article yuxin ge

8/3/2019 Article Yuxin Ge

1/35

ON THE 2-SCALAR CURVATURE

YUXIN GE, CHANG-SHOU LIN, AND GUOFANG WANG

Abstract. In this paper, we establish an analytic foundation for a fully non-linearequation 21 = f on manifolds with positive scalar curvature and apply it to give a

(rough) classification of such manifolds. A crucial point is a simple observation that thisequation is a degenerate elliptic equation without any condition on the sign of f and itis elliptic not only for f > 0 but also for f < 0. By defining a Yamabe constant Y2,1with respect to this equation, we show that a positive scalar curvature manifold admitsa conformal metric with positive scalar curvature and positive 2-scalar curvature if andonly if Y2,1 > 0. We give a complete solution for the corresponding Yamabe problem.Namely, let g0 be a positive scalar curvature metric, then in its conformal class there isa conformal metric with

2(g) = 1(g),

for some constant . Using these analytic results, we give a rough classification of thespace of manifolds with positive scalar curvature metrics.

1. Introduction

Let (M, g0) be a compact Riemannian manifold with metric g0 and [g0] the conformalclass of g0. Let Ricg and Rg denote the Ricci tensor and scalar curvature of a metric grespectively. The Schouten tensor of the metric g is defined by

Sg =

1

n 2 Ricg Rg2(n 1) g .The importance of the Schouten tensor in conformal geometry can be viewed in the fol-lowing decomposition of the Riemann curvature tensor

Riemg = Wg + Sg g,where is the Kulkani-Nomizu product. Note that g1 Wg is invariant in a givenconformal class. Therefore, in a conformal class the Schouten tensor is important.

The objective of this paper has two-folds. First, we study a class of fully nonlinearquotient equations relating the scalar curvature and the recently introduced 2-scalarcurvature. Then, we apply the analysis established for this class of equations to studythe space of metrics with positive scalar curvature. Let us first recall the definition of the

2-scalar curvature.For a given 1 k n, the k-scalar curvature or k-scalar curvature is defined by

k(g) := k(g1 Sg),

where g1 Sg is locally defined by (g1 Sg)ij =

k gik(Sg)kj and k is the kth elementary

symmetric function. Here for an nn symmetric matrix A we define k(A) = k(), where1


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2 YUXIN GE, CHANG-SHOU LIN, AND GUOFANG WANG

= (1, , n) is the set of eigenvalues ofA. It is clear that 1(g) is a constant multipleof the scalar curvature Rg. The k-scalar curvature k(g), which was first considered byViaclovsky [59], is a natural generalization of the scalar curvature.

In this paper, we focus on the 2-scalar curvature. One of the reason to restrict ourself

to 2-scalar curvature is that 2(g) still has a variational structure. This is an observationof Viaclovsky [59].1 The variational structure is very crucial for our paper.

Since [59] and [9], there has been an intensive study for a nonlinear Yamabe problemrelated to the k-scalar curvature k(g), namely, finding a conformal metric g in a givenconformal class [g0] satisfying

(1) k(g) = c,

where

(2) g [g0] +k .Here +k is a convex open cone -the Garding cone- defined by

+k = { = (1, 2, , n) R

n

| j() > 0, j k}.By g +k we mean that the Schouten tensor Sg(x) +k for any x M. g +1is equivalent to that g has positive scalar curvature. Note that the condition g +kguarantees that equation (1) is elliptic.

Equation (1) is a fully nonlinear conformal equation, but it admits many nice properties,which usually are only true for the semilinear equations. For example, the local a prioriestimates was established in [25], a Liouville type theorem was given in [44]. The existenceproblem of (1) has been intensively studied. For the existence results see [10], [17], [18],[21], [26], [28], [30], [32], [44], [45], [46] [53], [58], [61]. See also a recent survey by Viaclovsky[62] and references therein or the lecture notes of Guan [20]. There are many interestingapplications in geometry, especially in the 4-dimensional case, see for examples [9], [11],

and [63]. See also [21], [22].However, till now one can only deal with equations like

(3) k(g) = f

with condition f 0 or even f > 0. This has something to do with the ellipticity ofthe corresponding equation. With this restriction we could not use equation (3) to studynegative 2-scalar curvature. (cf. the work of Gursky and Viaclovsky [33] for metrics ink and a class of uniformly elliptic fully nonlinear equations [28].) In this paper, insteadof 2(g) we consider the quotient type equation

(4)21

= f.

When f > 0, it was studied recently in [18]. See also [27], [30], [23] and [53]. The crucialpoint of this paper is that we can also deal with the case with negative functions f for (4).In fact, we have

1For k > 2, k(g) has a variational structure if and only if the underlying manifold is locally conformallyflat ([59],[6]).


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ON THE 2-SCALAR CURVATURE 3

Observation. The operator 21 is elliptic in the cone +1 \ R1 and concave in +1 .

For the notation and the proof of this Observation, see Lemma 1. Recall that theordinary Yamabe constant is defined

Y1([g0]) = Y1([g0]) = inf gC1([g0])

1(g)dvol(g)

(vol(g))n2n

.

In this paper we will call it the first Yamabe constant. For simplicity of notation, wedenote +k [g0] by Ck([g0]), or even just Ck if there is no confusion. A conformal class haspositive first Yamabe constant if and only if this class contains a metric of positive scalarcurvature. This is a simple, but important fact in the study of the metrics of positivescalar curvature. As one of applications of our study of equation (4), we will prove asimilar result for the 2-scalar curvature. From now on we consider the conformal class[g] with C1 = and call a metric with positive scalar curvature psc metric. Now we firstdefine a nonlinear eigenvalue for 2(g), more precisely for a fully nonlinear operator

(5) 2(2u + du du |u|2

2g0 + Sg0).

Define

(6) (g0, 2) = (M, g0, 2) :=

infgC1([g0])

2(g)dvol(g)

e4udvol(g), if n > 4,

2(g)dvol(g), if n = 4,

supgC1([g0])

2(g)dvol(g)

e4udvol(g)

, if n = 3.

One can also define a similar constant (g0, k) for k. When k = 1, one can check that(g0, k) is the first eigenvalue of the conformal Laplacian. Hence we call the constant(g0, 2) nonlinear eigenvalue of operator (5). Such a nonlinear eigenvalue for fully non-linear operator was first considered in [48]. In the context of fully nonlinear conformaloperators, it was first considered in a preliminary version of [26].

Now we define the second conformal Yamabe constant for a conformal class with pscmetrics.

Y2,1([g0]) = Y2,1([g0]) :=

infgC1([g0])

2(g)dvol(g)

(

1(g)dvol(g))n4n2

, if n > 4,

2(g)dvol(g), if n = 4,sup

gC1([g0])

2(g)dvol(g)

1(g)dvol(g), if n = 3.

We emphasize that the infimum (or supremum) in the definition of 2 and Y2,1 is takenover C1, not over C2. It is well-known that when n = 4,

M2(g)dvol(g) is a constant in a


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given conformal class. The relationship between the sign of (g0, 2) and Y2,1(g0]) will bediscussed in the next section.

Now we state our main analytic results in this paper .

Theorem 1. The constant (g0, 2) is achieved, provided (g0, 2) 0. More precisely,we have1) If (g0, 2) > 0, then C2([g0]) is not empty and (g0, 2) > 0 is achieved by a

smooth metric g = e2ug0 C2 satisfying(7) 2(g) = e

4u.

2) If (g0, 2) = 0, then (g0, 2) = 0 is achieved by a C1,1 metric g = e2ug0 C1

satisfying2(g) = 0.

Here C1 is the closure ofC1. Namely, C1 is the space of metrics with non-negative scalarcurvature. With the help of Theorem 1 and results in [10], [32] and [18] we can solve the

Yamabe problem for

2

1 completely.Theorem 2. Let (Mn, g0) be a compact Riemannian manifold with g0 +1 and n 3.The followings hold

1) If Y2,1([g0]) > 0, then C2([g0]) is not empty. Moreover there is a smooth metric gin C2 satisfying

(8)2(g)

1(g)= 1.

