September 5, 2003; accepted for publication on November 21, 2003; presented byManfredo do Carmo

ABSTRACT

In this paper we prove a tangency principle (see Fontenele and Silva 2001) related with the length of the second fundamental form, for hypersurfaces of an arbitrary ambient space. As geometric applications, we make radius estimates of the balls that lie in some component of the comple-mentary of a complete hypersurface into Euclidean space, generalizing and improving analogous radius estimates for embedded compact hypersurfaces obtained by Blaschke, Koutroufiotis and the authors. The basic tool established here is that some operator is elliptic at points where the second fundamental form is positive definite.

Key words: hypersurfaces, tangency principle, second fundamental form, balls, radius estimates.

1 INTRODUCTION

In (Fontenele and Silva 2001), the same authors proved that if ann-dimensional embedded compact hypersurfaceMninto(n+1)-dimensional Euclidean space satisfies|Hk| ≥ _λ1k,for somek-mean

curvatureHk, 1≤ k≤n,and some positive constantλ,then the greatest ball that fits insideMn

has radius less thanλunlessMn_{is a sphere of radiusλ,generalizing results of (Blaschke 1956) and}

(Koutroufiotis 1973) for surfaces. Our basic tool for the proof of the above result was a tangency principle stated in (Fontenele and Silva 2001) as Theorem 1.1. This tangency principle turned out to be very useful to obtain other geometric applications. In this work we obtain a tangency principle (see Fontenele and Silva 2001) related with the length of the second fundamental form and improve

The first author dedicates this work to his parents José (in memoriam) and Herondina Correspondence to: Francisco X. Fontenele

(see Remark 3.1) the radius estimate for the greatest ball that fits inside a suitable component of the complementary of ann-dimensional complete hypersurface into(n₊1)-dimensional Euclidean space. In order to state our results we need the following.

As in (Fontenele and Silva 2001), given a hypersurfaceMnof a complete Riemannian manifold Nn+1with metric , and exponential mapping exp: T N →N,we parametrize a neighborhood ofMn_{containingpand contained in a normal ball ofN}n+1_putting

ϕ(x)₌expp(x+µ(x)ηo), (1)

wherex varies in a neighborhoodW of zero inTpM (the tangent space toMnatp),ηois a fixed

unitary vector normal toMn atpandµis an unique real function defined inW withµ(0)₌ 0. Letη: W → T_{ϕ(W )}⊥ M be a local orientation ofMn withη(0) =ηo.Denote byAη(x) the second

fundamental form ofMn_{in the directionη(x)and byσ the vector valued second fundamental form}

ofMn_{.The length of the second fundamental form atxis given by}

|σ|2(x)=traceA2η(x).

Ifλ1(x)≤λ2(x)≤ · · · ≤λn(x)are the principal curvatures ofMnatx ∈W,we have that

|σ_|2(x)₌

n

i=1

λ2_i(x).

Given hypersurfacesM₁nandM₂nofNn+1_withT

pM1=TpM2(tangent atp), parametrizeM1nand

M₂n as in (1) obtaining correspondent functionsµ1andµ2.As in (Fontenele and Silva 2001), we say thatM₁n remains aboveM₂nin a neighborhood ofpwith respect toηoifµ1(x) ≥ µ2(x)in a

neighborhood of zero.

Following the ideas in (Fontenele and Silva 2001), we obtain the following tangency principle:

Theorem_1.1. _{Consider hypersurfacesM}n

1 andM n 2 ofN

n+1_{tangent atpandη}

oa unitary vector

normal toM1natp.Denote by|σ1|2(x)and|σ2|2(x)the length of the second fundamental form of

respectivelyMn

1 andM2natx ∈W.Assume thatM1nremains aboveM2nin a neighborhood ofpwith

respect toηo,|σ1|2(x) ≤ |σ2|2(x)in a neighborhood of zero and that the principal curvatures of

M2at zero are all positive. Under these conditions,M1nandM

n

2 must coincide in a neighborhood

ofp.

For hypersurfaces with boundaries and as a consequence of proof of Theorem 1.1, we obtain the following tangency principle:

Theorem_1.2. _LetMn

1 andM2nbe hypersurfaces ofNn+1with boundaries∂M1and∂M2

respec-tively. Suppose thatM₁nandM₂nas well as∂M1and∂M2are tangent atp∈∂M1∩∂M2and letηo

be normal toM₁natp.Denote by_|σ1|2(x)and|σ2|2(x)the length of the second fundamental form

of respectivelyM₁nandM₂natx∈W.Assume thatM₁nremains aboveM₂nin a neighborhood ofp

ofM2at zero are all positive. Under these conditions,M1nandM2nmust coincide in a neighborhood

ofp.

