Research Article A Global Curvature Pinching Result of the ...downloads.hindawi.com/journals/aaa/2013/237418.pdfA Global Curvature Pinching Result of the First Eigenvalue of the Laplacian
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 237418 5 pageshttpdxdoiorg1011552013237418
Research ArticleA Global Curvature Pinching Result of the First Eigenvalue ofthe Laplacian on Riemannian Manifolds
Peihe Wang1 and Ying Li2
1 School of Mathematical Sciences Qufu Normal University Shandong Qufu 273165 China2 College of Science University of Shanghai for Science and Technology Shanghai 200093 China
Correspondence should be addressed to Peihe Wang peihewanghotmailcom
Received 8 December 2012 Revised 19 February 2013 Accepted 9 March 2013
Academic Editor Wenming Zou
Copyright copy 2013 P Wang and Y Li This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The paper starts with a discussion involving the Sobolev constant on geodesic balls and then follows with a derivation of a lowerbound for the first eigenvalue of the Laplacian onmanifolds with small negative curvatureThe derivation involves Moser iteration
1 Introduction
The Laplacian is one of the most important operator onRiemannian manifolds and the study of its first eigenvalue isalso an interesting subject in the field of geometric analysisIn general people would like to estimate the first eigenvalueof Laplacian in terms of geometric quantities of themanifoldssuch as curvature volume diameter and injectivity radius Inthis sense the first interesting result is that of Lichnerowiczand Obata which proved the following result in [1] let 119872119899be an 119899-dimensional compact Riemannian manifold withoutboundary with Ric (119872) ge (119899 minus 1) then the first eigenvalueof Laplacian on 119872
119899 will satisfy that 1205821(119872) ge 119899 and the
inequality becomes equality if and only if119872119899 cong 119878119899
The above result implies that the first eigenvalue of theLaplacian will have a lower bound less than 119899 if the Riccicurvature of manifolds involved has a lower bound 119899 minus 1
except on a small part where the Ricci curvature satisfiedthat Ric (119872) ge 0 Now a natural question arises what is thelower bound of the first eigenvalue of Laplacian on such amanifold In [2] Petersen and Sprouse gave a lower boundunder the assumption that the bad part of the manifolds issmall in the sense of 119871119901-norm where 119901 is a constant largerthan half of the dimension of the manifold In this paper weare interested in the lower bound of the first eigenvalue underthe global pinching of the Ricci curvature and we obtain auniversal estimate of this lower bound on a certain class ofmanifolds
2 A Sobolev Constant on the Geodesic Ball
The Sobolev inequality is one of the most important toolsin geometric analysis and the Sobolev constant plays animportant part in the study of this field In this section wewill obtain a general Sobolev constant only depending on thedimension of the manifold on the geodesic ball with smallradius
Definition 1 Let 119861119901(119877) sube 119872 be a geodesic ball with radius
119877 we define the Sobolev constant 119862119904(119877) on it to be the
infimum among all the constant 119862 such that the inequality||119891||2
2119899(119899minus2)le 119862||nabla119891||
2
2holds for all 119891 isin 119882
12
0(119861119901(119877))
Definition 2 Let 119861119901(119877) sube 119872 be a geodesic ball with radius
119877 we define the isoperimetric constant 1198620(119877) on it to be the
supremum among all the constant 120572 such that the inequalityArea (120597Ω) ge 120572Vol (Ω)1minus(1119899) holds for all Ω sube 119861
119901(119877) with
smooth boundaryFor any fixed point 119901 and radius 119877 Croke proves that the
where 120583119909is the standard surface measure of the unit sphere
119888119899minus1
is denoted to be the area of the unit sphere 119878119899minus1
Definition 3 Using the Notation above 120596 = inf119909isinΩ
120596119909is
called the visibility angle ofΩIf the manifold has Inj (119872) ge 119894 which ensures that any
minimal geodesic starting from any point in 119861119901(1198942) will
reach the boundary 120597119861119901(1198942) before it reaches its cut locus
then the visibility angle of 119861119901(1198942) for any point 119901 which we
denote by 120596(1198942) satisfies 120596(1198942) = 1
Lemma 4 Let 119872119899 be a closed Riemannian manifold withInj (119872) ge 119894 then for any 119901 isin 119872 the following Sobolevinequality holds on 119861
119901(1198942) ||119891||2