2) If Y2,1([g0]) = 0, then there is a C1,1 metric g in C1 satisfying satisfying

(9) 2(g) = 0.

3) If Y2,1([g0]) < 0, then there is a smooth metric g in

C1 satisfying

(10) 2(g) = 1(g).In fact, we can show that Y2,1 is achieved when n > 3. This will be carried out elsewhere.

We remark that in case 3) though the solution metric g is in C1, it is smooth. One canshow that any point x M with 2(g)(x) = 0 (and hence 1(g)(x) = 0) is Ricci-flat, i.e.,Ric(x) = 0. We believe that g C1.

A direct consequence of the main analytic results is

Corollary 1. Let (Mn, g0) be a closed Riemannian manifold of dimension n > 2 withpositive Yamabe constant Y1([g0]) > 0. If Y2,1([g0]) > 0, then there is a metric g [g0]with g +2 , i.e., with

Rg

> 0 and 2

(g) > 0.

Note that Y2,1[g0]) > 0 if and only if (g0, 2) > 0. See Lemma 3 below. When n = 4,the result was proved by Chang-Gursky-Yang [9]. In another direct proof was given in[31]. See also [21] for locally conformally flat manifolds. Many interesting applications togeometry were given in these papers, especially in [11]. In 3-dimension, together with theHamiltons result [34] obtained by the Ricci flow, the Corollary gives


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Corollary 2. Let(M3, g0) be a compact 3-dimensional Riemannian manifold. IfY1([g0]) >0 and Y2,1([g0]) > 0, then M is diffeomorphic to a quotient of the sphere.

We remark that the method given in [31] and [21] can be used to give another proof of

Corollary 1 at least for n > 4. In another direction, one may ask if the non-emptyness of+2 implies the positivity of Y2,1. This question is not easy to answer by only using themethods given in [31] and [21]. In [18], we showed that the non-emptyness of +2 impliesthe positivity of

infgC2

(1

1(g)dvol(g)))n4n2

2(g)dvol(g).

It is clear that Y2,1 is less than or equals to the above constant. With the analysisestablished here we can show that the non-emptyness of +2 implies the positivity of Y2,1.

Theorem 3. Let (M, g) be a compact Riemanniann manifold with positive scalar curva-ture. The second Yamabe constant is positive if and only if +2 is non-empty.

We remark again that the case n = 4 was given in [9]. As mentioned above, it is well-known that the positivity of the first Yamabe constant Y1 is equivalent to the existence ofa conformal metric of positive scalar curvature. Theorem 3 means that Y2,1 has a similarproperty. Motivated by this result, we hope to use 2-scalar curvature to give a furtherclassification of the manifolds admitting metric with positive scalar curvature. This isthe second aim of this paper. For the further discussion let us first recall the followingdefinition.

(1+) Closed connected manifolds with a Riemannian metric whose scalar curvature isnon-negative and not identically 0.

(10) Closed connected manifolds with a Riemannian metric with non-negative scalarcurvature, but not in class (10).

(1) Closed connected manifolds not in classes (1+) or (10).There is a remarkable result of Kazdan and Warner obtained in 1975.

Theorem A (Trichotomy Theorem) ([40], [41]) LetMn be a closed connected manifoldof dimension n 3.

1. If M belongs to class (1+), then every smooth function is the scalar curvaturefunction for some Riemannian metric onM.

2. IfM belongs to class (10), then a smooth functionf is the scalar curvature functionof some Riemannian metric on M if and only if f(x) < 0 for some point x M,or else f = 0. If the scalar curvature of some g vanishes identically, then g isRicci flat.

3. IfM belongs to class (1

), then a smooth functionf is the scalar curvature function

of some Riemannian metric on M if and only if f(x) < 0 for some point x M.The analysis used in the proof of Theorem A is based on the analysis for the eigenvalue

problem for the conformal Laplacian operator

(11) u + n 24(n 1) Rgu = u.


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And it is closely related to the famous Yamabe equation

(12) u + n 24(n 1) Rgu = u

n+2n2 .

The existence of solutions for (12) is the so-called Yamabe problem, which was solved byYamabe, Trudinger, Aubin and Scheon completely.

From Theorem A or an Earlier result of Aubin [2] we know that a negative function canalways be realized as a scalar function of a metric. See also the results of [16] and [49] forthe existence of negative Ricci curvature. The class of (10) is very small and consistingof very special manifolds, thanks to a result of Futaki [14]. Theorem A also implies thatclass (1+) is just the class of manifolds which admits a metric of positive scalar curvature.There are topological obstructions for the manifolds with positive scalar curvature, see[47] and [37]. This class attracts much attention of geometers for many years, especiallyafter the work of Gromov-Lawson[19] and Schoen-Yau [52]. The most important problemin this field is the Gromov-Lawson-Rosenberg conjecture which was proved by Stolz [54]in the simply connected case. For this conjecture, see for instance [50] and [56].

Now using 2-scalar curvature we divide (1+) further into 3 subclasses:

(2+) Closed connected manifolds admitting a psc metric whose 2-scalar curvature isnon-negative and not identically 0.

(20) Closed connected manifolds admitting a psc metric with non-negative 2-scalarcurvature, but not in class (2+).

(2) Closed connected manifolds in (1+), but not in classes (2+) or (20).Analog to the Trichotomy Theorem of Kazdan-Warner for the scalar curvature and in

view of the analysis established here, we propose the following

Conjecture (Trichotomy Theorem) Let Mn (n > 2) be a closed connected manifoldin class (1+).

1. IfM belongs to class (2+), then for every smooth function f there is a psc metricg such that f 1(g) is its 2-scalar curvature.2. If M belongs to class (20), then a smooth function f admits a psc metric g with

2(g) = f 1(g) if and only if f(x) < 0 for some point x M, or else f = 0.3. If M belongs to class (2), then a smooth function f admits a psc metric g with

2(g) = f 1(g) if and only if f(x) < 0 for some point x M.

Though we could not now prove this conjecture at moment, we have the following resultssupport this conjecture.

Theorem 4. Any Riemannian manifold Mn (n 4) in the class (1+) admits a metric inC1 with non-positive 2-scalar curvature.

The metric given in Theorem 4 satisfies

2(g)dvol(g) < 0 and has vanishing Riccicurvature at points with 1(g)(x) = 0.

Theorem 5. Let Mn be a closed connected manifold of dimension n > 3.

1. If M belongs to class (2+), then for every constant b there is a metric g C1 suchthat b1(g) is its 2-scalar curvature.


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2. If M belongs to class (20), then a constant b admits a metric g C1with 2(g) =b1(g) if and only if b 0.

3. If M belongs to class (2), then a constant b admits a metric g C1 with 2(g) =b1(g) if and only if b < 0.

The metrics found in Theorem 5 may have points with vanishing 1-scalar curvature. Atsuch points its Ricci tensor also vanishes. When b > 0, then the metric has positive scalarcurvature. It is trivial that the sphere belongs to (2+). By Corollary 2, in 3-dimension,only the sphere and its quotients belong to (2+) and there is no manifolds in (20). In4-dimension, many examples of manifolds with metrics in +2 can be found in [9]. Thosemanifolds certainly belong to (2+). One can prove that S

3 S1 belongs to (20), namelythere is no positive scalar metric on S3 S1 with positive 2-scalar curvature, see Section7 below. Like class (10), class (20) should be very small. However, when n > 4, we haveno example of manifolds in class (1+) which do not belong to (2+). This is somewhatstrange. A possible candidate is the manifold S6 H3, where H3 is a compact quotientof the hyperbolic space

H3

. It is easy to check that its product metric gP satisfies that1(gP) > 0 and 2(gP) = 0. However, we believe that that S6 H3 has a metric g with

1(g) > 0 and 2(g) > 0. See Example 2 in Section 7. Therefore we may ask

Problem. Is there a topological obstruction for the existence of psc metric with positive2-scalar curvature?

