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uniqueness results on a geometric pde in riemannian and cr geometry revisited xiaodong wang abstract we revisit some uniqueness results for a geometric nonlinear pde related to the scalar curvature ...

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                                       UNIQUENESS RESULTS ON A GEOMETRIC PDE IN
                                         RIEMANNIAN AND CR GEOMETRY REVISITED
                                                                 XIAODONG WANG
                                        Abstract. We revisit some uniqueness results for a geometric nonlinear PDE
                                        related to the scalar curvature in Riemannian geometry and CR geometry. In
                                        the Riemannian case we give a new proof of the uniqueness result assuming
                                        only a positive lower bound for Ricci curvature. We apply the same principle in
                                        the CR case and reconstruct the Jerison-Lee identity in a more general setting.
                                        As a consequence we prove a stronger uniqueness result in the CR case. We
                                        also discuss some open problems for further study.
                                                                1. Introduction
                                  Let (n;g) be a Riemannian manifold and ge= u4=(n2)g another metric confor-
                               mal to g, where u is a positive smooth function on . The scalar curvatures are
                               related by the following equation
                                                        4(n1)                 e (n+2)=(n2)
                                                       n2 gu+Ru=Ru                         :
                                      n
                               Let (S ;gc) be the sphere with the standard metric. A conformal metric ge =
                               u4=(n2)g has constant scalar curvature n(n1) i¤
                                         c
                                                          4                 (n+2)=(n2)       n
                               (1.1)                n(n2)u+u=u                      ; on S :
                                                                 n
                               Conformal di¤eomorphisms of S give rise to a natural family of solutions to the
                               above equation
                                                                                      (n2)=2
                                                     ut; (x) = (cosht + (sinht)x  )         ;
                                                   n
                               where t  0; 2 S . It is a remarkable theorem that these are all the positive
                               solutions to (1.1). There are now several proofs for this theorem. Analytically, by
                               the stereographic projection (1.1) is equivalent to the following equation
                                                       v=n(n2)v(n+2)=(n2) on Rn
                                                                    4
                               whose positive solutions were classi…ed by Gidas-Ni-Nirenberg [GNN] using the
                               moving plane method. Geometrically, it follows from the following more general
                               theorem of Obata.
                                                                  n                                                2
                               Theorem 1. ([O2]) Suppose ( ;g) is a closed Einstein manifold and g =  g
                               is a conformal metric with constant scalar curvature, where  is a positive smooth
                               function. Then  must be constant unless (n;g) is isometric to the standard sphere
                                 n                                                                         n
                               (S ;gc) up to a scaling and  corresponds to the following function on S
                                                                                        2
                                                          (x) = c(cosht+sinhtx)
                                                                n
                               for some c > 0;t  0 and  2 S .
                                                                          1
                                  2                                    XIAODONG WANG
                                     Obata’s proof is short and elegant and is based on the following formula
                                                                                1 2          
                                                            T =T +(n2)             D  n g ;
                                  where T and T are the traceless Ricci tensor of g and g, respectively. But this
                                                                                                                      2
                                  argument is quite subtle as it requires using the unknown metric g =  g as the
                                  background metric instead of the given Einstein metric g.
                                     There is parallel story in CR geometry. Let M2m+1 be a CR manifold and
                                  e     2=m
                                   = f      two pseudohermitian structures. The pseudohermitian scalar curvatures
                                            e
                                  of  and  are related by the following formula
                                                              2(m+1)                     e (m+2)=m
                                                                  m bf+Rf=Rf                       :
                                        2m+1             m+1           	
                                  On S         = z2C           : jzj = 1   the canonical pseudohermitian structure c =
                                   p          2 2m+1
                                   2 1@jzj       j       satis…es R     =(m+1)=2 and R = m(m+1)=2. There-
                                                   S                  
                                  fore  = f2=mc has scalar curvature m(m+1)=2 i¤
                                                               4                 (m+2)=m        2m+1
                                  (1.2)                      2bf+f =f                    on S      :
                                                              m
                                                                               2m+1
                                  Pseudoconformal di¤eomorphisms of S                yield a natural family of solutions to
                                  the above equation
                                                                                             1=m
                                                            f   (z) = cosht+(sinht)z             :
                                                             t;
                                  It is a remarkable result of Jerison and Lee [JL] that these are all the positive
                                                                                                                  2m+1       
                                  solutions of (1.2). The proof is based on a highly nontrivial identity on S            ; c ,
                                  with  = f2=m
                                  (1.3)
                                   RegD +gE 3 p1U 
                                                        0         ;
                                      1     1                    2
                                                           2        
                                   =      +         jD j +E 
                                        2    2                  
                                       h            2                      2               2    1              1        2i
                                   + jD U j +jU +E D j +jU +E j +  D +  E  :
                                                                                               
