Vanishing theorems for L2 harmonic 1forms on complete submanifolds in a Riemannian manifold

Let M be an ndimensional complete orientable noncompact hypersurface in a complete Riemannian manifold of nonnegative sectional curvature. For 2 ≤ n ≤ 6, we prove that if M satisfies the δstability inequality (0 < δ ≤ 1), then there is no nontrivial L2β harmonic 1form on M for some constant β. We also provide sufficient conditions for complete hypersurfaces to satisfy the δstability inequality. Moreover, we prove a vanishing theorem for L2 harmonic 1forms on M when M is an ndimensional complete noncompact submanifold in a complete simplyconnected Riemannian manifold N with sectional curvature KN satisfying that −k2 ≤ KN ≤ 0 for some constant k.

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

Nguyen Thac Dunga, Keomkyo Seob, ∗

a Department of Mathematics, Mechanics, and Informatics (MIM), Hanoi University of Sciences

(HUS–VNU), Vietnam National University, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

bDepartment of Mathematics, Sookmyung Women’s University, Hyochangwongil 52, Yongsan-ku,

Seoul, 140-742, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:

Received 30 June 2014

Available online xxxx

Submitted by H.R Parks

Keywords:

δ-Stabilityinequality

L2 harmonic 1-form

Traceless second fundamental form

First eigenvalue

Let M be an n-dimensional complete orientable noncompact hypersurface in a complete Riemannian manifold of nonnegative sectional curvature For 2≤ n ≤ 6,

we prove that ifM satisfiesthe δ-stabilityinequality (0 < δ ≤ 1), then there is

no nontrivial L 2β harmonic 1-form on M for some constant β. We also provide sufficient conditions for complete hypersurfaces to satisfy theδ-stabilityinequality Moreover, we prove a vanishing theorem forL2 harmonic 1-forms onM when M isan

n-dimensionalcomplete noncompact submanifold in a complete simply-connected Riemannian manifoldN withsectional curvatureK Nsatisfying that−k2≤ K N ≤ 0

for some constantk.

© 2014 Elsevier Inc All rights reserved.

LetM nbeann-dimensionalorientableminimalhypersurfaceinaRiemannianmanifoldN ofnonnegative

f ∈ C ∞

0 (M )



M

|∇f|2−|A|2+ Ric(ν, ν)

anddv isthevolumeform onM

* Corresponding author.

E-mail addresses:dungmath@yahoo.co.uk (N.T Dung), kseo@sookmyung.ac.kr (K Seo).

URL:http://sookmyung.ac.kr/~kseo (K Seo).

http://dx.doi.org/10.1016/j.jmaa.2014.10.076

0022-247X/© 2014 Elsevier Inc All rights reserved.

Ontheotherhand,foranumber0< δ ≤ 1,itiscalledδ-stableifanyfunctionf ∈ C ∞



M

|∇f|2− δ|A|2+ Ric(ν, ν)

It isobviousthatδ1-stability impliesδ2-stability for0< δ2< δ1≤ 1.Inparticular,ifM is stable,then M

is δ-stablefor0< δ ≤ 1.

(See[3,8,10,19,27]andreferencesthereinformoredetails.)Itiswell-knownthattheonlycompleteorientable

M |A|2dv < ∞,then M is a hyperplane

M |A| n dv < ∞ is a hyperplane Recently, Tam and Zhou proved that a complete

n−2

For an n-dimensional complete orientable noncompact (not necessarily minimal) hypersurface M ina

that if M satisfies the stability inequality (1.1), then there is no nontrivial L2 harmonic 1-form on M ,

−k2≤ KN ≤ 0 forsomeconstantk,itturnsoutthatiftheL2normφnofthetraceless secondfundamental

parabolic.WenotethatM isnon-parabolicprovidedithasanon-constantpositivesuperharmonicfunction

B p (r) ⊂ M,



1

r

then M is parabolic.