For the scalar curvature as mentioned above there are topological obstructions. Whatwe ask is to find further conditions to distinguish the manifolds between (2+), (20) and(2+) for higher dimensional manifolds. There is a similar and related problem proposed byStolz in [55] for further obstructions to the existence of metrics with positive Ricci tensor.For some relationship between the positive Ricci tensor and positive k-scalar curvature,

see [24]. With our analysis, the problem to find a topological obstruction for 2-scalarcurvature perhaps might be not very difficult.

The paper is organized as follows. In Section 2, we show the Observation and discussthe relationship between (g0, 2) and Y2,1. We introduce a class of perturbated equations(27) and a Yamabe type flow (30) in Section 3 and establish local priori estimates for theseequations and flows in Section 4. In Section 5 we prove the global existence of the Yamabetype flow and Theorem 1 and Theorem 2. One of another crucial points of this paper, theYamabe type flow preserves the positivity of the scalar curvature will also be proved in thissection. In Section 6, we show Theorem 3. We give the geometric applications in Section7 by proving Theorems 4 and 5. In the last section, we mention further applications insimilar equations.

2. Some preliminary facts

Let Sn be the space of n n real symmetric matrices and F a smooth function in Sn.By extending F to the whole space of n n real matrices by F(A) = F( 12 (A + At)), we


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view F as a function of n n variables wij and define

Fij =F

wij.

The quotient function 21 can be viewed as a function in Sn as follows. Let W be asymmetric matrix and W = {1, 2, , n} its the eigenvalues. Then we define 21 (W) =2(W)1(W)

. A symmetric matrix is said to be in +k if W +k . Let R1 be the subspace ofSn consisting of symmetric matrices of rank 1.Lemma 1. For 1 < k n set F = kk1 . We have

1) the matrix (Fij)(W) is semi-positive definite forW +k1 and is positive definitefor W +k1\R1.

2) The function F is concave in the cone +k1. Whenk = 2, for all W +1 and forall R = (rij)

Sn, we have

(13)

ijkl

2

wijwkl

2(W)

1(W)

rijrkl =

ij(1(W)rij 1(R)wij)2

31(W).

Proof. For = (1, 2, , n) and i {1, 2, , n} let i = {1, 2, , i, , n} be the(n 1)-tuple obtained from without the ith-component. A direct calculation gives

(14)F

i=

k1(i)k1() k()k2(i)k1()2

.

Since

+k

1, i

+k

2. For the proof see for instance [38]. From two identities

k() = ik1(i) + k(i) and k1() = ik2(i) + k1(i),

(14) becomes

(15)F

i=

2k1(i) k(i)k2(i)k1()2

.

Note that we set 0() = 1. By the Newton-McLaughlin inequality

(k 1)(n k)2k1(i) k(n k + 1)k(i)k2(i),we have

F

i 0

and equality implies that i = {0, 0, , 0}. This proves 1).2) was proved in [38].


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Though Lemma 1 is a rather simple fact2, as mentioned in the introduction it is oneof crucial points of our paper. It means that 21 is a concave, degenerate elliptic operator

in +1 . With this observation in mind, we consider a family of perturbed operators for apositive function

R+

(16)

F : +1 R

W F(W) = 2(W) 1(W)

.

As a direct consequence of Lemma 1, we have the following:

Lemma 2. The matrix (Fij )(W) is positive definite for all W +1 and the function Fis strictly concave in +1 . Moreover, for all W +1 and for all R = (rij) Sn(R)

(17)ijkl

2F(W)

wijwkl rijrkl 2

31(W)

irii2

.

Lemma 2 means that for any function > 0, the operator F is elliptic and strictlyconcave.

Now we discuss the relationship between the sign of the invariants Y2,1([g0]) and thatof (g0, 2) defined in the introduction.

Lemma 3. The eigenvalue (g0, 2) is a finite number. Moreover, we have

(1) if n 4, then (g0, 2) > 0 (resp. = 0, < 0 ) if and only if Y2,1([g0]) > 0 (resp.= 0, < 0);(2) if n = 3, then (g0, 2) > 0 (resp. 0 ) if and only if Y2,1([g0]) > 0 (resp. 0).

Proof. The following formula was given in the proof of the Sobolev inequality in [17]: forany g = e2ug0 [g0] we have(18)

2

2(g)dvol(g) =

n 42

1(g)|u|2g0e2udvol(g) +

n 44

|u|4g0e4udvol(g)

+

i,j

TijS(g0)ijdvol(g) +

i,j

S(g0)ijuiujdvol(g)

+ 12

1(g0)|u|2g0e4udvol(g),

2We remark that this Lemma might be observed also by other mathematicians. For example in [13]there is a rather similar formula. We would like to thank Tobias Lamm who told us this reference.


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where Tij = 1(W)gij Wij is the first Newton transformation associated with W. From

this formula, we have(19)

2

2(g)dvol(g) =

n

4

2

1(g)|u|2

g0e2u

dvol(g) +

n

4

4|u|4g0e4udvol(g)

+

e2u1(g)1(g0)dvol(g)

e4u|S(g0)|2g0dvol(g)

+(4 n)

i,j

S(g0)ijuiujdvol(g) +

1(g0)|u|2g0e4udvol(g)

+

e4uu, 1(g0)g0dvol(g).

We first consider the case n 5. By (19), there exists some c > 0 such that

(20) 2(g)dvol(g) n 4

16|u|

4g0e

4udvol(g)

ce4udvol(g),

provided that g +1 . Here we have used the fact

i,j

S(g0)ijuiujdvol(g)

1

8

|u|4g0e4udvol(g) + 2 sup

M|S(g0)|2g0

e4udvol(g)

and

e4uu, 1(g0)g0 1(g0)|u|2g0e

4udvol(g) + ce4udvol(g).As a consequence, we conclude

(g0, 2) c.Similarly, in case n = 3, we have for all g C1([g0])

(21)

2(g)dvol(g) = 1

4

1(g)|u|2g0e2udvolg

1

8

|u|4g0e4udvol(g)

14

e4u|u|2g01(g0)dvol(g)

+1

2e4u(21(g0) |S(g0)|

2g0

)dvol(g)

+1

2

i,j

S(g0)ijuiujdvol(g)

116

|u|4g0e4udvol(g) + c

e4udvol(g),


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which yields (g0, 2) c. Thus, we finish the proof of the first part of Lemma. It is easyto see that for all g C1([g0])

(22)M1(g)dvol(g) =

n 22 |u|2 + 1(g0) e2udvol(g) Y1([g0])(V ol(g))n2n .Now we consider the case n 5. Suppose (g0, 2) > 0, i.e.,

M2(g) (g0, 2)

M

e4udvol(g)

for any g = e2ug0 +1 . From (20), (22) and Holders inequality, we deduce

(23)

2(g)dvol(g) c(

|u|4g0e4udvol(g) +

e4udvol(g))

c(M

1(g)dvol(g))2(V ol(g))1

cY1([g0])n

n2 (

M

1(g)dvol(g))n4n2 ,

which implies Y2,1([g0]) > 0. Inversely, using Holders inequality and (22), we have

(24)

2(g)dvol(g) c(

M

1(g)dvol(g))n4n2

cY1([g0])n4n2 (V ol(g))

n4n ce

4udvol(g).

This gives the desired result. Clearly, we have (g0, 2) < 0 if and only if Y2,1([g0]) < 0.Consequently, (g0, 2) = 0 if and only if Y2,1([g0]) = 0. In the cases n = 4 and n = 3, theresult is trivial.

Remark 1. In the case n = 3, if (g0, 2) = < 0, then Y2,1([g0]) < 0. To see this, forany g C1([g0]), it follows from (21) and (22) that there holds

(25)

1(g)dvol(g)

2(g)dvol(g) 1(g)dvol(g) e4udvol(g)

c

eudvol(g0)

eudvol(g0) c < 0.


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Remark 2. In the case n 5, the invariant Y2,1([g0]) is finite real number. To see this, for anyg C1([g0]) it follows from the Holders inequality, (19) and (22) that

(26)

2(g)dvol(g)(

1(g)dvol(g))n

4n2 c e4udvol(g)

(

1(g)dvol(g))n

4n2

c (V ol(g))n4n

(

1(g)dvol(g))n4n2

c > .

