Supported blow-up and prescribed scalar curvature on S^n

112 Pages·2011·0.838 MB·English

by  Man Chun Leung

MEMOIRS
of the
American Mathematical Society
Number 1002
Supported Blow-Up and Prescribed
Scalar Curvature on Sn
Man Chun Leung
September 2011 • Volume 213 • Number 1002 (third of 5 numbers) • ISSN 0065-9266
American Mathematical Society
Number 1002
Supported Blow-Up and Prescribed
Scalar Curvature on Sn
Man Chun Leung
September2011 • Volume213 • Number1002(thirdof5numbers) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Supportedblow-upandprescribedscalarcurvatureonSn /ManChunLeung.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 1002)
“September2011,volume213,number1002(thirdof5numbers).”
Includesbibliographicalreferences.
ISBN978-0-8218-5337-5(alk. paper)
1. Blowing up (Algebraic geometry). 2. Curvature. 3. Transformations (Mathematics).
4.Differentialequations,Elliptic. I.Title.
QA571.L48 2011
515(cid:2).3533—dc23 2011020094
Memoirs of the American Mathematical Society
Thisjournalisdevotedentirelytoresearchinpureandappliedmathematics.
Publisher Item Identifier. The Publisher Item Identifier (PII) appears as a footnote on
theAbstractpageofeacharticle. Thisalphanumericstringofcharactersuniquelyidentifieseach
articleandcanbeusedforfuturecataloguing,searching,andelectronicretrieval.
Subscription information. Beginning with the January 2010 issue, Memoirs is accessi-
ble from www.ams.org/journals. The 2011 subscription begins with volume 209 and consists of
six mailings, each containing one or more numbers. Subscription prices are as follows: for pa-
per delivery, US$741 list, US$592.80 institutional member; for electronic delivery, US$667 list,
US$533.60institutional member. Uponrequest, subscribers topaper delivery ofthis journalare
also entitled to receive electronic delivery. If ordering the paper version, subscribers outside the
United States and India must pay a postage surcharge of US$69; subscribers in India must pay
apostagesurchargeofUS$95. ExpediteddeliverytodestinationsinNorthAmericaUS$58;else-
whereUS$167. Subscriptionrenewalsaresubjecttolatefees. Seewww.ams.org/help-faqformore
journalsubscriptioninformation. Eachnumbermaybeorderedseparately;pleasespecifynumber
whenorderinganindividualnumber.
Back number information. Forbackissuesseewww.ams.org/bookstore.
Subscriptions and orders should be addressed to the American Mathematical Society, P.O.
Box 845904, Boston, MA 02284-5904USA. All orders must be accompanied by payment. Other
correspondenceshouldbeaddressedto201CharlesStreet,Providence,RI02904-2294USA.
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
[email protected].
MemoirsoftheAmericanMathematicalSociety(ISSN0065-9266)ispublishedbimonthly(each
volume consisting usually of more than one number) by the American Mathematical Society at
201CharlesStreet,Providence,RI02904-2294USA.PeriodicalspostagepaidatProvidence,RI.
Postmaster: Send address changes to Memoirs, American Mathematical Society, 201 Charles
Street,Providence,RI02904-2294USA.
(cid:2)c 2011bytheAmericanMathematicalSociety. Allrightsreserved.
Copyrightofindividualarticlesmayreverttothepublicdomain28years
afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles.
(cid:2) (cid:2) (cid:2)
ThispublicationisindexedinScienceCitation IndexR,SciSearchR,ResearchAlertR,
(cid:2) (cid:2)
CompuMath Citation IndexR,Current ContentsR/Physical,Chemical& Earth Sciences.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 161514131211
Contents
Chapter 1. Introduction 1
Chapter 2. The Subcritical Approach 5
Chapter 3. Simple, Towering, Aggregated and Clustered Blow-ups 11
Chapter 4. Supported and Collapsed Blow-ups 21
Chapter 5. Toward Isolated Blow-ups 25
Chapter 6. Toward Supported Blow-up for ΔK˜(0)>0
– Excluding Simple Blow-up 33
Chapter 7. Excluding Collapsed Isolated Blow-up
(Hess K˜(0) Positive Definite) 45
o
Chapter 8. Close Up 75
Chapter 9. Single Simple Blow-up and the Proof of the Main Theorem 81
Bibliography 97
iii