2119899(119899minus2)le 119862||nabla119891||
2
2 where
119891 isin 11988212
0(119861119901(1198942)) and 119862 = 119862(119899)
infin and120596 is just the visibility angleof the domainΩ
As discussed above we will have120596(Ω) = 1 ifΩ sube 119861119901(1198942)
then according to Crokersquos inequality we obtain 1198620(1198942) ge
119888119899minus1
(1198881198992)1minus(1119899) The relation between 119862
0(1198942) and 119862
119904(1198942)
tells us that 119862119904(1198942) le 119862(119899) where 119862(119899) is a constant only
depending on the dimension 119899
Proposition 5 Let119872119899 be a closed 119899-dimensional Riemannianmanifold with Inj (119872) ge 119894 then for all 119901 isin 119872 Vol (119861
119901(1198942))
ge 119862119899119894119899 where119862
119899is a constant only depending on the dimension
119899
Proof Also take the inequality of Croke
Area (120597Ω)Vol (Ω)1minus(1119899)
ge119888119899minus1
(1198881198992)1minus(1119899)
1205961+(1119899)
(4)
then the result can easily be derived from the fact that120596(1198942) = 1 and Area (120597119861
119901(119903)) = 119889Vol (119861
119901(119903))119889119903 after we
integrate both sides of the inequality
3 The First Eigenfunction and Eigenvalue
Let 119872119899 be a closed 119899-dimensional Riemannian manifoldsuppose that120582
1(119872) is the first eigenvalue of the Laplacian and
119906 is the first eigenfunction In other words they will satisfythat Δ119906 + 120582
1(119872)119906 = 0 By linearity we can assume that
minus1 le 119906 le 1 and inf119909isin119872
119906 = minus1 for the linearity For theconvenience we call it the normalized eigenfunctionNextwewill study some properties of the normalized eigenfunctionand the eigenvalue
Lemma 6 Let 119872119899 be a closed 119899-dimensional Riemannianmanifold with Ric (119872) ge 0 and Inj (119872) ge 119894 Then a constant1198621(119899 119894) gt 0 can be found such that 120582
1(119872) le 119862
1(119899 119894)
Proof One of the theorems of Yau and Schoen [1] shows that1205821(119872) le 119864
1198991198892
le 1198641198991198942
≜ 1198621(119899 119894) if Ric (119872) ge 0 where 119889 is
the diameter of the manifold and 119864119899is a constant depending
only on 119899We will now introduce some notation Let Ric
minus(119909) denote
the lowest eigenvalue of the Ricci curvature tensor at 119909 Fora function 119891(119909) on 119872
119899 we denote 119891+(119909) = max119891(119909) 0
Notice that a Riemannian manifold satisfies Ric ge 119899 minus 1 ifand only if ((119899 minus 1) minus Ric
minus)+equiv 0
The well-known Myers theorem shows that a closedmanifold with Ric ge 119899 minus 1 would have a bounded diameter119889 le 120587 In other words one can deduce that 119889 le 120587 if one has(1Vol(119872)) int
119872
((119899 minus 1) minus Ricminus)+119889vol = 0 We will show next
a result analogous to the one in [5] which we will use in ourestimation of the eigenvalueThe proof follows identically soit will be omitted (the reader can refer to the aforementionedarticle)
Lemma 7 Let 119872119899 be a closed 119899-dimensional Riemannianmanifold with Ric (119872) ge 0 then for any 120575 gt 0 there exists1205980= 1205980(119899 120575) gt 0 such that if
1
Vol (119872)int119872
((119899 minus 1) minus Ricminus)+119889vol le 120598
0(119899 120575) (5)
then the diameter will satisfy 119889 lt 120587 + 120575 In particular thereexists 120598
1= 1205981(119899) such that if
1
Vol (119872)int119872
((119899 minus 1) minus Ricminus)+119889vol le 120598
1(119899) (6)
then the diameter will satisfy 119889 lt 2120587 This fact together withthe volume comparison theorem implies that Vol (119872) le
1198622(119899) where 119862
2(119899) is also a constant only dependent of 119899
Now we can get a rough lower bound for the first eigen-value
Lemma 8 For 119899 isin N let 1205981= 1205981(119899) gt 0 as above and suppose
that119872119899 is a closed manifold with
Ric (119872) ge 0
1
Vol (119872)int119872
((119899 minus 1) minus Ricminus)+119889vol le 120598
1(119899)
(7)
Abstract and Applied Analysis 3
then there exists a constant 1198623(119899) gt 0 such that 120582
1(119872) ge
1198623(119899)
Proof The proof mainly belongs to Li and Yau [6] Let 119906 bethe normalized eigenfunction of119872 set V = log (119886 + 119906) where119886 gt 1 Then we can easily get that
ΔV =minus1205821(119872) 119906
119886 + 119906minus |nablaV|
2
(8)
Denote that 119876(119909) = |nablaV|2(119909) and we then have by theRicci identity on manifolds with Ric (119872) ge 0
Δ119876 = 2V2
119894119895+ 2V119895V119895119894119894ge 2V2