In the case n = 3, we do not know if Y2,1([g0]) < + or not, although it is always truethat Y2,1([g0]) > 0 if and only if (g0, 2) > 0, and that (g0, 2) is finite. However, webelieve that it is true.

3. Yamabe type flows

Now we want to consider the existence of the following equation

(27) F(g) =2(g) e4u

1= constant,

with g = e2ug0 for > 0. Note that is a positive function. In this paper we will choose as a small positive constant. Following [26], [17] and [18] we will introduce a suitableYamabe type flow to study equation (27).

For any (0, +) and for g = e2ug0, consider the following perturbed functional

E(g) :=

2

n 4M

(2(g) e4u)dvol(g), if n = 4,

1

0M

(2(gt)

2e4tu)udvol(gt)dt, if n = 4,

where gt = e2tug0. When = 0, the functional was considered in [59], [9] and [7]. Set

F1(g) =M

1(g)dvol(g) and F2(g) =M

2(g)dvol(g)

From the variational formula given in [59], [9] and [7], we have

(28)d

dtE(g) =

(2(g) e4u)g1 d

dtgdvol(g)

and

(29)d

dtF1(g) =n

2

2

1(g)g1 d

dt gdvol(g).

Now we introduce a Yamabe type flow, which non-increases E and preserves F1.

(30)du

dt= 1

2g1

d

dtg = e2u

2(g) e4u1(g)

r(g)e2u + s(g),


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where r(g) and s(g) are space constants, given by

(31) r(g) :=F2(g)

Me

4udvolg

F1(g)

and

(32)

M

1(g)

e2u

2(g) e4u1(g)

r(g)e2u + s(g)

dvol(g) = 0.

Lemma 4. Flow (30) preserves F1 and non-increases E. Hence when n 4, then r isnon-increasing along the flow, and when n = 3, then r is non-decreasing along the flow.

Proof. By the definition of s(g) and (29), flow (30) preserves F1. By the definition of sand r, we can compute as follows

(33)

d

dtE(g) =

M

(2(g) e4u)g1 ddt

gdvol(g)

= 2

e2u1(g)

e2u2(g) e4u

1(g) r(g)e2u

2

dvol(g).

Therefore, the desired result yields.

4. Local estimates

In this section, we will establish a priori estimates for flow (30) and equations (8), (9)and (10). Local estimates for this class of fully nonlinear conformal equations was firstgiven in [25]. Since then there are many extensions. See for instance [12] and the surveypaper [62]. It is important to note that the a priori estimates established below do not

depend on the perturbation > 0.Given > 0, assume g0 C1([g0]). By Lemma 2, (30) is parabolic. By the standard

implicit function theorem we have the short-time existence result. Let T (0, ] so that[0, T) is the maximum interval for the existence of the flow g(t) +1 .Theorem 6. Assume that n 3, > 0 and g0 +1 . Let u be a solution of (30) in ageodesic ball BR [0, T] for T < T and R < 0, the injectivity radius of M.

(1) Assume that t [0, T], there holdsr(t) 0.

Then there is a constant C depending only on (BR, g0) (independent of and T)

such that for any (x, t) BR/2 [0, T](34) |u|2 + |2u| C.

(2) Assume that t [0, T], there holdsr(t) > 0.


14/35


Then there is a constant C depending only on (BR, g0) (independent of and T)such that for any (x, t) BR/2 [0, T]

(35)

|u

|2 +

|2u

| C(1 + sup

t[0,T]r(t)

e2inf(x,t)BR[0,T] u(x,t)).

In particular, if we assume n 4, we have(36) |u|2 + |2u| C(1 + e2inf(x,t)BR[0,T] u(x,t)).

Proof. Let W = (wij) be an n n matrix with wij = 2iju + uiuj |u|2

2 (g0)ij + (Sg0)ij.Here ui and uij are the first and second derivatives of u with respect to the background

metric g0. Recall F(W) =2(W)

1(W). Set

(37)

(Fij (W)) := F

wij

(W)=

1(W)T

ij 2(W)ij + ij21(W)

where (Tij) = (1(W)ijwij) is the first Newton transformation associated with W, and

ij is the Kronecker symbol. In view of Corollary 2 we see that (Fij ) is positive definiteand F is concave in

+1 . For the simplicity of notations, we now drop the index , if there

is no confusion. We try to show the local estimates for first and second order derivativestogether. Let S(T M) denote the unit tangent bundle ofM with respect to the background

metric g0. We define a function G : S(T M) [0, T] R(38) G(e, t) = (

2u +

|u|2g0)(e, e)

Without loss of generality, we assume R = 1. Let C0 (B1) be a cut-off function definedas in [25] such that

(39)

0, in B1, = 1, in B1/2,

|(x)| 2b01/2(x), in B1,|2| b0, in B1.

Here b0 > 1 is a constant. Since e2ug0 +1 , to bound |u| and |2u| we only need to

bound (2u + |u|2g0)(e, e) from above for all e S(T M) and for all t [0, T]. For thispurpose, denote G(e, t) = (x)G(e, t). Assume (e1, t0) S(Tx0M) [0, T] such that

G(e1, t0) = maxS(TM)[0,T]

G(e, t),(40)

t0 > 0,(41)

G(e1, t0) > n maxB1

1(g0).(42)


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Let (e1, , en) be a orthonormal basis at point (x0, t0). It follows from the fact W +1nG(e1, t0) (u + n|u|2) (n|u|2 + n 2

2|u|2 1(g0)),

3n 22

|u|2 1n

G(e1, t0),

so that

G(e1, t0) 3n2

2

n + 1n|u|2 21

20|u|2.

Consequently, we obtain

(43) 211u(x0, t0) 1

20|u|2(x0, t0)

Set for any i = j = 1, , n

e =1

2 (ei ej).We have

(44) G(e, t0) =1

2(G(ei, t0) + G(ej, t0)) 2iju(x0, t0).

Thus, there holds

(45) |2iju(x0, t0)| G(e1, t0) 1

2(G(ei, t0) + G(ej, t0)).

On the other hand, we have i = 1, , n

(46) (n 1)G(e1, t0) + G(ei, t0) (u + n|u|2

) (3n

2

2 |u|2

1(g0)),which implies

(47) G(ei, t0) ( 3n 22

|u|2 1(g0)) (n 1)G(e1, t0).

Together with (45), we deduce

(48) |2iju(x0, t0)| nG(e1, t0) 3n 2

2|u|2 + 1(g0) (n + 1)G(e1, t0).

(Indeed, at all point (x, t), the estimate |2iju| (n + 1)G(e1, t0) holds). Now choose thenormal coordinates around x0 such that at point x0

x1= e1

and consider the function on M [0, T]G(x, t) = (x)(u11 + |u|2)(x, t).


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(without the confusion, we denotes also this function by G). Clearly, (x0, t0) is a maximumpoint of G(x, t) on M [0, T]. At (x0, t0), we have

0

Gt = (u11t + 2

l

ulult),(49)

0 = Gj =j

G + (u11j + 2l1

ululj), for any j,(50)

0 (Gij) =ij 2ij

2G + (u11ij +

l1

(2uliulj + 2ululij))

.(51)

Recall that (Fij) is definite positive. Hence, we have

(52)

0 i,j1

FijGij Gt

i,j1

Fij ij 2ij

2G +

i,j1

Fij(u11ij +l1

(2uliulj + 2ululij))

(u11t + 2l1

ulult).