Abstract
We expound the notion of supported blow-up and apply it to study the
renownedNirenberg/Kazdan-WarnerproblemonSn. Whenn≥5andundersome
mildconditions, weshowthatblow-upatapointwithpositivedefiniteHessianhas
to be a supported isolated blow-up, which, when combined with a uniform volume
bound,isaremovablesingularity. Anewasymmetricconditionisintroducedtoex-
cludesingle simple blow-up. Theseenableustoobtainageneralexistencetheorem
for n≥5 with rather natural condition.
ReceivedbytheeditorAugust22,2007and,inrevisedform,October21,2010.
ArticleelectronicallypublishedonMarch2,2011;S0065-9266(2011)00636-2.
2000 MathematicsSubjectClassification. Primary35J60;Secondary53C21.
Key wordsand phrases. Noncompactness,blow-up,removablesingularity,scalarcurvature.
Affiliation at time of publication: Department of Mathematics, National University of
Singapore, 10, Lower Kent Ridge Road, Singapore 119076, Republic of Singapore; email:
[email protected].
(cid:2)c2011 American Mathematical Society
v

CHAPTER 1
Introduction
The simplicity and subtlety of curvature and its global nature draw many at-
tention. In particular, in the 1960’s, people start to wonder what kind of function
K canbethescalarcurvatureofaconformalmetriconthestandardunitsphereSn.
This is widely known as the Nirenberg/Kazdan-Warner problem, and is linked to
theYamabeproblem([3][47]). Thequestionamountstofindingapositivesolution
u of the equation
(1.1) Δ1u−cnn(n−1)u+cnKunn−+22 =0 in Sn.
Here Δ is the Laplacian on Sn with the standard metric g , and
1 1
n−2
c = , n≥3.
n 4(n−1)
ForgeneralK, equation(1.1)provestoberemarkablyflexibleanddifficulttosolve,
largely due to the blow-up phenomenon. R. Schoen initiates the study on isolated
blow-up and simple blow-up (commonly referred as simple isolated blow-up) [46].
In a series of papers [40] [41], Y.-Y. Li develops the notions and applies them to
obtain existence and compactness results. The study is furthered by C.-C. Chen
andC.-S.Lin,andothers,whoapplythepowerfulandnaturalmethodonreflection
uponmovingplanes(cf. [16][17][42])forsomenonconstantK. Clusteredblow-up
(see Definition 3.33), with new constructions by S. Taliaferro [50] and the author
[38], remains a topic for future endeavor.
Equation (1.1) is a focus of research for decades, and continues to inspire new
thoughts. Recent existence results mainly use
• the degree counting methods (e.g. [13] [14] [15] [20] [40] [41] [44] [48]),
• the gradient flow methods (e.g. [5] [6] [7] [8], cf. also [49]), or
• the reduced finite dimension variational method (e.g. [2] [43] [53]).
The majority of these require the solution set to be uniformly bounded. We refer
to the article [43] by A. Malchiodi for an overview on existence results.
Whilerecognizingtheusefulnessofcompactnessinfindingsolutionsofequation
(1.1),oneislefttoponderthedilemma: ByselectingthosefunctionsK sothatblow-
ups are impossible (i.e., compactness regained), we naturally miss functions that
can afford a bounded and a blow-up subcritical sequences. This intriguing thought
breathes the idea that blow-ups need not always be harmful in finding solutions.
Undersuitableconditions,westillcanuseablow-upsubcriticalsequencetoproduce
asolutionbyremovingthesingularities. Anecessaryconditionforthistohappenis
thatthesequencemustbesupported(Definition4.2). Broadlyspeaking,suchlower
boundexistswhentheblow-upoccursatapointQatwhichK haspositivedefinite
1
2 MANCHUNLEUNG
Hessian and n ≥ 5 (refer to Theorem 7.1). Accompanied by the finite volume
property , which is endogenous in the subcritical approach when K is positive
(Lemma 2.14), we show that Q is a removable singularity (full cause appeared in
Theorem 8.1).
Our analysis on supported isolated blow-ups leads to the following.
Main Theorem. For n ≥ 5, let K ∈ Cn+3(Sn) be a positive function on
Sn with finite number of critical points. Separate the critical points by the sign
of Δ K: {p+,···, p+} with positive Δ K; {Q ,···, Q } with non-positive
g1 1 (cid:2) g1 1 m
Δ K. Assume the following conditions (A) and (B).
g1
(A) Hess K is positive definite at each p+, 1≤k ≤(cid:2).
g1 k
(B) K satisfies the HSn-condition (defined below) at each Q , where each
βj j
integer βj satisfies n ≤ βj ≤ n+2; and via the stereographic projection PQj in
which Q serves as the north pole,
j
(cid:2)
(1.2) (cid:5)[K◦(PQj)−1]dy (cid:7)= (cid:4)0.
Rn
