Proceedings of the International Congress of Mathematicians
Hyderabad, India, 2010
Scalar Curvature, Conformal Geometry,
and the Ricci Flow with Surgery
Fernando Codá Marques∗
Abstract
In this note we will review recent results concerning two geometric problems
associated to the scalar curvature. In the first part we will review the solution
to Schoen’s conjecture about the compactness of the set of solutions to the
Yamabe problem. It has been discovered, in a series of three papers, that the
conjecture is true if and only if the dimension is less than or equal to 24. In the
second part we will discuss the connectedness of the moduli space of metrics
with positive scalar curvature in dimension three. In two dimensions this was
proved by Weyl in 1916. This is a geometric application of the Ricci flow with
surgery and Perelman’s work on Hamilton’s Ricci flow.
Mathematics Subject Classification (2010). Primary 53C21; Secondary 83C05.
Keywords. Scalar curvature; Yamabe problem; Ricci flow with surgery.
1. The Compactness Conjecture
The celebrated Riemann’s Uniformization Theorem states that any compact
Riemannian surface can be conformally deformed to a surface of constant Gauss
curvature. We will begin by describing the Yamabe Problem, which is a way of
generalizing uniformization to higher-dimensional manifolds.
1.1. The Yamabe problem. Let (M n , g) be a smooth compact Riemannian manifold of dimension n ≥ 3. The conformal class of g is the set
[g] = {g̃ = φ2 g : φ ∈ C ∞ (M ), φ > 0}.
The Yamabe Problem consists of finding a metric g̃ ∈ [g] of constant scalar
curvature.
∗ Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460320, Rio de Janeiro - RJ, Brazil. E-mail: [email protected].
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If we write g̃ = u n−2 g, u > 0, the transformation law for the scalar curvature
is
Rg̃ = −
n+2
n−2
4(n − 1) − n−2
u
Rg u .
∆g u −
n−2
4(n − 1)
Here Rg and Rg̃ denote the scalar curvatures of g and g̃, respectively, and
∆g is the Laplace-Beltrami operator associated with g. The linear operator
n−2
Lg = ∆g − 4(n−1)
Rg is usually called the conformal Laplacian of g.
Therefore the Yamabe Problem is equivalent to finding a positive solution
u to the partial differential equation
n+2
Lg (u) + c(n)Ku n−2 = 0
(1)
n−2
for some constant K, where c(n) = 4(n−1)
.
It turns out that the constant scalar curvature metrics g̃ ∈ [g] are the critical
points of the functional
R
Rg̃ dvg̃
Q(g̃) = M
,
n−2
R
n
dvg̃
M
known as the normalized total scalar curvature functional, when restricted to
the conformal class [g]. This variational structure played a prominent role in the
solution of the Yamabe Problem. In fact, after the initial paper of Yamabe [66],
which contained a gap, the combined works of Trudinger [63], Aubin [2], and
Schoen [52] established the existence of a minimizing solution in the conformal
class [g] of any given (M, g).
It is natural to ask if such solution is unique. In that respect, the conformal
classes of compact Riemannian manifolds should be classified in three types,
according to the sign of the Yamabe quotient:
Q(M, g) = inf Q(g̃).
g̃∈[g]
If the Yamabe quotient is negative, it follows from the Maximum Principle,
applied to equation (1), that the solution (of negative constant scalar curvature)
is unique. If the Yamabe quotient is zero, the solution (of zero scalar curvature)
is unique up to a constant factor. The structure of the set of solutions in the
positive Yamabe quotient case can be very rich though.
1.2. The set of solutions and some conjectures. It is convenient, in the positive Yamabe quotient case, to normalize the scalar curvature
of the solutions to be n(n − 1), for example. Hence let
Mg = {g̃ ∈ [g] : Rg̃ = n(n − 1)}
be the space of solutions with such normalization.
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The simplest and most important example is given by the standard sphere
(S n , g0 ). Here g0 denotes the metric induced by the Euclidean metric on the
unit sphere S n ⊂ Rn+1 . This case is special because the standard sphere is the
only compact manifold, up to conformal equivalence, which admits a noncompact group of conformal transformations Conf (S n ). By looking at the transformation law for the Einstein tensor (traceless Ricci) under conformal changes,
Obata (see [44]) proved that
Mg0 = {ψ ∗ (g0 ) : ψ ∈ Conf (S n )}.
Since these metrics are all isometric to each other, every solution is minimizing
in this particular example. Notice that Mg0 is noncompact.
The example of S 1 (L) × S n−1 with the product metric (L denotes the
length of the circle factor) was analyzed by R. Schoen in [54] . The set of
solutions in this case can be described as a finite union of one-parameter families, parametrized by circles, and depending on L. If L is big, there exists a large
number of high energy solutions with high Morse index. In fact, a theorem of
Pollack ([49]) shows that every compact Riemannian manifold of positive scalar
curvature can be perturbed, in the C 0 topology, to have as many solutions as
desired. These solutions generally have high energy and index.
In order to obtain more refined information about Mg , through the Morse
theory of Palais and Smale (see [45]), one needs to prove a priori estimates (or
compactness) for the set of solutions. The difficulty of that has its origin in the
fact that these estimates fail in the case of the standard sphere (S n , g0 ).