First, from the definition of , we have

(53)i,j1

Fijij 2ij

2G C

i,j1

|Fij|1

G,

and

(54)

i,j1

|Fij| i

Fii

=

n 1 n2(W)

21(W)

+

n

21(W) C

i,j1

|Fij|

since W is positive definite . Using the facts that

(55) ukij = uijk +m

Rmikjum,

(56) ukkij = uijkk +m

(2Rmikjumk Ricmjumi Ricmiumj Ricmi,jum + Rmikj,kum)

and

(57) (l

u2l )11 = 2l

(u11lul + u21l) + O(|u|2),


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we have

(58)

i,j1

Fiju11ij

i,j1

Fijwij11 (u11)iuj ui(u11)j +

l1(u21l + u11lul)(g0)ij

2

i,j1

Fijui1uj1 C(1 + |2u| + |u|2)i,j1

|Fij|

and

(59)

i,j,l

Fijululij i,j,l

Fijulwijl i,j,l

Fij(uluiluj + uluiujl)

+1

2i,j

Fij

u,

(|

u|2)

(g0)ij

C(1 +

|u|2)

i,j1 |Fij

|.

Combining (58) and (59), we deduce(60)

i,j1Fij(u11ij + 2

l1

(uliulj + ululij))

i,j1

Fij(wij11 + 2l1

wijlul) + 2i,j1

Fijl2

uliulj +i,j,l1

u21lFij(g0)ij

i,j

Fij

(u11 + |u|2)iuj + ui(u11 + |u|2)j u, (u11 + |u|2)(g0)ij

C(1 + |2u| + |u|2) i,j1

|Fij|

i,j

Fij(wij11 + 2l

wijlul) + u211

i,j

Fij(g0)ij

+i,j

Fij (iuj + jui , u(g0)ij) G2

C(1 + |2u| + |u|2)i,j1

|Fij|.

Now, we want to estimate

i,j,l Fijwijlul and

i,j F

ijwij11 respectively. Using Lemma1, there holds

(61)i,j,l

Fij

wijlul =l

Flul,

and

(62)i,j

Fijwij11 = F11

i,j,k,m

2F

wijwkmwij1wkm1 F11.


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Therefore, these estimates give

(63)

i,j1

Fij(wij11 + 2l1

wijlul) F11 + 2l

Flul.

Recall from (30) that

(64) F = ut + r(g)e2u s(g),

so that

(65) F11 = u11t + r(g)e2u(2u11 + 4u21),

(66) Fl = ult + r(g)e2u(2ul), l = 1, , n.

Gathering (43), (52), (53), (54), (60) (63), (65) and (66), we obtain(67)

0

C

i,jFij G

+ i

Fii u211 Ci,j Fij (1 + |u|

2 +

|2u

|)

+i,j

Fij (iuj + jui , u(g0)ij) G

+ r(g)e2u

2u11 4

nl=2

u2l

C

i

Fii

G

+

i

Fii

u211

C

i

Fii

G|u|

C

i

Fii

( + G) ,

since r(g) 0 for any t [0, T]. As a consequence, there holds(68) C2 CG CG

G + 2u211.

Recall (43) holds at point (x0, t0) so that

(69) G(x0, t0) = (u11 + |u|2)(x0, t0) 21(u11)(x0, t0).Together with (68), we deduce

(70) C2 CG CG

G +G2

212,

which implies

(71) C

CG +G2

2 212.

This yields

(72) G C.Here C is a constant independent ofT and . Therefore, we have finished the proof of thefirst part of the Theorem. The second part yields from (67) and Lemma 4.


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The same proof gives the local estimates for the elliptic equation.

Corollary 3. Assume n 3, > 0 andg0 +1 . LetBR be a geodesic ball for R < 0, theinjectivity radius of M. Assume that e2ug0 +1 is a solution of the following equationin BR

(73)2(g) e4u

1(g)= ,

for some constant .

(1) If 0, then there is a constant C depending only on (BR, g0) (independent of and T) such that for any (x, t) BR/2 [0, T]

(74) |u|2 + |2u| C.(2) If > 0, then there is a constantC depending only on (BR, g0) and (independent

of and T) such that for any (x, t) BR/2 [0, T]

(75) |u|2 + |2u| C(1 + e2inf(x,t)BR[0,T] u(x,t)).

Corollary 4. Under the same hypotheses as in Theorem 6 there is a constant C dependingonly on g0 (independent of ) such that for any t [0, T)(76) uC(M) + 2uC(M) C.

Corollary 5. Under the same hypotheses as in Theorem 6 with r 0, then there is aconstant C depending only on g0 (independent of ) such that for any t [0, T)(77)

u

C2(M)

C.

Proof. First, for any t [0, T) we have(78) F1(g(t)) F1(g(0)).Hence the following Sobolev inequality

M1(g)dvol(g) Y1([g0])(V ol(g))

n2n , g +1 ,

implies that the volume V ol(g(t)) along the flow is bounded from above, i.e.,

(79)

M

enudvol(g0) < C,

for some constant C > 0. On the other hand, applying Corollary 4, we haveM

e2udvol(g) CM

1(g)dvol(g) = CF1(g(0)) > 0.This, together with (79) and Corollary 4, implies u is uniformly bounded. In view ofCorollary 4, we prove the result.

We remark that in this paper we only use (1) of Theorem 6.


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5. Proof of Theorems 1 and 2

Now for > 0 we define

(80) a :=

infgC1([g0]) E(g)

(M1(g)dvol(g))

n4n2

if n = 4;M

2(g)dvol(g) if n = 4;

If a is achieved by a metric g = e2ug0, the g satisfies

(81)2(g) e4u

1(g)= ,

for some constant . Equivalently, we will consider the energy functional E on the nor-malized cone C1([g0])

(82) C1([g0]) :=

g C1([g0]) |M

1(g)dvol(g) = 1

.

The first step is to solve the perturbed equation (81).

Proposition 1. Let > 0. In the following three cases

1) when n 5, a < 0,2) when n = 4, a < 0, which is equivalent to

M

2(g)dvol(g) < 0,3) when n = 3, a > 0,

flow (30) globally converges to a solution of (81). As a direct application, a is achievedby a function u satisfying (81) for = 1. Moreover, such a solution is unique.Proof. The proof follows closely the proof given in [17]. When n 5, without loss ofgenerality we choose g0 +1 such that E(g0) < 0 and

M

1(g0)dvol(g0) = 1. Using

Lemma 4, we have r(g(t)) r(g(0)) < 0, t [0, T). If n = 3, or n = 4, by thehypotheses, we have r(g(t)) < c < 0, t [0, T). Applying Corollary 5, the solution uof flow (30) has a uniform C2 bound, which is independent of t. We divide the proof into3 steps.

Step 1. The flow preserves the positivity of the scalar curvature. This is another crucialpoint of this paper.

Proposition 2. There is a constant C0 > 0, independent of T [0, T) such that1(g(t)) > C0 for any t [0, T].


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Proof. The proof follows closely the proof given in [26] and [17]. Recall

W = (wij) = (2iju + uiuj |u|2

2(g0)ij + (Sg0)ij),

F(W) =2(W)

1(W).

Hence, F = ut + r(g(t))e2u s(g(t)). Without loss of generality, we assume that the

minimum of F is achieved at (x0, t0) M (0, T]. Near (x0, t0), we have

(83)d

dtF =

ij

Aij(2g(ut))ij =ij

Aij

(2g(F))ij r(g)(2g(e2u))ij

,

where

A

ij

:=

F

wij =

(21(W)

2(W) + )

ij

1(W)W

ij

21(W)

is positive definite. To simplify the notations, we drop the index as before. Since (x0, t0)

is the minimum of F in M [0, T], at this point, we have dFdt

0, Fl = 0 l and(Fij) is non-negative definite. Note that

(2g)ijF = Fij + uiFj + ujFi l

ulFlij = Fij,

at (x0, t0), where Fj and Fij are the first and second derivatives with respect to the back-ground metric g0. From the positivity of A and (83), we have(84)

0 Ft i,j

AijFij

r(g)i,j

Aij{(e2u)ij + ui(e2u)j + uj(e2u)i l

ul(e2u)lij}

= r(g)e2ui,j

Aij{2wij + 2uiuj + 2S(g0)ij + |u|2ij}

r(g)e2u22(W) 2

1(W)

r(g)e2u

i,j

Aij(2uiuj + 2S(g0)ij + |u|2ij).