∞



1

r

theδ-stability inequality holdsonM for0< δ ≤ 1 ifany f ∈ C ∞



M

|∇f|2− δ|A|2+ Ric(ν, ν)

f2dv ≥ 0.

nonnegative sectional curvature If the δ-stability inequality holds on M for 0< δ ≤ 1, then the volume of

M is infinite.



M

|∇ϕ|2≥ δ



M



Ric(ν, ν) + |A|2

ϕ2

Let q := δ( |A|2+ Ric(ν, ν)) and let D ⊂ M be any bounded domain with smooth boundary Denote

Let M n be an n-dimensional orientable submanifold in an (n + p)-dimensional Riemannian manifold

N n+p Fixapointx ∈ M andchooseanylocal orthonormalframe{e1, · · · , e n+p} suchthat{e1, · · · , e n} is

Aα X, Y = ¯ ∇X Y, e α,

where X, Y are tangent vector fields and ∇ denotes¯ the Levi-Civita connection on N Then the

H = n+p

α=n+1 (trace A α )e α.

φα X, Y = Aα X, Y − X, Y H, eα

φ(X, Y ) =

n+p

α=n+1 φαX, Y eα.

|φ|2=|A|2− H2

n .

H2

|A|2 ≤ n.

φ = A − H

n g,

√ n−1

2n( √

n −1+1)2 Then we have

2(n − 1)|H|2− (n − 2) √ n − 1|H|n |A|2− |H|2≥ −bn2|A|2. (2.2)

n2

|H|

|A|



n − |A| H22 − 2(n − 1)

n2

H2

|A|2 ≤ b.

n ] by

n − t2− 2(n − 1)

n2 t2.

n ] fn (t). Then

(n − 2) √ n − 1tn − t2≤ 2(n − 1)t2+ Bn2, ∀t ∈ [0, √ n ]

(n − 2)2(n − 1)x(n − x) ≤ 4(n − 1)2x2+ 4B(n − 1)n2x + B2n4, (2.3)

B ≥ (n − 2)2

√

n − 1

n − 1 + 1)2 = b,

curvature K N satisfying that K ≤ KN where K is a constant Then the Ricci curvatureRicM of M satisfies

RicM ≥ (n − 1)K + 1

n2

2(n − 1)|H|2− (n − 2) √ n − 1|H|n |A|2− |H|2 − n − 1

n |A|2.

cur-vature Then

√

n − 1

n − 1

n − 1

n − 1 + 1)2

|A|2

√

n − 1

N with nonnegative sectional curvature If the δ-stability inequality holds on M for some 2√ n−2

n−1 ≤ δ ≤ 1, then there is no nontrivial L 2β harmonic 1-form on M for any constant β satisfying

2δ

√

n − 1



n − 1

< β < √ 2δ

n − 1

1 +



n − 1

.



|ω| 2β dv < ∞.

We shallusethesamenotation ω foraharmonic1-formand itsdual harmonicvectorfieldinanabuseof

Δ|ω|2= 2

|∇ω|2+ RicM (ω, ω)

.

Δ|ω|2= 2

|ω|Δ|ω| + ω |2

.

|ω|Δ|ω| − RicM (ω, ω) = |∇ω|2− ω |2

.

|∇ω|2− ω |2≥ 1

n − 1 ω |

2

|ω|Δ|ω| ≥ 1

n − 1 ω |

2

|ω|Δ|ω| ≥ 1

n − 1 ω |

2

−

√

n − 1

2 |A|2|ω|2.

|ω| αΔ|ω| α=|ω| α

α(α − 1)|ω| α−2 ω | 2

+ α |ω| α−1Δ|ω|

α ω | α 2+ α |ω| 2α −2 ω |Δ|ω

≥ α − 1

α ω | α 2+ α |ω| 2α −2

1

n − 1 ω |2

√

n − 1

2 |A|2|ω|2

≥

(n − 1)α α

2

− α

√

n − 1

(n − 1)α



M

|ω| 2qα φ2 ω | α 2

≤ M

|ω| (2q+1)α φ2Δ|ω| α + α

√

n − 1

2



M

|A|2φ2|ω| 2(q+1)α

= α

√

n − 1

2



M

|A|2φ2|ω| 2(q+1)α − (2q + 1)

M

|ω| 2qα ω | α2φ2

− 2



φ |ω| (2q+1)α

∇φ, ∇|ω| α

.