When the ambient is the(n₊1)-dimensional Euclidean space and the hypersurfacesM₁nand M2nhave the same constant length of the second fundamental form, Theorems 1.1 and 1.2 are the

analogous of the well known maximum principles for hypersurfaces with the same constantk-mean curvature in the Euclidean space.

For enunciate our geometric applications let us fix some notation. Given an oriented hyper-surfaceMnof the(n+1)-dimensional Euclidean spaceRn+1_{,thek-mean curvatureH}

k(p)ofMn

atpis given by

Hk(p)=

1 n k

i1<i2<···<ik

λi1λi2. . . λik,

whereλ1≤λ2≤ · · · ≤λnare the principal curvatures ofMnatp.In particular,H1(p)is denoted

by h(p)and called the mean curvature of M atp andH2(p)is denoted byR(p)and called the

scalar curvature of M atp.

The Ricci curvature ofMn_{at a pointpin the direction of an unitary vectorvis given by}

Ricp(v)=

1 n₋1

n

i=2

K(v, wi),

wherew1=v, w2, . . . , wnis an orthonormal basis ofTpMandK(v, wi)is the sectional curvature

of the plane generated byvandwi.

Definition_{1.3. IfUis an open subset of the Euclidean space}Rn+1_{,we define}

ρU :=sup{r >0, such thatUcontains a closed ball inRn+1of radiusr},

whereU denotes the closure ofUinRn+1_.

Condition I. _Mn_{is a complete connected euclidean hypersurface that splitsR}n+1_{into two disjoint}

regions of whichMn_{is the common boundary.}

Theorem_1.4. _{Suppose thatM}n_{satisfies Condition I. Assume further that|σ|}2_≥ n

λ2 and|h|> n−2

n λ

overMn_{,whereλis a positive constant. Under these conditions, if we denote byUthe component}

ofRn+1_{\M that contains the normals for whichhis positive, thenρU ≤λ.Moreover, ifU∪M}

contains a closed ball of radiusλthenMncoincides with a sphere inRn+1_{of radiusλ.}

Corollary_{1.5.Suppose thatM}n_{satisfies Condition I. Assume further that|σ|}2_≥ n

λ2andR > n−4 n λ2

overMn_{,whereλis a positive constant. Then, the mean curvature functionhis positive for some}

suitable orientation ofMnand, denoting byUthe component ofRn+1_{\Mthat contains the normals}

for whichhis positive, we haveρU ≤λ.Moreover, if in the closure ofU there exists a closed ball

Corollary _1.6. _{Suppose that M}n _{satisfies Condition I and that |Hk| ≥} 1

λk on M

n _{for some}

k,1 _≤ k _≤ n,and some positive constantλ > 0.Fork _≥ 2,assume further that there exists at

least one point inMn_{where the second fundamental form is definite. Then there exists a component}

UofRn+1_{\Msuch thatρU ≤λ.Moreover, ifU∪Mcontains a closed ball of radiusλ,thenM}n

coincides with a sphere inRn+1_{of radiusλ.}

Theorem_1.7. _{Suppose thatM}n_{satisfies Condition I and is oriented. Assume thath≥0and that}

|σ_|2

≥ λn2 overMn,whereλis a positive constant. Assume further thatn≥3and RicM ≥ −_(n−1_1)λ2.

Then, if we denote byUthe component ofRn+1_{\Mthat contains the normals, we have ρU ≤λ.}

Moreover, ifU_∪Mncontains a closed ball of radiusλthen eithernmust be even orMncoincides

with a sphere inRn+1_{of radiusλ.}

Corollary_1.8. _{Suppose thatM}n _{satisfies Condition I and that |σ|}2 _≥ n

λ2 overMn, for some

positive constantλ.Consider also that RicM >−_(n−1_1)λ2.Then, the mean curvature functionhis

positive for some suitable orientation ofMn _{and denoting byUthe component ofR}n+1

\Mthat

contains the normals for whichhis positive, we haveρU ≤λ.Furthermore, if in the closure ofU

there exists a closed ball of radiusλ,thenMnmust be a sphere inRn+1_{of radiusλ.}

In the following result,Mn_{is a connected and complete manifold isometrically immersed in}

Rn+1_.