Here we have usedi,j

Aijwij = 2(W) + 1(W)

. On the other hand, we have

(85)

i,j

AijS(g0)ij =(21(W) 2(W))1(g0)

21(W) 1

1(W)

i,j

WijS(g0)ij +1(g0)

21(W).


22/35


Going back to (84), we have

(86)

0 Ft i,j

AijFij

r(g)e2u22(W) 2

1(W)+

2(21(W) 2(W))1(g0)21(W)

21(W)

i,j

WijS(g0)ij +21(g0)

21(W)

,

since (Aij) is positive definite and r(g) < c is negative. Let us use O(1) denote termswith a uniform bound. One can check 2(g) = O(1) for uC2 is uniformly bounded andi,j

WijS(g0)ij = O(1). Also the term 21(W) 2(W) is always non-negative. From (86),

we conclude that there is a positive constant C2 > 0 (independent of T) such that

(87) 1(W)(x0, t0) > C2 > 0.

Now for any (x, t) M [0, T], we have2(W)(x, t)

1(W)(x, t) O(1)

so that there is a positive constant C > 0, independent of T, such that

1(W)(x, t) Csince we have 2(W) 12 21(W) provided 2(W) 0. This finishes the proof of theProposition and this step.

Step 2. Now we can prove equation (81) admits a solution. From Step 1, we know thatthe flow is uniformly parabolic. And, Krylovs theory implies u(t) has a uniform C2,

bound. Hence, T = . One can also show that u(t) globally converges to u(), whichclearly is a solution of (81) for k = r(g()). (Note that r(g(t)) is monotone and boundedso that r(g()) exists) (see [26]). So u = u() 1

2log |r(g())| solves (81) for = 1.

Step 3. The solution to equation (81) for k = 1 is unique.Assume u1 and u2 are two solutions. We consider the function v = u1 u2, which solvesthe following second order elliptic equation

(88)i,j

Aij(x)vij +i

Bivi + m(x)v = 0,

where

Aij(x) =

10

Fij(tW2 + (1 t)W1)dt,


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Wl(x) =

(ul)ij + (ul)i(ul)j 1

2|ul|2(g0)ij + (S(g0))ij

, for l = 1, 2

and

m(x) = 21

0 e2((1

t)u1(x)+tu2(x))

dt < 0.

This a uniformly elliptic equation. Now applying the classical strong maximum principle,we deduce that

v 0.Hence we have the uniqueness. It is clear that the unique solution achieves a. Thus, wefinish the proof.

In the following, we study a nonlinear eigenvalue problem for the operator

(89) 2(2u + du du |u|2

2g0 + Sg0)

in +1 . The nonlinear eigenvalue problem for the operator (89) in +2 was considered inthe first version of [26]. In that paper, the nonlinear eigenvalue problem can only beconsidered in +2 . With our analysis established for 2/1, we can consider the nonlineareigenvalue problem for the operator (89) in a larger class +1 .

Proposition 3. Let(Mn, g0) be a compact Riemannian manifold withg0 +1 andn 3.Assume that the first eigenvalue (g0, 2) > 0. ThenC2([g0]) is not empty. Namely thereexists a metric in [g0] with

1(g) > 0 and 2(g) > 0.

Moreover, there is a regular metric g = e2ug0 C2([g0]) satisfying(90) 2(g) = e4u,

for = (g0, 2) > 0.

Proof. We define a function h

h : R (0, +]

h() = a :=

infgC1([g0])

E(g)(M1(g)dvol(g))

n4n2

if n = 4;M

2(g)dvol(g) if n = 4;

Thus, h is a non-increasing function if n 4 and a non-decreasing function ifn = 3. First,we consider the case n 4. Define a set A(91) A := { (0, +)| h() < 0}.By the assumption that (g0, 2) > 0, it is easy to check that

(92) ((g0, 2), +) A and (0, (g0, 2)) A = .


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Hence for any > (g0, 2) we have h() < 0. By Proposition 1 we have a smoothmetric gu = e

2ug0 C1([g0]), which solves equation (81) for = h() and satisfies

M1(gu )dvol(gu ) = 1. From Corollary 5, the set of solutions{u, ((g0, 2), (g0, 2) + 1)}

is uniformly bounded in C2 norm. From equation (81), 1(gu ) is uniformly bounded frombelow by a positive constant for ((g0, 2), (g0, 2) + 1), since > (g0, 2) > 0 andalso we have

2(W)

1(W) 1

21(W), provided that 1(W) > 0. By the Krylovs result, the set

of solutions {u, ((g0, 2), (g0, 2) + 1)} is also uniformly bounded in C2, normfor > 0. Define

(93) 0 := lim(g0,2)+

h().

When

(g0, 2), u (by taking a subsequence) converges in C

2 norm to u0

+1 ,

which is a solution of (81) for = (g0, 2) and = 0. Clearly, 0 0. We claim(94) 0 = 0.

Otherwise, E(g0,2)(u0) < 0, which implies that there is some small > 0 such that ((g0, 2), (g0, 2)], we have E(u0) < 0. Hence h() < 0 for all ((g0, 2), +),that is, ((g0, 2) , +) A. This contradicts to (92). Hence we have (94), which isequivalent to say that

2(gu0) (g0, 2)e4u0 = 0.This means that u0 +2 solves (90) for = (g0, 2). In particular, u0 +2 .

The proof for the case n = 3 is similar by consider a set A defined by

A :={

(0, +

)|

h() > 0}

.

We leave the proof to the interested reader.

Remark 3. In the case n 4, the function h() is Lipschitz continuous. Indeed, for all < and for all g C1([g0]), we have

(95)

0 < E(u) E(u) = ( )

e4udvol(g) c( )(V ol(g))n4n

cY1([g])4nn2 ( )

1(g)dvol(g)

n4n2

This proves the claim.

Proof of Theorem 2. First we discuss the case Y2,1([g0]) > 0. Using Lemma 3, we have(g0, 2) > 0. By Proposition 3, the cone

+2 is not empty. Hence, from the results in [18]

for the cases n 5, in [10] for n = 4 and [32] for the case n = 3, there is a solution of (8).Now suppose Y2,1([g0]) = 0. As in the proof of Proposition 3, we consider the function

h() for all > 0. With the same arguments, for any small positive > 0, equation (90)


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admits the unique solution u, since h() < 0 > 0 if n 4 and h() > 0 > 0 ifn = 3. This fact is clear for n 4. And in the case n = 3, we have

inf

gC1([g0])1(g)dvol(g) e4udvol(g) > 0.

Moreover, as 0, the family of solution {u} is bounded in C2 norm. Thus, as 0,u converges in C

1, norm for any (0, 1) to some C1,1 function u. Consequently, thisC1,1 conformal metric gu = e

2ug0 [g0] +1 solves equation (9).

Let us consider the last case Y2,1([g0]) < 0. As in the previous case, there holds h() h(0) = Y2,1([g0]) < 0 > 0 if n 4, and h() h(0) = Y2,1([g0]) > 0 > 0 if n = 3and the family of solution {u} is bounded in C2 norm as 0. We claim this family isalso bounded in C2, norm for > 0. For this purpose, we will write equation (81) in the

new form. Recall wij = 2iju + uiuj |u|2

2 (g0)ij + (Sg0)ij. We define a new second order

tensor W = (wij) with

wij = wij + (g0)ij,

where is some small positive constant to be fixed later. It is clear that

(96) 1(W) = 1(W) + n

and

(97) 2(W) = 2(W) + 2(n 1)1(W) + n(n 1)2.

Denote u the unique solution of (81) with

1(g)dvol(g) = 1 for k = h() and for all

small > 0. Using (96) and (97), u satisfies the following equation

(98)

2(W)

n(n

1)2

2(n

1)(1(W)

n)

1(W) n = h()e2u

,

that is, u solves

(99)2(W)

1(W)+

+ nh()e2u + n(n 1)21(W)

= h()e2u + 2(n 1).