             

                                  where
                                                             1                1                  1
                                                   D =  ;D =  D ;E =  E ;
                                                                ;                              
 

                                                             1           2          1 1
                                                    E =       (g) ;
                                                                ;               2                
                                                               2                1     1       1     2
                                                     U =           D      ; g =    + + j@j +i :
                                                           m+2 ;            2     2                     0
                                  Here and throughout this paper we always work with a local unitary frame fT :
                                   = 1; ;mg for T1;0M and T0 = T is the Reeb vector …eld. It should be
                                  emphasized that in all these formulas covariant derivatives are calculated w.r.t. the
                                  unknown pseudoconformal structure c.
                                     The Jerison-Lee identity is in fact valid on any closed Einstein pseudohermitian
                                  manifold. Here by Einstein we mean R =  and A = 0 (torsion-free).
                                  The following more general uniqueness result, which is the analogue of the Obata
                                  theorem, was proved in [W].
                                                                           UNIQUENESS RESULTS REVISITED                                                  3
                                         Theorem 2. ([W]) Let M2m+1; be a closed Einstein pseudohermitian mani-
                                         fold. Suppose  =  is another pseudohermitian structure with constant pseudo-
                                         hermitian scalar curvature. Then  must be constant unless M2m+1; is CR
                                                           2m+1        
                                         isometric to S             ; c   up to a scaling and  corresponds to the following function
                                               2m+1
                                         on S                                                                       2
                                                                          (z) = c cosht+(sinht)z 
                                                                                      m
                                                                           2m+1
                                         for some t  0 and  2 S                 .
                                            WenotethatliketheObataargumentallcalculationshavetobecarriedoutwith
                                         respect to the unknown pseudehermitian structure  = . Complicated formulas
                                         relating the curvature tensors of  and  as well as various Bianchi identities are
                                         also heavily used in the proof.
                                            TheJerison-Lee identity is truly remarkable and a better understanding is highly
                                         desirable. In this paper, we propose a di¤erent approach to reconstruct the formula.
                                         Thebasic idea is to study the model case carefully and then come up with the right
                                         quantities to apply the maximum principle. We …rst revisit the Riemannian case
                                         and give a new(?) proof of the uniqueness results. In fact, this new proof does
                                         not require the Einstein condition. A positive lower bound for Ricci curvature
                                         su¢ ces. Suppose (Mn;g) is a compact Riemannian manifold with Ric  n1 and
                                         u2C1(M)ispositive and satis…es the following equation
                                                                   u+n(n2)u= n(n2)u(n+2)=(n2):
                                                                                    4                  4
                                         If we write u = v(n2)=2, then v satis…es
                                                                                      n 1          2            2
                                                                             v= 2v            jrvj +1v :
                                                                                                                1         2      2      
                                         By the study of the model case, we consider  = v                            jrvj +v +1 . A simple
                                         calculation shows that
                                                                               (n2)hrlogv;ri
                                         and therefore the maximum principle comes into play. This simple argument yields
                                         the following result which is more general than Obata’s theorem.
                                         Theorem 3. Let (Mn;g) be a smooth compact Riemannian manifold with a (pos-
                                         sibly empty) convex boundary. Suppose u 2 C1(M) is a positive solution of the
                                         following equation
                                                                        u+u=u(n+2)=(n2) on                        M;
                                                                                     @u = 0                    on @M;
                                                                                     @
                                         where  > 0 is a constant. If Ric  (n 1)g and   n(n2)=4, then u must be
                                                                                                                              n            n      
                                         constant unless  = n(n2)=4 and (M;g) is isometric to (S ;gc) or S ;gc . In
                                                                                                                                             +
                                                                                  n        n
                                         the latter case u is given on S             or S     by the following formula
                                                                                           +
                                                                                                                 (n2)=2
                                                                     u(x) = c (cosht+(sinht)x)                             :
                                                                                 n
                                                                           n
                                         for some t  0 and  2 S .