Corollary_1.9. _{Assume thatKM ≥ 0and|σ|}2 _≥ n

λ2 overMn,for some positive constantλ.If

Mn is not compact, assume further that there exists at least one point inMnwhere all sectional

curvatures are positive. Then there exists a componentUofRn+1

\Msuch thatρU ≤λ.Moreover,

ifU_∪Mcontains a closed ball of radiusλthenMn_{coincides with a sphere inR}n+1_{of radiusλ.}

2 SKETCH OF PROOF OF THEOREM 1.1

Fix an orthonormal basise1, e2, . . . , eninTpM1=TpM2and introduce coordinates, forxinTpM1,

settingx =ni=1xiei.As in (1), parametrizeM1andM2in a neighborhood ofpby respectively

ϕ1_andϕ2_{, obtaining respectively functionsµ}1_andµ2_.Letηl

: W _→T⊥

ϕl_{(W )}Ml, l = 1,2, be

a local orientation ofMl withηl(0) = ηo and denote byAηl_(x)the second fundamental form of

Ml in the directionηl(x).Denote byϕil(x)the vector ∂ϕl

∂xi(x)and byA

l_{(x)the matrix ofA} ηl_(x)in

the basisϕ_il(x), 1 ≤ i ≤ n.In (Fontenele and Silva 2001), it is proved the existence of a matrix valued functionA˜ defined inRn(n2+1)+n×N,whereN is a connected open set ofRn+1containing

the origin, such that

˜

A(µl_ij(x), µl_i(x), µl(x), x)=Al(x), x ∈W, l =1,2, (2)

where(µl

ij(x), µli(x), µl(x), x), 1≤i ≤j ≤n,is a point ofRd, d = n(n+ 1)

2 +2n+1.We write

an arbitrary point ofRd _as(r

ij, ri, z, x), 1≤i≤j ≤n,andx =(x1, . . . , xn).Define a function

_: Rn(n2+1)+n×N →Rby

Now using the derivatives ofA,˜ with respect to therkl’s, given in (Fontenele and Silva 2001), we

obtain that

n

k≤l=1

∂ ∂rkl

( (1−t ) µ2ij(0)+t µ 1

ij(0) , 0, 0, 0)ξkξl =2At(0) ξ, ξ,

where At₍₀₎

= ˜A( (1₋t ) µ2

ij(0)+t µ1ij(0) , 0, 0, 0).Using thatM1n remains aboveM2n in a

neighborhood ofpwith respect toηo,|σ1|2(x)≤ |σ2|2(x)in a neighborhood of zero and that the

principal curvatures ofM2at zero are all positive, one can prove thatis elliptic in((1−t ) µ2ij(0)+

t µ1_ij(0) , 0, 0, 0)fort _{∈ [}0,1_],and, restrictingW if necessary, conclude thatis elliptic with respect to the functions(1₋t )µ2+t µ1, t ∈ [0,1_](see Fontenele and Silva 2001).

Recalling that_|σl|2(x)=trace[Al(x)]2,it follows from (2), (3) and our assumptions that

(µ2_ij(x), µ2_i(x), µ2(x), x)_{= |}σ2|2(x)≥ |σ1|2(x)=(µ1ij(x), µ 1 i(x), µ

1_{(x), x), x} ∈W.

Now the conclusion of the theorem is obtained from the following maximum principle ( Alexandrov 1962):

Maximum Principle. _{Letf, g: U →}R_beC2_{-functions defined in an open setUofR}n_{and let}

: Ŵ⊂Rd

→R_{be a function of classC}1_{.Suppose thatis elliptic with respect to the functions}

(1₋t )f ₊tg, t _{∈ [}0,1_].Assume also that

(fij(x), fi(x), f (x), x)≥(gij(x), gi(x), g(x), x), ∀x ∈U,

and thatf _≤gonU.Then,f < g onU unlessf andgcoincide in a neighborhood of any point

xo∈U such thatf (xo)=g(xo).

Now we will prove Theorem 1.7 for give an idea of how one can use Theorem 1.1 to obtain geometric applications.