119894119895+ 2 ⟨nablaV nablaΔV⟩ (9)
For the term V2119894119895 we have
sum
119894119895
V2
119894119895ge(ΔV)2
119899ge1
119899(1198762
+21205821(119872) 119906
119886 + 119906) (10)
and for the term ⟨nablaV nablaΔV⟩ we have
⟨nablaV nablaΔV⟩ = minus1198861205821(119872)
119886 + 119906119876 minus ⟨nablaV nabla119876⟩ (11)
Therefore assume 1199090isin 119872 to be the maximum of 119876 then at
1199090 we have
0 ge2
119899119876 (1199090) + (
41205821(119872)
119899minus2 (119899 + 2) 119886120582
1(119872)
119899 (119886 minus 1)) (12)
Therefore
119876 (119909) le 119876 (1199090) le
(119899 + 2) 1198861205821(119872)
119886 minus 1 (13)
Denote 120574 to be the minimizing unit speed geodesic join-ing themaximum andminimumpoints of 119906 then integrating11987612 along 120574 one will get
log( 119886
119886 minus 1) le log(119886 +max 119906
119886 minus 1) le 119889(
(119899 + 2) 1198861205821(119872)
119886 minus 1)
12
(14)
Let 119905 = (119886 minus 1)119886 then for any 119905 isin (0 1) we have (119899 +2)1205821(119872) ge 119905(119889
minus2
(log (1119905))2)Considering the maximum of the right hand and the
upper bound of the diameter derived in Lemma 7 we candeduce that a positive constant 119862
3(119899) can be found such that
1205821(119872) ge
4119890minus2
(119899 + 2) 1198892ge 1198623(119899) (15)
where 119889 is the diameter of the manifold
Corollary 9 If the manifold one discussed satisfies all theconditions in Lemma 8 and its injectivity radius satisfiesInj (119872) ge 119894 and if one let 119906 to be the normalized eigenfunctionthen there exists a constant 119862
4(119899 119894) gt 0 such that |nabla119906|2 le
1198624(119899 119894)
Proof Set 119886 = 2 in the (13) from above Then applyingLemma 6 one obtains
1
9|nabla119906|2
le |nablaV|2
(119909) le 21198621(119899 119894) (119899 + 2) (16)
therefore
|nabla119906|2
le 1198624(119899 119894) (17)
Proposition 10 Let 119872119899 be a closed 119899-dimensional Rieman-nian manifold 119906 the first eigenfunction of the Laplacian and1205821(119872) the corresponding eigenvalue thenΔ|119906|+120582
1(119872)|119906| ge 0
holds in the sense of distribution Moreover if 119872119899 is compactwith boundary then the same conclusion holds for its Neumannboundary value problem
Proof From the definition we know that Δ119906 + 1205821(119872)119906 = 0
holds on119872 Denote
119872+
= 119909 isin 119872 | 119906 (119909) gt 0
119872minus
= 119909 isin 119872 | 119906 (119909) lt 0
1198720
= 119909 isin 119872 | 119906 (119909) = 0
(18)
According to the maximum principle of elliptic equation andthe discussion about nodal set and nodal regions in [1] wecan conclude that 120597119872+ = 120597119872
minus
= 1198720 is a smooth manifold
with dimension 119899 minus 1For all 120601 isin 119862
infin
(119872) 120601 ge 0 integrating by parts we thenhave
where 119899+ and 119899minus denote the outward normal direction withrespect to the boundaries of 119872+ and 119872minus respectively Notethat 120597119906120597119899minus ge 0 on 120597119872
minus and 120597119906120597119899+
le 0 on 120597119872+ for the
definition of119872minus and119872+ This completes the proof
When119872 has boundary we can apply the same reasoningexcept that the test function will require 120601 isin 119862
infin
0(119872) This
gives the proofAs long as the given manifold is compact one knows
that the first normalized eigenfunction is then determinedThis indicates that the first normalized eigenfunction ofthe Laplacian has a close relation with the geometry of
4 Abstract and Applied Analysis
the manifold In particular one would hope to bound the1198712-norm of first normalized eigenvalue of Laplacian from
below by the geometric quantities In this sense we have thefollowing result
Theorem 11 Let 119872119899 be a closed 119899-dimensional Riemannianmanifold with Ric (119872) ge 0 and Inj (119872) ge 119894 If 119906 is thenormalized eigenfunction of the Laplacian then there exists aconstant 119862
5(119899 119894) gt 0 such that int
119872
1199062
ge 1198625(119899 119894)
Proof We use Moser iteration to get the result From Propo-sition 10 we know that Δ|119906| + 120582
1(119872)|119906| ge 0 holds on 119872 in
the sense of distribution Set V = |119906| and take the point 119901 isin 119872
such that 119906(119901) = minus1For 119886 ge 1 denote 119877 = 1198942 120601 is a cut-off function on
119861119901(119877) then we have by integrating by parts
then 1205821(119872) ge 119899 minus 120575 as long as (1Vol (119872)) int
119872
((119899 minus 1) minus