Choose a small > 0 such that (0, 0) and x M, we have(100) h()e2u(x) + 2(n 1) < 0.Now equation (100) is uniformly elliptic and concave for all (0, 0). From the classicalKrylovs result, the set of solutions {u} is also bounded in C2, norm for > 0. Passingto the limit, u converges in C

2, norm for any

(0, ) to some C2, function u

+

1,

which is a solution of (10). Finally, writing equation in the form (100) for = 0, theuniqueness comes from the maximum principle as in the proof of Proposition 1. Therefore,we finish the proof.

Proof of Theorem 1. When (g0, 2) > 0, the result follows from Proposition 3. When(g0, 2) = 0, the proof is the same as in the case Y2,1([g0]) = 0 in the previous proof.


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Proof of Corollary 1. It follows from Proposition 3 directly.

Proof of Corollary 2. From Theorem 2, the cone

+

2 is not empty. Set g C2([g0]). Froma result due to Gursky-Viaclovsky [29] the Ricci tensor of g is pointwise positive. Thanksto the result of Hamilton [34] by using the Ricci flow, M is diffeomorphic to a compact 3-dimensional Riemannian manifold with constant curvature. Moreover, M is diffeomorphicto S3/, where is a finite isometry subgroup ofS3 in the standard metric.

6. The Yamabe invariants Y2,1([g0])

In this section, We prove Theorem 3.

Proposition 4. Let(Mn, g0) be a compact Riemannian manifold withg0

+1 andn

3.

The followings hold1) If there is a regular conformal metric g [g0] +2 satisfying equation (8), thenY2,1([g0]) > 0;2) If there is a C1,1 conformal metric g [g0] +1 satisfying equation (9), thenY2,1([g0]) = 0;3) If there is a regular conformal metric g [g0] +1 solving equation (10), thenY2,1([g0]) < 0.

Theorem 3 follows from this Proposition. To prove the Proposition, we need the fol-lowing lemma.

Lemma 5. Let (Mn

, g0) be a compact Riemannian manifold with g0 +1 and n 3. At

most one of the followings holds:

1) equation (8) admits a regular solution g [g0] +2 ;2) equation (9) admits a C1,1 solution g [g0] +1 ;3) equation (10) admits a regular solution g [g0] +1 .

Proof. Let g [g0] +2 be a solution of equation (8). Without loss of generality, assumeg = g0. From a result of Guan-Wang [27], there is no C

1,1 solution of equation (9) in +2 .On the other hand, for any g = e2ug0 C1([g0]), let x0 be a minimum point of u. Atthis point, we get u = 0 and hessian matrix 2u is non-negative definite. Thus, theSchouten tensor at point x0 is in

+2 . As a consequence, equation (10) does not admit a

regular solution g

[g0]

+1 .

Now suppose g1 = e2u1g0 is a regular solution of equation (10) in [g0] +1 . As above,equation (8) does not admit a regular solution g [g0] +2 . As in the proof of Theorem2, u1 solves

(101)2(W)

1(W)+

ne2u1 + n(n 1)21(W)

= e2u1 + 2(n 1),


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where > 0 is a small positive number such that f(x) := e2u1 + (n 1) < 0 andwij = wij + (g0)ij. Let g2 = e

2u2g0 [g0] +1 be a C1,1 solution of equation (9). Thus,u2 is a subsolution to (101), that is,

(102) 2(

W)1(W)

+ nf(x)1(W)

e2u1 + 2(n 1)

since u2 solves

(103)2(W)

1(W)+

n(n 1)21(W)

= 2(n 1)

and

(104) 1(W) n.

Set v = u2 u1 and H(W, x) = 2(W) + nf(x)1(W)

. Denote

Hij := H(W, x)wij.

On the other hand, for any c R, u2 + c is also a C1,1 solution of equation (9). Withoutloss of generality, we could suppose v 0 and max v = 0. From (101) and (102), v is asubsolution of some uniformly elliptic second order operator, that is,

Lv :=i,j

Aij2ijv +i

Biiv 0,

where

Aij =

10

Hij(sW2 + (1 s)W1)ds.From the strong maximum principle, v

0. This contradiction yields the desired result

and we finish the proof of Lemma.

Corollary 6. Let (M, g0) be a compact Riemannian manifold with positive scalar curva-ture and of dimension n 4. The sign of (g0, 2), equivalently the sign of Y2,1([g0]), isa conformal invariant.

Proof of Proposition 4. It is a direct result of Theorem 2 and Lemma 5.

7. Geometric Applications

First we show that any manifold in class (1+) admits a psc metric with negative Yamabeconstant Y2,1 < 0. Therefore we can deform this metric in its conformal class to a psc

metric with negative 2-scalar curvature. Then we show that Y2,1 is continuous in asuitable sense.

Proof of Theorem 4. With the analysis established above, in order to show the Theo-rem, We need only to show that for any manifold in (1+) there is a psc metric g with

2(g)dvol(g) < 0.


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Here we use the well-known construction of Gromov-Lawson [19] for positive scalarcurvature metrics. See also [50]. Let Sp be an embedded sphere in M with trivial normalbundle of codimension q = n p 3. Let Sp Dq(r) be an embedding into M for somesmall constant r > 0. Let r0 be a constant fixed later as small as we want.

By the construction of Gromov-Lawson, we have a manifold (N1, h1) with an end Sp Sq1 [0, +) such that N1 (Sp Sq1 [1, ], h1) is isometric to M Sp Dq(r)and (Sp Sq1 [2, +), h1) is isometric to Sp Sq1(r) [2, +) with the productmetric. Here Sp is standard sphere with radius 1 and Sq1(r) is the standard spherewith a small radius r. The crucial point in the construction of Gromov-Lawson is thatthe scalar curvature of h1 is positive. Now we glue on N1 Sp Sq1(r) [0, +) aproduct manifold Sp Dq, where Dq is not equipped with the flat metric, but a productmetric ofSq1 [0, b] in a neighborhood of the boundary and of positive scalar curvatureon Dq. The result manifold is of positive scalar curvature and is diffeomorphic to M. Itcontains a product Sp Sq1(r) [2, 0] for a small r and a large 0.

Now we consider the case q = 3. In this case, its Ricci curvature, written as a diagonal

matrix, is diag{n4, , n4, r2

, r2

, 0}. Its Schouten tensor, also written as a diagonalmatrix and up to a multiple constant independent of r, is

diag{n 4 , , n 4 , r2 , r2 , }with

=1

2(n 1) ((n 3)(n 4) +2

r2).

It can be written as1

n 1 r2 diag{1, , 1, n 2, n 2, 1} + C,

Where C is a matrix whose entries are independent of r. To decide the sign of 2-scalarcurvature for this product, we only need to compute 2(

1,

,

1, n

2, n

2,

1) =

12 (n 1)(n 2). Hence choosing r small, we have negative 2-scalar curvature in thisproduct part. By choosing 0 large enough, we obtain a metric g on M with

M2(g) < 0.

Therefore for the conformal class [g] we have Y2,1([g]) < 0. By Theorem 2, we have a

conformal metric h with 2(h) 0.

When n = 4, we have another proof. In this case,

(2(g) +1

16|W|2)dvol(g)

is the Euler characteristic ofM up to a constant multiple. Since|W|2dvol(g) is invariant

in a conformal class, so is

2(g)dvol(g). In [1] the authors constructed a sequence ofmetrics gi satisfying that

Y1([gi]) Y1([g0]) and

|W|2dvol(gi) ,


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as i . Since Rg0 > 0, from this sequence we can find a psc metric gi with

2(gi)dvol(gi) 0 find gu = e2ug C1([g]) such that

2(gu)dvol(gu) < Y2,1([g]) +

1

2 and

1(gu)dvol(gu) = 1.

For sufficiently large j, we have gu C1([gj]). Since

2 and 1 are continuous for thisfixed function u, we have

Y2,1([gj]) Y2,1([g]) + ,for a large j, and hence (105).

Now we show that Y0 Y2,1([g]). Let Y2,1(Sn) be the second Yamabe constant for thestandard sphere Sn. Using a method given in [53], one can show that

Y2,1(Sn) Y2,1([g]),

for any metric psc metric g. Without loss of generality, we may assume Y0 < Y2,1(Sn).