Then the conformal scalar curvature equation (1.1) has a positive C2 solution.
In the above, Hess K is the Hessian of K with respect g , and
g1 1
(cid:2) (cid:3)(cid:2) (cid:2) (cid:4)
(cid:5)[K◦(PQj)−1]dy = ∂K◦(PQj)−1 dy, ···, ∂K ◦(PQj)−1 dy
Rn Rn ∂y|1 Rn ∂y|n
is interpreted as a vector (y| is the k-th component of the point). In particular,
k
condition (1.2) implies that K is not rotationally symmetric above Q . See [10]
j
[21] [23] [25] for existenceresults when K isrotationally symmetric; compare also
with [1]. Condition (1.2) is invariant under generative conformal transformations
(Proposition 9.70).
Definition 1.3 (HSn-Condition). Given an integer β˜ ≥ 1, a function K ∈
β i
C1(Sn)issaidtofulfilltheHSn-conditionatapointQ∈Sn ifviathestereographic
β
projection P in which Q serves as the south pole, the function K˜ := K ◦P−1
Q Q
satisfies the following: In an open ball B (ρ) we can express
o
(1.4) K˜(y)=K˜(0)+Q(β)(y)+R(y),
where Q(β) is a C1 homogeneous function of degree β, satisfying
(1.5) C1|y|β˜i−1 ≤|(cid:5)Q(β)(y)|≤C2|y|β˜i−1 for y ∈Bo(ρ),
and the remainder R∈C[β](B (ρ)) satisfying
o
(cid:5)[β]
(1.6) |(cid:5)sR(y)|·|y|−β+s −→ 0 as |y|→0.
s=0
Here [β] is the integer part of β.
1. INTRODUCTION 3
Indeed a weaker condition (see Remark 9.69) is sufficient for the proof of the
MainTheorem,whichappearsin§9g. ConditionsofthetypeHSn areusedexten-
β
sively to study equation (1.1), e.g. [16] [17] [20] [40] [41] [42]. Yet our approach
contrasts, and in some sense complements, the known methods. Significant ad-
vances are achieved in recent years for low dimensions (n ≤ 5; see, however, the
brilliant works of Escobar-Schoen [26] and Li [40], cf. also [8] [22] [23] [31] [43]).
Our central theorem concerns higher dimensions. We note that condition (1.2),
whichis global as stipulatedby the Kazdan-Warner formula [11] [45]and thenon-
existence results [9] [10] [19], is also new and can be readily checked.
Weoutlinethekeystepsintheproof. UsingthefactthatPohozaevidentityfor
thesubcritical equations(see(6.4))hasanextrapositiveterm,weshowthatsimple
blow-upcannotappearatacriticalpointwithpositiveLaplacian(seeTheorem6.2
for the complete statement). By considering primary bubbling with possible offset
center, apoint Qat which K has positive definite Hessian can only be a supported
towering blow-up point (refer to Chapter 7 for details).
Not surprisingly, the positive term in the Pohozaev identity (6.4) becomes a
hindrance when the Laplacian is non-positive (see the remark in §6f). We observe
that a single simple blow-up can be modeled by the Green function. Through
conformal deformation, the Green function is used to obtain a scalar flat manifold
(theEuclideanspaceinthepresentsituation). Weestimatetheuniformclosenessof
theGreenfunctiontothesimpleblow-up(Lemma9.22). Thekeyisthatthecenters
ofthebubblescannotbe‘toofar’fromtheorigin(cf. thecondition n≤β ≤n+2,
j
contrastingβ˜ ∈(n−2, n)in[40]&[53]). Coupledwiththetranslational Pohozaev
i
identity, we arrive at the asymmetric condition (1.2) (Theorem 9.1).
Conventions 1.7. • Throughout this paper, the dimension n ≥ 3, and
1
c = (n−2)/[4(n−1)]. We observe the practice of using C, possibly with sub-
n
indices, to denote various positive constants, which may be rendered differently
from line to line according to the contents, unless otherwise is mentioned. Whilst
weuse C¯, possiblywith sub-index, to denotea fixed positiveconstantwhich always
keeps the same value as it is first defined.
• Let {r } and {R } be sequences of non-negative numbers. We use the
2 i i
notations
“r ≤ O(R )” to denote “r ≤CR for all i(cid:9)1”; and
i i i i
“r ≤ o(R )” to denote “r ≤c R for all i(cid:9)1, where lim c =0”.
i i i i i i
i→∞
Likewise, we define r ≥ O(R ). A term depending on i is written O(R ) if its
i i i
absolute value is smaller then a positive constant times R for all i(cid:9)1. Likewise,
i
we define a term being o(R ).
i
• Astatementinvolvingasequenceissaidtohold“modulo a subsequence” ifwe
3
can select a subsequence (from the original sequence in the statement) so that the
statement is valid for this subsequence. As a rule, we assume that the statement is
true for the original sequence so that the notations remain clean.
• Denote by B (r) the open ball in Rn with center at y and radius r >0, and
4 y
Vol(Sn) the measure of Sn in Rn+1 with respect to the standard metric.