Ricminus)+119889vol le 120598 and this proves the theorem
Acknowledgments
The authors owe great thanks to the referees for theircareful efforts to make the paper clearer Research of thefirst author was supported by STPF of University (noJ11LA05) NSFC (no ZR2012AM010) the Postdoctoral Fund(no 201203030) of Shandong Province and PostdoctoralFund (no 2012M521302) of China Part of this work was donewhile the first author was staying at his postdoctoral mobileresearch station of QFNU
References
[1] S T Yau and R Schoen Lectures in Differential GeometryScientific Press 1988
[2] P Petersen andC Sprouse ldquoIntegral curvature bounds distanceestimates and applicationsrdquo Journal of Differential Geometryvol 50 no 2 pp 269ndash298 1998
[3] D Yang ldquoConvergence of Riemannian manifolds with integralbounds on curvature IrdquoAnnales Scientifiques de lrsquoEcole NormaleSuperieure Quatrieme Serie vol 25 no 1 pp 77ndash105 1992
[4] I Chavel Riemannian Geometry A Mordern IntroductionCambridge University Press Cambridge UK 2000
[5] C Sprouse ldquoIntegral curvature bounds and bounded diameterrdquoCommunications in Analysis andGeometry vol 8 no 3 pp 531ndash543 2000
[6] P Li and S T Yau ldquoEstimates of eigenvalues of a compactRiemannianmanifoldrdquo inAMS Proceedings of Symposia in PureMathematics pp 205ndash239 American Mathematical SocietyProvidence RI USA 1980
where 120583119909is the standard surface measure of the unit sphere
119888119899minus1
is denoted to be the area of the unit sphere 119878119899minus1
Definition 3 Using the Notation above 120596 = inf119909isinΩ
120596119909is
called the visibility angle ofΩIf the manifold has Inj (119872) ge 119894 which ensures that any
minimal geodesic starting from any point in 119861119901(1198942) will
reach the boundary 120597119861119901(1198942) before it reaches its cut locus
then the visibility angle of 119861119901(1198942) for any point 119901 which we
denote by 120596(1198942) satisfies 120596(1198942) = 1
Lemma 4 Let 119872119899 be a closed Riemannian manifold withInj (119872) ge 119894 then for any 119901 isin 119872 the following Sobolevinequality holds on 119861
119901(1198942) ||119891||2
2119899(119899minus2)le 119862||nabla119891||
2
2 where
119891 isin 11988212
0(119861119901(1198942)) and 119862 = 119862(119899)
infin and120596 is just the visibility angleof the domainΩ
As discussed above we will have120596(Ω) = 1 ifΩ sube 119861119901(1198942)
then according to Crokersquos inequality we obtain 1198620(1198942) ge
119888119899minus1
(1198881198992)1minus(1119899) The relation between 119862
0(1198942) and 119862
119904(1198942)
tells us that 119862119904(1198942) le 119862(119899) where 119862(119899) is a constant only
depending on the dimension 119899
Proposition 5 Let119872119899 be a closed 119899-dimensional Riemannianmanifold with Inj (119872) ge 119894 then for all 119901 isin 119872 Vol (119861
119901(1198942))
ge 119862119899119894119899 where119862
119899is a constant only depending on the dimension
119899
Proof Also take the inequality of Croke
Area (120597Ω)Vol (Ω)1minus(1119899)
ge119888119899minus1
(1198881198992)1minus(1119899)
1205961+(1119899)
(4)
then the result can easily be derived from the fact that120596(1198942) = 1 and Area (120597119861
119901(119903)) = 119889Vol (119861
119901(119903))119889119903 after we
integrate both sides of the inequality
3 The First Eigenfunction and Eigenvalue
Let 119872119899 be a closed 119899-dimensional Riemannian manifoldsuppose that120582
1(119872) is the first eigenvalue of the Laplacian and
119906 is the first eigenfunction In other words they will satisfythat Δ119906 + 120582
1(119872)119906 = 0 By linearity we can assume that
minus1 le 119906 le 1 and inf119909isin119872
119906 = minus1 for the linearity For theconvenience we call it the normalized eigenfunctionNextwewill study some properties of the normalized eigenfunctionand the eigenvalue
Lemma 6 Let 119872119899 be a closed 119899-dimensional Riemannianmanifold with Ric (119872) ge 0 and Inj (119872) ge 119894 Then a constant1198621(119899 119894) gt 0 can be found such that 120582
1(119872) le 119862
1(119899 119894)
Proof One of the theorems of Yau and Schoen [1] shows that1205821(119872) le 119864
1198991198892
le 1198641198991198942
≜ 1198621(119899 119894) if Ric (119872) ge 0 where 119889 is
the diameter of the manifold and 119864119899is a constant depending
only on 119899We will now introduce some notation Let Ric
minus(119909) denote
the lowest eigenvalue of the Ricci curvature tensor at 119909 Fora function 119891(119909) on 119872
119899 we denote 119891+(119909) = max119891(119909) 0