(n − 1)α



M

|ω| 2qα ω | α 2φ2

≤ α

√

n − 1

2



M

|ω| 2(q+1)α |A|2φ2− 2

M

φ |ω| (2q+1)α

∇φ, ∇|ω| α



M

|∇φ|2≥ δ



M



Ric(ν, ν) + |A|2

φ2≥ δ



M

|A|2φ2.

δ



M

|ω| 2(q+1)α |A|2φ2≤ (q + 1)2



M

|ω| 2qα ω | α 2φ2+



M

|ω| 2(q+1)α |∇φ|2

+ 2(1 + q)



M

|ω| (2q+1)α φ

∇φ, ∇|ω| α

(n − 1)α



M

|ω| 2qα ω | α 2φ2

≤ α δ

√

n − 1

2

(q + 1)2



M

|ω| 2qα ω | α 2φ2+



M

|ω| 2(q+1)α |∇φ|2

+



2α δ

√

n − 1

 

M

|ω| (2q+1)α φ

∇φ, ∇|ω| α



2α δ

√

n − 1

 

M

|ω| (2q+1)α φ

∇φ, ∇|ω| α

δ

√

n − 1



M

2|ω| (2q+1)α φ |∇φ| ω | α

≤ |D|

ε



M

|ω| 2qα ω | α2φ2+1

ε



M

|ω| 2(q+1)α |∇φ|2

where

D := 1 − α

δ

√

n − 1

From theinequalities(2.9)and (2.10),itfollowsthat



(n − 1)α −

√

n − 1

2

α(q + 1)2

δ

 

M

|ω| 2qα ω | α 2φ2

≤

√

n − 1

2

α δ



M

|ω| 2(q+1)α |∇φ|2

M

|ω| 2qα ω | α2φ2+|D|

ε



M

|ω| 2(q+1)α |∇φ|2,



(n − 1)α −

√

n − 1

2

α(q + 1)2

δ − |D|ε

 

M

|ω| 2qα ω | α 2φ2

≤

√

n − 1

2

α

δ +

|D|

ε

 

M

Now letβ := (1 + q)α andchoosethenumbersα and q suchthat

(n − 1)α −

√

n − 1

2

α(q + 1)2

δ > 0.



M

|ω| 2qα ω | α2φ2≤ C

M

2(q + 1) − (n n − 1)α − 2 −

√

n − 1

2

α(q + 1)2

n − 1 −

√

n − 1

2

β2

δ > 0,

2δ

√

n − 1



n − 1

< β < √ 2δ

n − 1

1 +



n − 1

.

on[0, R] and φ= 0 in[2R, ∞) with |φ  | ≤ 2



|ω| 2qα ω | α 2

≤ 4C

R2



|ω| 2β

Letting R → ∞, we concludethat|ω| is constant sinceω is an L 2β harmonic 1-form However,since the

manifold N with nonnegative sectional curvature Let α1 and b be the same constants as in Theorem 2.6

If the stability inequality (1.1) holds on M , then there is no nontrivial L 2β harmonic 1-form on M for any constant β satisfying

2

√

n − 1



n − 1

< β < √ 2

n − 1

1 +



n − 1

.

√ n−1

2n < δ ≤ 1 inTheorem 2.6,wehavethefollowingvanishingtheoremforL2

manifold N with nonnegative sectional curvature If M satisfies δ-stability inequality for some (n −1)

√ n−1

2n <

δ ≤ 1, then there is no nontrivial L2 harmonic 1-form on M

√ n−1

2n < δ ≤ 1,wesee

2δ

√

n − 1



n − 1

< 1 < √ 2δ

n − 1

1 +



n − 1

.