Otherwise, we have limj

Y2,1

([gj

]) = Y0

= Y2,1

(Sn)

Y2,1

and hence we are done.Consider the following perturbed energy functional

J,(g) :=

(2(g) e2u)dvol(g))

(

e2u1(g)dvol(g))n4

n22

for small > 0 and > 0 in the cone C1. The analysis established above implies thatthe above functional admits a minimizer in C1([g]) for any psc metric. It is easy to checkthat for given j N, there are small constants j > 0 and j > 0 such that a minimizergj = e

2ujgj C1 of the perturbed functional Jj ,j satisfies

Jj ,j(gj) Y2,1([gj]) +1

j.

As a minimizer of J,, gj = e2ujgj satisfies an equation similar to (8), for which wehave local estimates Since gj converges to g in C

4,, the local estimates and Y0 < Y2,1(Sn)

imply that gj converges to a metric g = e2ug C1([g]) in C0-topology, and hence in

C1,1-topology. If g C1, we can show thatlimj

Y2,1([gj]) = limj

Jj ,j (gj) Y2,1([g]).


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Otherwise, we first show that

(106) limj

2(e

2ujgj)dvol(e2ujgj) = limj

2(e

2ujg)dvol(e2ujg)

and

(107) limj

1(e

2ujgj)dvol(e2ujgj) = limj

1(e

2ujg)dvol(e2ujg).

This is clear, for uj has a uniform C2 bound and gj conevrges to g. Then, we claim that

there exists metrics e2ujg C1([g]) such that

(108) limj

2(e

2ujg)dvol(e2ujg) = limj

2(e

2ujg)dvol(e2ujg)

and

(109) limj

1(e2uj

g)dvol(e2uj

g) = limj

1(e2uj

g)dvol(e2uj

g).

Given j N, we choose tj (0, 1) with lim tj = 1 such that e2tjujg C1([g]). To seethis, we compute(110)

e2tjuj1(e2tjujg) = tjguj t2jn 2

2|uj|2g + 1(g)

= tje2uj1(e2ujgj) + tj(guj gjuj + 1(g) 1(gj))

+tj(n 2

2|uj |2gj

n 22

|uj|2g)

+(1 tj)(tjn

2

2 |uj|2

g + 1(g)).

It is clear that

(111)

tj(guj gjuj + 1(g) 1(gj)) + tj( n 22 |uj|2gj n 22 |uj|2g)

Cg gjC1(|uj |2g + 1(g)).Thus, we can choose tj close to 1 such that e

2tjujg C1([g]). From (19), (22) andfact that uj has uniform bound C

0 norm, we have (108) and (109), which implies thatY0 = limj Y2,1([gj ]) Y2,1([g]). Hence we finish the proof of the Lemma.

In fact, the Lemma is true for C4-topology.Proof of Theorem 5. If M belongs to class (2+), then there is a psc metric with positive2-scalar curvature. By Theorem 3, Y2,1([g]) > 0. From Theorem 4 we have another pscmetric g with Y2,1([g]) < 0. Using the previous Lemma we have the third psc metric gwith Y2,1([g]) = 0. Now 1. follows from Theorem 2. The proof for 2. and 3. is the same.


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In order to prove the Conjecture (Trichotomy Thereom) as in [40] and [41], we need animplicit function theorem for the linearization operator of 2(g) in L

2,p. This seems to berather difficult for us, at least at the moment.

Example 1. Let us consider the manifold M = S3 S1. It is a locally conformally flatmanifold with 1 > 0 and 2 = 0 for the product metric. From

22(M) =

M

2(g) +1

16|W|2,

we haveS3S1 2(g) = 116

|W|2 0 for any metric g on S3 S1. Hence, S3 S1 belongsto class (20).

Example 2. For n > 4 let us consider the locally conformally flat manifold Sp Hnp,where p > n/2 and Hnp is a compact quotient of the hyperbolic space Hnp with sectionalcurvature 1. Let g0 be the product metric. It is clear that 1(g0) > 0, for p > n/2. Its2-scalar curvature, up to a positive constant multiple, is

(p n2

)2 n4

.

Hence, p > n+n

2 if and only if 2(g0) > 0. Now we consider such p with 2(g0) = 0,

namely p = n+n

2 . It is clear that n = (2m + 1)2 for some m N. For example m = 1,

M = S6 H3. In general Mm := S(m+1)(2m+1) Hm(2m+1). We conjecture that on sucha manifold there is a psc metric with positive 2-scalar curvature. A rough idea to checkthis can be made as follows. By Theorem 2, if it is not true, we know that any psc metricg on Mm has Y2,1([g]) 0, which is equivalent to

infgC1([g])M2(g)dvol(g) 0.

Hence 0 is be a min-max value ofM2(g)dvol(g) and the product metric g0 would be

a critical point of

2 on the space of all metrics. If this is true, then a result in [39]implies that g0 is a metric of constant sectional curvature. This certainly is false. Fromthis example, we propose a

Conjecture. When n > 4, class (20) is empty.

8. Further applications in fully nonlinear equations

With the method considered in this paper, we can also deal with the following problems.

1. When (M, g0) is a locally conformally flat manifold with g0 Ck1 (k > 1), themethod presented here can be used to study the equation

(112)k(g)

k1(g)= f.


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Similarly we define a Yamabe type invariant in [21] as follows

Yk,k1([g0]) = infMk(g)dvol(g)

(Mk1(g)dvol(g))n2k

n2k+2

,

for k n/2. We can show that if Yk,k1 > 0 ( Yk,k1 = 0 ans Yk,k1 < 0 resp.), then thereis a conformal metric g with k > 0 (k = 0 and k 0 resp.)

2. Let (M4, g0) be a compact 4-dimensional manifold with positive scalar curvature.One can consider the equation

(113)2 s|W|2

1= f,

where W is the Weyl tensor and s is a non-negative number. In this case for a fixed s

(2 s|W|2)dvol(g)

is a constant in a given conformal class. With the method presented here, we can showthat the number

(2 s|W|2)dvol(g) is positive, null or negative resp. if and only if

there is a conformal metric with

2 s|W|2 > 0, = 0 or 0 resp.The positive and null cases were studied already in [11] as mentioned above. This can beseen as a generalization of the following classical result: Let (M2, g0) be a closed surface.Its Euler characteristic (M2) = 12

MRgdvol(g) is positive, negative or null resp. if and

only if there is a conformal metric g with positive, negative or null scalar curvature resp.It is clear that equation (113) can also be considered on a higher dimensional manifold.

3. Our methods can also be applied to study the following fully nonlinear equationsk(2u)

k1(2u) = f

andk(2u + ug)

k1(2u + ug) = f,in the class of (k 1)-admissible functions.

4. Another interesting problem is a generalization of the prescribed scalar curvatureproblem. Let Mn = Sn and f : Sn R1 be a smooth function. We would like to ask ifthere is a conformal metric g

C1 such that

2(g) = f 1(g).

A Kazdan-Warner type neccesary condition could be obtained as Han [35] did for theprescribed k-scalar curvature problem. Here, and also in the previous problem, thefunction f need not be positive.


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Acknowledgement. This project was started several years ago together with Pengfei Guan.The key observation was discovered by Pengfei Guan and the second named author intheir study of fully nonlinear equations. We would like to thank Pengfeng Guan for manyhelpful discussions and encouragement. A part of the work was carried out while the first

author was visiting McGill University and he would like to thank the department andPengfei Guan for warm hospitality.

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(1960), 2137.

Laboratoire dAnalyse et de Mathematiques Appliquees, CNRS UMR 8050, Departement

de Mathematiques, Universite Paris XII-Val de Marne,, 61 avenue du General de Gaulle,

94010 Creteil Cedex, France

E-mail address: [email protected]

Department of Mathematics, National Taiwan University and Taida Institute for Math-

ematical Science, National Taiwan University, Taiwan


Faculty of Mathematics, University Magdeburg, D-39016, Magdebrug, Germany


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