Notice that a Riemannian manifold satisfies Ric ge 119899 minus 1 ifand only if ((119899 minus 1) minus Ric
minus)+equiv 0
The well-known Myers theorem shows that a closedmanifold with Ric ge 119899 minus 1 would have a bounded diameter119889 le 120587 In other words one can deduce that 119889 le 120587 if one has(1Vol(119872)) int
119872
((119899 minus 1) minus Ricminus)+119889vol = 0 We will show next
a result analogous to the one in [5] which we will use in ourestimation of the eigenvalueThe proof follows identically soit will be omitted (the reader can refer to the aforementionedarticle)
Lemma 7 Let 119872119899 be a closed 119899-dimensional Riemannianmanifold with Ric (119872) ge 0 then for any 120575 gt 0 there exists1205980= 1205980(119899 120575) gt 0 such that if
1
Vol (119872)int119872
((119899 minus 1) minus Ricminus)+119889vol le 120598
0(119899 120575) (5)
then the diameter will satisfy 119889 lt 120587 + 120575 In particular thereexists 120598
1= 1205981(119899) such that if
1
Vol (119872)int119872
((119899 minus 1) minus Ricminus)+119889vol le 120598
1(119899) (6)
then the diameter will satisfy 119889 lt 2120587 This fact together withthe volume comparison theorem implies that Vol (119872) le
1198622(119899) where 119862
2(119899) is also a constant only dependent of 119899
Now we can get a rough lower bound for the first eigen-value
Lemma 8 For 119899 isin N let 1205981= 1205981(119899) gt 0 as above and suppose
that119872119899 is a closed manifold with
Ric (119872) ge 0
1
Vol (119872)int119872
((119899 minus 1) minus Ricminus)+119889vol le 120598
1(119899)
(7)
Abstract and Applied Analysis 3
then there exists a constant 1198623(119899) gt 0 such that 120582
1(119872) ge
1198623(119899)
Proof The proof mainly belongs to Li and Yau [6] Let 119906 bethe normalized eigenfunction of119872 set V = log (119886 + 119906) where119886 gt 1 Then we can easily get that
ΔV =minus1205821(119872) 119906
119886 + 119906minus |nablaV|
2
(8)
Denote that 119876(119909) = |nablaV|2(119909) and we then have by theRicci identity on manifolds with Ric (119872) ge 0
Δ119876 = 2V2
119894119895+ 2V119895V119895119894119894ge 2V2
119894119895+ 2 ⟨nablaV nablaΔV⟩ (9)
For the term V2119894119895 we have
sum
119894119895
V2
119894119895ge(ΔV)2
119899ge1
119899(1198762
+21205821(119872) 119906
119886 + 119906) (10)
and for the term ⟨nablaV nablaΔV⟩ we have
⟨nablaV nablaΔV⟩ = minus1198861205821(119872)
119886 + 119906119876 minus ⟨nablaV nabla119876⟩ (11)
Therefore assume 1199090isin 119872 to be the maximum of 119876 then at
1199090 we have
0 ge2
119899119876 (1199090) + (
41205821(119872)
119899minus2 (119899 + 2) 119886120582
1(119872)
119899 (119886 minus 1)) (12)
Therefore
119876 (119909) le 119876 (1199090) le
(119899 + 2) 1198861205821(119872)
119886 minus 1 (13)
Denote 120574 to be the minimizing unit speed geodesic join-ing themaximum andminimumpoints of 119906 then integrating11987612 along 120574 one will get
log( 119886
119886 minus 1) le log(119886 +max 119906
119886 minus 1) le 119889(
(119899 + 2) 1198861205821(119872)
119886 minus 1)
12
(14)
Let 119905 = (119886 minus 1)119886 then for any 119905 isin (0 1) we have (119899 +2)1205821(119872) ge 119905(119889
minus2
(log (1119905))2)Considering the maximum of the right hand and the
upper bound of the diameter derived in Lemma 7 we candeduce that a positive constant 119862
3(119899) can be found such that
1205821(119872) ge
4119890minus2
(119899 + 2) 1198892ge 1198623(119899) (15)
where 119889 is the diameter of the manifold
Corollary 9 If the manifold one discussed satisfies all theconditions in Lemma 8 and its injectivity radius satisfiesInj (119872) ge 119894 and if one let 119906 to be the normalized eigenfunctionthen there exists a constant 119862
4(119899 119894) gt 0 such that |nabla119906|2 le
1198624(119899 119894)
Proof Set 119886 = 2 in the (13) from above Then applyingLemma 6 one obtains
1
9|nabla119906|2
le |nablaV|2
(119909) le 21198621(119899 119894) (119899 + 2) (16)
therefore
|nabla119906|2
le 1198624(119899 119894) (17)
Proposition 10 Let 119872119899 be a closed 119899-dimensional Rieman-nian manifold 119906 the first eigenfunction of the Laplacian and1205821(119872) the corresponding eigenvalue thenΔ|119906|+120582
1(119872)|119906| ge 0
holds in the sense of distribution Moreover if 119872119899 is compactwith boundary then the same conclusion holds for its Neumannboundary value problem
Proof From the definition we know that Δ119906 + 1205821(119872)119906 = 0
holds on119872 Denote
119872+
= 119909 isin 119872 | 119906 (119909) gt 0