M

|∇f|2− δ|A|2f2dv ≥ 0.

stability inequality holds on M for some (n −1) 2n √ n −1 < δ ≤ 1, then there is no nontrivial L2harmonic 1-form

on M

λ (Ω) > 0 denotethefirsteigenvalueoftheDirichletboundaryvalueproblem

λ1(M ) = inf

Ω λ1(Ω),

δ-stabilityinequality.Moreprecisely,weprovethatiftheL n or L ∞ normofthesecond fundamentalform

curvature K N satisfying that K N ≤ K ≤ 0 where K ≤ 0 is a constant Assume that for0< δ ≤ 1

|A|2≤ λ1(M )

δ − nK.

Then the δ-stability inequality holds on M

C0∞ (M )

λ1(M ) ≤



M |∇f|2



M f2 ,



M

|∇f|2− δ|A|2+ Ric(ν, ν)

f2≥



M



λ1(M ) −δ |A|2+ nK

f2≥ 0.

simply-connected manifold with nonpositive sectional curvature Then for any φ ∈ W 1,2



M

|φ| 2n

n−2 dv

n

≤ CS M

|∇φ|2+

φ |H|2

where CS is the Sobolev constant which depends only on n ≥ 3.

Theorem 3.3 Let M n (n ≥ 3) be an n-dimensional complete hypersurface in a complete simply connected Riemannian manifold N with nonpositive sectional curvature Assume that for0< δ ≤ 1



M

|A| n

2

n

,

where CS is a Sobolev constant in (3.1) Then the δ-stability inequality holds on M



M

|f| 2n

n−2

n

≤ CS



M

|∇f|2+

f |H|2

≤ CS M

|∇f|2+ C S



M

|H| n

2

n 

M

|f| 2n

n−2

n

.

Thus



M

|f| 2n

n−2

n

1− CSH2

n



M

|∇f|2,

since

H2

n ≤ nA2

(n + δ)C S <

1

C S



M

|∇f|2− δ|A|2+ Ric(ν, ν)

f2≥1− CS H2n

C S



M

|f| 2n

n−2

n

− δ M

|A|2f2

≥

n

CS − δA2

n



M

|f| 2n

n−2

n −2 n

≥

1− nCS A2

n

CS − δA2

n



M

|f| 2n

n−2

n

≥

1

C S − (n + δ)A2

n



M

|f| 2n

n−2

n

≥ 0,

Δω = −(dδ + δd)ω = 0 and



|ω|2< ∞.

Evenwithoutassumingstabilityofminimalhypersurfaces,ifM ⊂ R n+1isacompleteminimalhypersurface

onM (See[21,25,32]fordetails.)Theanalogueofthisresultisalsotrueforacompleteminimalhypersurface

satisfies

λ1(M ) > (n − 1)2

n



k2− inf |H|2

,

inf|H| intheirresult

simply-connected Riemannian manifold N with sectional curvature KN satisfying that −k2≤ KN ≤ 0 where k is

a constant Assume that the traceless second fundamental form φ satisfies

φn <



1

n(n − 1)CS .

In the case k = 0, assume further that the first eigenvalue λ1(M ) of M satisfies

2(n − 1)2k2

n3− (n − 2)(n − 1)n(n − 1)CS φn − 2n(n − 1)CSφ2

n ,

where CS is a Sobolev constant in (3.1) Then there is no nontrivial L2 harmonic 1-form on M

1-form ω

|ω|Δ|ω| ≥ 1

n − 1 ω |

2

+ RicM (ω, ω).

Ric≥ (n − 1)

|H|2

n2 − k2

− n − 2

n2



n(n − 1)|φ||H| − n − 1

n |φ|2.