3 PROOF OF THEOREM 1.7

Proof of Theorem _1.7. _{Consider in} Rn+1 _{an arbitrary closed ballB}

r[po], centered atpo and

radiusr,contained inU _∪M.MoveB_{r[po]until its boundary}S_{r[po] touchesM}n _{the first time.}

Letpbe a tangency point betweenMn_andS

r[po].Denote byλ1 ≤ λ2 ≤ · · · ≤λnthe principal

curvatures of Mn atp. It is well known thatλi ≤ 1_r for all i.Since λ1+λ2+ · · · +λn ≥ 0 by

assumption, we have

λ1+λ2+ · · · +λi+ · · · +λn≥ −λi,

whereλi means thatλihas been omitted on the sum. Therefore, for a negativeλi,we deduce that

1

whereei stands for an unitary eigenvector with eigenvalueλi.The above inequality, gives−1_λ ≤

λi <0 for a negativeλi.We consider two possibilities:

P.1. There exists atpat least one negativeλi.Denoting bytthe number ofλi,s that are negative

and using our assumption on the length of the second fundamental form, we get

n λ2 ≤

t

j=1

λ2_{j +}

n

j=t+1

λ2_{j ≤} t λ2+

n₋t r2 .

Thus,λ≥ r.In caser =λ,we have thatλ1 =λ2 = · · · =λt = −1_λ andλt+1 = · · · =λn = _λ1.

Sinceh_≥0,we obtain thatn_≥2t.On the other hand, we have

1

λ2 ≥ −(n−1)Ric(e1)= −λ1( λ2+ · · · +λn)=

n₋2t₊1 λ2 ,

which implies thatn_≤2t.Hence,n₌2tandnis even. Notice also thatMn_{is minimal atpand}

the Ricci curvature atpis constant and equal to_−(n−1_1)λ2.

P.2.Atp,allλi,sare nonnegative. In this case,

n λ2 ≤λ

2 1+λ

2

2+ · · · +λ 2 n ≤

n r2.

Thusλ_≥r.Ifr ₌λ,it follows easily thatλ1=λ2= · · · =λn= 1_λ.Using Theorem 1.1 and noting

that_λn2 is the constant value of the length of the second fundamental form of a sphere having radius

r,we obtain thatMnandS_{r[po]coincide in a neighborhood ofp.By an argument of connectness,}

we conclude thatMnis equal toS_r[po].

Remark_3.1. _{We point out that Corollary 1.6 extends to complete hypersurfaces Theorem 1.3 in}

(Fontenele and Silva 2001) and that its hypothesis are stronger than those in Theorem 1.4. In fact, fork=1,this follows from the well known inequality_|σ|2

≥n h2and, fork >1,this follows from Lemma 1 in (Montiel and Ros 1991). Furthermore, the estimate forρU in Theorem 1.4 improves

the one given by Corollary 1.6. For if inf_|h_{| = H}1

o >0 and inf|σ| 2

= λn2

o with|h|> n−2

nλo,the upper

bound given forρU in Theorem 1.4 isλoand the one given by Corollary 1.6 isHo.Thatλo≤ Ho

follows from the well known inequality|σ|2

≥n h2.For example, in the cilinderC =S1 ×R_in R3_{,oriented by the normals pointing to the componentU ofR}3

\C containing the origin,where S1_{is the unitary circle, the mean curvature and length of the second fundamental form are given}

respectively by 1₂and 1. The estimate given by Theorem 1.4 isρU ≤ √

2,while the estimate given by Corollary 1.6 isρU ≤2.

RESUMO

Neste trabalho nós provamos um princípio de tangência (veja Fontenele and Silva 2001) para

componente conexa do complemento de uma hipersuperfície completa do espaço Euclidiano, generalizando

e melhorando estimativas de raios análogas obtidas por Blaschke, Koutroufiotis e os autores. O fato básico

estabelecido aqui é que um determinado operador é elíptico nos pontos onde a segunda forma fundamental

é positiva definida.

Palavras-chave: hipersuperfícies, princípio de tangência, segunda forma fundamental, bolas, estimativas

de raios.

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2, 21: 341-416.

Blaschke W._{1956. Kreis und Kugel. 2. Auflage, Berlin.}

Fontenele F and Silva SL._{2001. A tangency principle and applications, Illinois J Math, 45: 213-228.}

Koutroufiotis D._{1973. Elementary geometric applications of a maximum principle for nonlinear elliptic}

operators, Arch Math, 24: 97-99.

Montiel S and Ros A._{1991. Compact hypersurfaces: the Alexandrov theorem for higher order mean}