119872minus
= 119909 isin 119872 | 119906 (119909) lt 0
1198720
= 119909 isin 119872 | 119906 (119909) = 0
(18)
According to the maximum principle of elliptic equation andthe discussion about nodal set and nodal regions in [1] wecan conclude that 120597119872+ = 120597119872
minus
= 1198720 is a smooth manifold
with dimension 119899 minus 1For all 120601 isin 119862
infin
(119872) 120601 ge 0 integrating by parts we thenhave
where 119899+ and 119899minus denote the outward normal direction withrespect to the boundaries of 119872+ and 119872minus respectively Notethat 120597119906120597119899minus ge 0 on 120597119872
minus and 120597119906120597119899+
le 0 on 120597119872+ for the
definition of119872minus and119872+ This completes the proof
When119872 has boundary we can apply the same reasoningexcept that the test function will require 120601 isin 119862
infin
0(119872) This
gives the proofAs long as the given manifold is compact one knows
that the first normalized eigenfunction is then determinedThis indicates that the first normalized eigenfunction ofthe Laplacian has a close relation with the geometry of
4 Abstract and Applied Analysis
the manifold In particular one would hope to bound the1198712-norm of first normalized eigenvalue of Laplacian from
below by the geometric quantities In this sense we have thefollowing result
Theorem 11 Let 119872119899 be a closed 119899-dimensional Riemannianmanifold with Ric (119872) ge 0 and Inj (119872) ge 119894 If 119906 is thenormalized eigenfunction of the Laplacian then there exists aconstant 119862
5(119899 119894) gt 0 such that int
119872
1199062
ge 1198625(119899 119894)
Proof We use Moser iteration to get the result From Propo-sition 10 we know that Δ|119906| + 120582
1(119872)|119906| ge 0 holds on 119872 in
the sense of distribution Set V = |119906| and take the point 119901 isin 119872
such that 119906(119901) = minus1For 119886 ge 1 denote 119877 = 1198942 120601 is a cut-off function on
119861119901(119877) then we have by integrating by parts
then 1205821(119872) ge 119899 minus 120575 as long as (1Vol (119872)) int
119872
((119899 minus 1) minus
Ricminus)+119889vol le 120598 and this proves the theorem
Acknowledgments
The authors owe great thanks to the referees for theircareful efforts to make the paper clearer Research of thefirst author was supported by STPF of University (noJ11LA05) NSFC (no ZR2012AM010) the Postdoctoral Fund(no 201203030) of Shandong Province and PostdoctoralFund (no 2012M521302) of China Part of this work was donewhile the first author was staying at his postdoctoral mobileresearch station of QFNU
References
[1] S T Yau and R Schoen Lectures in Differential GeometryScientific Press 1988
[2] P Petersen andC Sprouse ldquoIntegral curvature bounds distanceestimates and applicationsrdquo Journal of Differential Geometryvol 50 no 2 pp 269ndash298 1998
[3] D Yang ldquoConvergence of Riemannian manifolds with integralbounds on curvature IrdquoAnnales Scientifiques de lrsquoEcole NormaleSuperieure Quatrieme Serie vol 25 no 1 pp 77ndash105 1992
[4] I Chavel Riemannian Geometry A Mordern IntroductionCambridge University Press Cambridge UK 2000
[5] C Sprouse ldquoIntegral curvature bounds and bounded diameterrdquoCommunications in Analysis andGeometry vol 8 no 3 pp 531ndash543 2000
[6] P Li and S T Yau ldquoEstimates of eigenvalues of a compactRiemannianmanifoldrdquo inAMS Proceedings of Symposia in PureMathematics pp 205ndash239 American Mathematical SocietyProvidence RI USA 1980
then there exists a constant 1198623(119899) gt 0 such that 120582
1(119872) ge
1198623(119899)
Proof The proof mainly belongs to Li and Yau [6] Let 119906 bethe normalized eigenfunction of119872 set V = log (119886 + 119906) where119886 gt 1 Then we can easily get that
ΔV =minus1205821(119872) 119906
119886 + 119906minus |nablaV|
2
(8)
Denote that 119876(119909) = |nablaV|2(119909) and we then have by theRicci identity on manifolds with Ric (119872) ge 0
Δ119876 = 2V2
119894119895+ 2V119895V119895119894119894ge 2V2
119894119895+ 2 ⟨nablaV nablaΔV⟩ (9)
For the term V2119894119895 we have
sum
119894119895
V2
119894119895ge(ΔV)2
119899ge1
119899(1198762
+21205821(119872) 119906
119886 + 119906) (10)
and for the term ⟨nablaV nablaΔV⟩ we have
⟨nablaV nablaΔV⟩ = minus1198861205821(119872)
119886 + 119906119876 minus ⟨nablaV nabla119876⟩ (11)
Therefore assume 1199090isin 119872 to be the maximum of 119876 then at
1199090 we have
0 ge2
119899119876 (1199090) + (
41205821(119872)
119899minus2 (119899 + 2) 119886120582
1(119872)
119899 (119886 minus 1)) (12)
Therefore