Combiningthese twoinequalities,wehave

|ω|Δ|ω| ≥ 1

n − 1 ω |

2

|H|2

n2 − k2

|ω|2

− n − 2

n2



n(n − 1)|φ||H||ω|2− n − 1

f ≡ 1 on B(R), f ≡ 0 on M \ B(2R), and |∇f| ≤ 1



B R

f2|ω|Δ|ω| ≥

B R

1

n − 1 ω |

2

f2+



B R

|H|2

n2 − k2

|ω|2f2

−



B R

n − 2

n2



n(n − 1)|φ||H||ω|2f2−



B R

n − 1

n |φ|2|ω|2f2.



B R

f |ω|∇f, ∇|ω|− n

n − 1



B R

ω |2f2+n − 1

n



B R

|φ|2|ω|2f2

n2



n(n − 1)

B R

|φ||H||ω|2f2+ (n − 1)

B R

k2− |H|2

n2

|ω|2f2.

2



B R

f |ω|∇f, ∇|ω| 

B R

f2 ω |2

α



B R

and

2



B R

|φ||H||ω|2f2≤ β

B R

|H|2|ω|2f2+ 1

β



B R

|φ|2|ω|2f2.

α − n

n − 1



B R

f2 ω |2

α



B R

|∇f|2|ω|2

+

β(n − 2) 2n2



n(n − 1) − n − 1

n2



B R

|H|2|ω|2f2

+

n − 2



n(n − 1) + n − 1

n



B R

|φ|2|ω|2f2



B R

|φ|2|ω|2f2≤



B R

|φ| n

2

n 

B R



|ω||f|n−2 2n

n −2 n

≤ CS



B R

|φ| n

2

n 

B R

ω |f 2

+



B R

|H|2|ω|2f2

≤ CS



B R

|φ| n

2

n

(1 + α)



B R

f2 ω | 2

+

α



B R

|∇f|2|ω|2





B R

|φ| n

2

n 

B R

Inthecasek = 0, weneedanadditionalestimate.Usingthemonotonicityofthefirsteigenvalueλ1(B R) of

λ1(M ) ≤ λ1(B R)≤



B R |∇f|2



λ1(M )



B R

|ω|2f2≤ (1 + α)

B R

f2 ω |2

+

α



B R

A



B R

f2 ω | 2+ B



B R

|H|2|ω|2f2≤ C



B R

A = n

n − 1 − α −



k2(n − 1)

λ1(M ) + C S



B R

|φ| n

2

n

n − 2



n(n − 1) + n − 1

n



(1 + α)

B = n − 1

n2 − n − 2

2n2



n(n − 1)β − CS



B R

|φ| n

2

n

n − 2



n(n − 1) + n − 1

n

C = 1

α+



k2(n − 1)

λ1(M ) + C S



B R

|φ| n

2

n

n − 2



n(n − 1) + n − 1

n



α

.

β + CS φ2

n

1

we see that B > 0 for any β > 0. Take β = √

Now lettingR → ∞ in theinequality(4.7), weobtain|∇|ω|| ≡ 0 and |H||ω| ≡ 0. Since|∇|ω|| ≡ 0, we get|ω| ≡ constant.Supposethat|ω| isanonzeroconstant.Fromthefactthat|H||ω| ≡ 0,itfollowsthatM

spaceRN If the traceless second fundamental form φ satisfies

φn <

 1

n(n − 1)CS , then there is no nontrivial L2 harmonic 1-form on M

simply-connected Riemannian manifold N with sectional curvature KN satisfying that −k2 ≤ KN ≤ 0 for some constant k = 0 If

φn <

 1

n(n − 1)CS and

λ1(M ) > 2n

2(n − 1)2k2

n3− n2+ 3n − 4 , then there is no nontrivial L2 harmonic 1-form on M

ofM andthecurvature oftheambientspace,whichisdifferentfrom [2]

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(NRF-2013R1A1A1A05006277)

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