                                            The above theorem is actually not new. It is a special case of a theorem by
                                         Bidaut-Véron and Véron [BVV] and Ilias [I]. Their method is based on a sophis-
                                         ticated integration by parts which can handle the subcritical case as well. We will
                                         say more about their result in the last section.
                                         4                                            XIAODONG WANG
                                            We apply the same principle to the CR case. Here the …rst di¢ culty is that
                                         there is no natural …rst order quantity and therefore we have to go to the 2nd order.
                                         There are three natural tensors of order 2 to consider and we must take a suitable
                                         contraction and combination to apply the maximum principle. As our argument
                                         in the Riemannian case, this approach has the advantage that the calculations are
                                         done on a …xed pseudohermitian manifold 2m+1; which does not have to be
                                                                                                                       e
                                         Einstein. The unknown pseudohermitian structure  =  and its curvature tensor
                                         do not enter the discussion at all. All it takes is to do covariant derivatives. Of
                                         course we are using a lot of hindsight from Jerison-Lee. Besides the identity (1.3)
                                         Jerison and Lee [JL] gave three additional divergence formulas on the Heisenberg
                                         group. The formula we obtain can be viewed as the generalization of their …rst
                                         formula ((4.2) in [JL]) to any pseudohermitian manifold with torision zero. (One
                                         can even drop this condition, but the additional terms involving the torsion A
                                         and its divergence seem too complicated). The calculations are still formidable.
                                         But we hope that this approach sheds more light on the Jerison-Lee work. We
                                         do get a more general identity, see Theorem 6. As a result we prove a stronger
                                         uniqueness theorem.
                                         Theorem 4. Let M2m+1; be a closed pseudohermitian manifold with A = 0
                                         and R        m+1. Suppose f > 0 satis…es the following equation on M
                                                         2
                                                                                 f+f=f(m+2)=m;
                                                                                      b
                                                                                            2                                                   2
                                         where  > 0 is a constant. If   m =4, then f is constant unless  = m =4 and
                                                                        2m+1         
                                         (M;) is isometric to S                  ; c  and in this case
                                                                                                                1=m
                                                                          f = c cosht+(sinht)z 
                                                                                  m
                                                                     2m+1
                                         for some t > 0; 2 S               .
                                            The paper is organized as follows. In the 2nd section we discuss the Riemannian
                                         case. In Section 3 we study the model case in CR geometry as a guide for …nding
                                         the right quantities. In Section 4 we present our reconstruction of the Jerison-Lee
                                         identity and prove the above uniqueness result. We discuss some open problems in
                                         the last section.
                                                                               2. The Riemannian case
                                                     n
                                            On(S ;gc) we consider the equation
                                         (2.1)                                   4      u+u=u(n+2)=(n2):
                                                                             n(n2)
                                         If u is positive, the equation simply means that u4=(n2)g has the same scalar
                                                                                                                            c
                                                                                          n                        n        n
                                         curvature n(n1). For t  0; 2 S the map t; : S ! S de…ned by
                                                  t; (x) =                  1              (x(x))+ sinht+cosht(x)
                                                                 cosht+sinht(x)                                   cosht+sinht(x)
                                         is a conformal di¤eomorphism with  g = u4=(n2)g with
                                                                                             t; c       t;        c
                                                                                                                 (n2)=2
                                                                     ut; (x) = (cosht + (sinht)x  )                       :
                                         Therefore these are solutions of the equation (2.1).
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...Uniqueness results on a geometric pde in riemannian and cr geometry revisited xiaodong wang abstract we revisit some for nonlinear related to the scalar curvature case give new proof of result assuming only positive lower bound ricci apply same principle reconstruct jerison lee identity more general setting as consequence prove stronger also discuss open problems further study introduction let n g be manifold ge u another metric confor mal where is smooth function curvatures are by following equation e gu ru s gc sphere with standard conformal has constant i c di eomorphisms rise natural family solutions above ut x cosht sinht t it remarkable theorem that these all there now several proofs this analytically stereographic projection equivalent v rn whose were classied gidas ni nirenberg using moving plane method geometrically follows from obata suppose closed einstein then must unless isometric up scaling corresponds sinhtx obatas short elegant based formula d traceless tensor respectiv...

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