119876 (119909) le 119876 (1199090) le
(119899 + 2) 1198861205821(119872)
119886 minus 1 (13)
Denote 120574 to be the minimizing unit speed geodesic join-ing themaximum andminimumpoints of 119906 then integrating11987612 along 120574 one will get
log( 119886
119886 minus 1) le log(119886 +max 119906
119886 minus 1) le 119889(
(119899 + 2) 1198861205821(119872)
119886 minus 1)
12
(14)
Let 119905 = (119886 minus 1)119886 then for any 119905 isin (0 1) we have (119899 +2)1205821(119872) ge 119905(119889
minus2
(log (1119905))2)Considering the maximum of the right hand and the
upper bound of the diameter derived in Lemma 7 we candeduce that a positive constant 119862
3(119899) can be found such that
1205821(119872) ge
4119890minus2
(119899 + 2) 1198892ge 1198623(119899) (15)
where 119889 is the diameter of the manifold
Corollary 9 If the manifold one discussed satisfies all theconditions in Lemma 8 and its injectivity radius satisfiesInj (119872) ge 119894 and if one let 119906 to be the normalized eigenfunctionthen there exists a constant 119862
4(119899 119894) gt 0 such that |nabla119906|2 le
1198624(119899 119894)
Proof Set 119886 = 2 in the (13) from above Then applyingLemma 6 one obtains
1
9|nabla119906|2
le |nablaV|2
(119909) le 21198621(119899 119894) (119899 + 2) (16)
therefore
|nabla119906|2
le 1198624(119899 119894) (17)
Proposition 10 Let 119872119899 be a closed 119899-dimensional Rieman-nian manifold 119906 the first eigenfunction of the Laplacian and1205821(119872) the corresponding eigenvalue thenΔ|119906|+120582
1(119872)|119906| ge 0
holds in the sense of distribution Moreover if 119872119899 is compactwith boundary then the same conclusion holds for its Neumannboundary value problem
Proof From the definition we know that Δ119906 + 1205821(119872)119906 = 0
holds on119872 Denote
119872+
= 119909 isin 119872 | 119906 (119909) gt 0
119872minus
= 119909 isin 119872 | 119906 (119909) lt 0
1198720
= 119909 isin 119872 | 119906 (119909) = 0
(18)
According to the maximum principle of elliptic equation andthe discussion about nodal set and nodal regions in [1] wecan conclude that 120597119872+ = 120597119872
minus
= 1198720 is a smooth manifold
with dimension 119899 minus 1For all 120601 isin 119862
infin
(119872) 120601 ge 0 integrating by parts we thenhave
where 119899+ and 119899minus denote the outward normal direction withrespect to the boundaries of 119872+ and 119872minus respectively Notethat 120597119906120597119899minus ge 0 on 120597119872
minus and 120597119906120597119899+
le 0 on 120597119872+ for the
definition of119872minus and119872+ This completes the proof
When119872 has boundary we can apply the same reasoningexcept that the test function will require 120601 isin 119862
infin
0(119872) This
gives the proofAs long as the given manifold is compact one knows
that the first normalized eigenfunction is then determinedThis indicates that the first normalized eigenfunction ofthe Laplacian has a close relation with the geometry of
4 Abstract and Applied Analysis
the manifold In particular one would hope to bound the1198712-norm of first normalized eigenvalue of Laplacian from
below by the geometric quantities In this sense we have thefollowing result
Theorem 11 Let 119872119899 be a closed 119899-dimensional Riemannianmanifold with Ric (119872) ge 0 and Inj (119872) ge 119894 If 119906 is thenormalized eigenfunction of the Laplacian then there exists aconstant 119862
5(119899 119894) gt 0 such that int
119872
1199062
ge 1198625(119899 119894)
Proof We use Moser iteration to get the result From Propo-sition 10 we know that Δ|119906| + 120582
1(119872)|119906| ge 0 holds on 119872 in
the sense of distribution Set V = |119906| and take the point 119901 isin 119872
such that 119906(119901) = minus1For 119886 ge 1 denote 119877 = 1198942 120601 is a cut-off function on
119861119901(119877) then we have by integrating by parts
then 1205821(119872) ge 119899 minus 120575 as long as (1Vol (119872)) int
119872
((119899 minus 1) minus
Ricminus)+119889vol le 120598 and this proves the theorem
Acknowledgments
The authors owe great thanks to the referees for theircareful efforts to make the paper clearer Research of thefirst author was supported by STPF of University (noJ11LA05) NSFC (no ZR2012AM010) the Postdoctoral Fund(no 201203030) of Shandong Province and PostdoctoralFund (no 2012M521302) of China Part of this work was donewhile the first author was staying at his postdoctoral mobileresearch station of QFNU
References
[1] S T Yau and R Schoen Lectures in Differential GeometryScientific Press 1988
[2] P Petersen andC Sprouse ldquoIntegral curvature bounds distanceestimates and applicationsrdquo Journal of Differential Geometryvol 50 no 2 pp 269ndash298 1998
[3] D Yang ldquoConvergence of Riemannian manifolds with integralbounds on curvature IrdquoAnnales Scientifiques de lrsquoEcole NormaleSuperieure Quatrieme Serie vol 25 no 1 pp 77ndash105 1992
[4] I Chavel Riemannian Geometry A Mordern IntroductionCambridge University Press Cambridge UK 2000
[5] C Sprouse ldquoIntegral curvature bounds and bounded diameterrdquoCommunications in Analysis andGeometry vol 8 no 3 pp 531ndash543 2000
[6] P Li and S T Yau ldquoEstimates of eigenvalues of a compactRiemannianmanifoldrdquo inAMS Proceedings of Symposia in PureMathematics pp 205ndash239 American Mathematical SocietyProvidence RI USA 1980
the manifold In particular one would hope to bound the1198712-norm of first normalized eigenvalue of Laplacian from
below by the geometric quantities In this sense we have thefollowing result
Theorem 11 Let 119872119899 be a closed 119899-dimensional Riemannianmanifold with Ric (119872) ge 0 and Inj (119872) ge 119894 If 119906 is thenormalized eigenfunction of the Laplacian then there exists aconstant 119862
5(119899 119894) gt 0 such that int
119872
1199062
ge 1198625(119899 119894)
Proof We use Moser iteration to get the result From Propo-sition 10 we know that Δ|119906| + 120582
1(119872)|119906| ge 0 holds on 119872 in
the sense of distribution Set V = |119906| and take the point 119901 isin 119872
such that 119906(119901) = minus1For 119886 ge 1 denote 119877 = 1198942 120601 is a cut-off function on
119861119901(119877) then we have by integrating by parts
then 1205821(119872) ge 119899 minus 120575 as long as (1Vol (119872)) int
119872
((119899 minus 1) minus
Ricminus)+119889vol le 120598 and this proves the theorem
Acknowledgments
The authors owe great thanks to the referees for theircareful efforts to make the paper clearer Research of thefirst author was supported by STPF of University (noJ11LA05) NSFC (no ZR2012AM010) the Postdoctoral Fund(no 201203030) of Shandong Province and PostdoctoralFund (no 2012M521302) of China Part of this work was donewhile the first author was staying at his postdoctoral mobileresearch station of QFNU
References
[1] S T Yau and R Schoen Lectures in Differential GeometryScientific Press 1988
[2] P Petersen andC Sprouse ldquoIntegral curvature bounds distanceestimates and applicationsrdquo Journal of Differential Geometryvol 50 no 2 pp 269ndash298 1998
[3] D Yang ldquoConvergence of Riemannian manifolds with integralbounds on curvature IrdquoAnnales Scientifiques de lrsquoEcole NormaleSuperieure Quatrieme Serie vol 25 no 1 pp 77ndash105 1992
[4] I Chavel Riemannian Geometry A Mordern IntroductionCambridge University Press Cambridge UK 2000
[5] C Sprouse ldquoIntegral curvature bounds and bounded diameterrdquoCommunications in Analysis andGeometry vol 8 no 3 pp 531ndash543 2000
[6] P Li and S T Yau ldquoEstimates of eigenvalues of a compactRiemannianmanifoldrdquo inAMS Proceedings of Symposia in PureMathematics pp 205ndash239 American Mathematical SocietyProvidence RI USA 1980
then 1205821(119872) ge 119899 minus 120575 as long as (1Vol (119872)) int
119872
((119899 minus 1) minus
Ricminus)+119889vol le 120598 and this proves the theorem
Acknowledgments
The authors owe great thanks to the referees for theircareful efforts to make the paper clearer Research of thefirst author was supported by STPF of University (noJ11LA05) NSFC (no ZR2012AM010) the Postdoctoral Fund(no 201203030) of Shandong Province and PostdoctoralFund (no 2012M521302) of China Part of this work was donewhile the first author was staying at his postdoctoral mobileresearch station of QFNU
References
[1] S T Yau and R Schoen Lectures in Differential GeometryScientific Press 1988
[2] P Petersen andC Sprouse ldquoIntegral curvature bounds distanceestimates and applicationsrdquo Journal of Differential Geometryvol 50 no 2 pp 269ndash298 1998
[3] D Yang ldquoConvergence of Riemannian manifolds with integralbounds on curvature IrdquoAnnales Scientifiques de lrsquoEcole NormaleSuperieure Quatrieme Serie vol 25 no 1 pp 77ndash105 1992
[4] I Chavel Riemannian Geometry A Mordern IntroductionCambridge University Press Cambridge UK 2000
[5] C Sprouse ldquoIntegral curvature bounds and bounded diameterrdquoCommunications in Analysis andGeometry vol 8 no 3 pp 531ndash543 2000
[6] P Li and S T Yau ldquoEstimates of eigenvalues of a compactRiemannianmanifoldrdquo inAMS Proceedings of Symposia in PureMathematics pp 205ndash239 American Mathematical SocietyProvidence RI USA 1980