In a topics course at Stanford in 1988 (see also [55] and [56]), motivated by
the study of the locally conformally flat case, R. Schoen proposed the following
conjecture, together with an outline of a strategy to prove it:
Compactness Conjecture
The set Mg of solutions to the Yamabe Problem, in the positive Yamabe quotient case, is compact (in any C k topology) unless the manifold is conformally
equivalent to the standard sphere.
The cases which were covered in the Stanford notes are the locally conformally flat case, published in [55], and the three dimensional case, the argument
for which is in the paper of Schoen and Zhang [61] (used there to establish a
single simple point of blow-up for the prescribed scalar curvature problem on
S 3 ). In dimensions 4 and 5, the conjecture was proved by O. Druet (see [19]).
It follows from basic arguments in blow-up analysis that non-converging
sequences of solutions to the Yamabe Problem have to concentrate and form
bubbles (spheres) at some points of the manifold, referred to as blow-up points.
This phenomenon can be explicitly illustrated in the case of the standard sphere
(S n , g0 ). If π : S n − {p} → Rn denotes the stereographic projection, the metrics
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g̃ε = π ∗ (4uεn−2 δ), where
uε (x) =
ε
ε2 + |x|2
n−2
2
,
lie in Mg0 . Notice that uε blows-up at the origin as ε → 0.
The main step in Schoen’s program to establish compactness in dimensions
greater than or equal to 6 consisted in proving a related statement, known as
the Weyl Vanishing Conjecture, concerning the location of possible blow-up
points:
4
If x ∈ M is a blow-up point of a sequence of solutions g̃ν = uνn−2 g to the
Yamabe Problem, then the Weyl tensor of the metric g should satisfy
∇k Wg (x) = 0
for all 0 ≤ k ≤ [ n−6
2 ].
Over the past several years many people have worked on these problems. It
follows from the works of the author ([37]) and Y. Y. Li and L. Zhang ([33])
that compactness holds for n ≤ 7 in general, and for arbitrary n under the
assumption that the Weyl tensor vanishes nowhere to second order. In [34], Li
and Zhang proved compactness for n ≤ 11.
We should also point out that non-smooth blow-up examples were obtained
by A. Ambrosetti and A. Malchiodi in [1], and by M. Berti and Malchiodi in
[4]. In [20] O. Druet and E. Hebey have also obtained blow-up examples for
Yamabe-type equations.
In the past few years the Compactness Conjecture was completely solved in
a series of three articles ([11], [12], and [28]). The results of [11], [12], and [28]
put together give the following answer to the Compactness Conjecture:
The Compactness Conjecture is true if and only if n ≤ 24.
In the next sections we will give an overview of the results in these papers
and of their proofs. We refer the reader to [9] and [38] for related accounts (see
also [8]).
1.3. A compactness theorem. Throughout this section (M n , g) will
be a smooth compact Riemannian manifold of dimension n ≥ 3 and of positive
Yamabe quotient.
n+2
] we define
For any p ∈ [1, n−2
Φp = {u > 0, u ∈ C ∞ (M ) : Lg u + Kup = 0 on M }.
n+2
Although the geometric problem corresponds to the exponent p = n−2
, critical
with respect to the Sobolev embeddings, the consideration of the subcritical
Scalar Curvature, Conformal Geometry, and the Ricci Flow
815
solutions is useful for the purposes of applying Morse theory and computing
the total Leray-Schauder degree of the problem.
In [28], M. Khuri, R. Schoen, and the author proved the following theorem:
Theorem 1.1. Suppose 3 ≤ n ≤ 24. If (M n , g) is not conformally diffeomorphic to (S n , g0 ), then for any ε > 0 there exists a constant C > 0 depending
only on g and ε such that
C −1 ≤ u ≤ C
and
k u kC 2,α ≤ C,
for all u ∈ ∪1+ε≤p≤ n+2 Φp , where 0 < α < 1.
n−2
The following compactness result is a corollary of the previous theorem and
standard elliptic regularity theory:
Corollary. Suppose 3 ≤ n ≤ 24. If (M n , g) is not conformally diffeomorphic
to (S n , g0 ), then the set Mg is compact in any C k topology.
The proof of Theorem 1.1 is by contradiction and follows the strategy outlined in the notes of R. Schoen ([53]). In order to illustrate the ideas let us
n+2
. Suppose then
for simplicity restrict ourselves to the critical exponent p = n−2
that there exists a sequence uν ∈ Φ n+2 such that maxM uν = uν (xν ) → ∞ as
n−2
ν → ∞. Suppose x = lim xν .
The first step is to obtain sharp approximations of the blowing-up sequence
of solutions in a neighborhood of the blow-up point. This is achieved by establishing optimal pointwise estimates which generalize the ones obtained by
the author in [37]. These estimates assume the blow-up point is isolated simple
(one bubble only). After the strategy is carried out with success for that particular case, the results can be used to handle the more general case of multiple
blow-up by scaling arguments.
The important point of the estimates of [28] is that in high dimensions
it is necessary to have an expansion of uν that goes beyond the rotationally
symmetric first approximation (standard bubble). The approximate solutions
used are the same ones introduced by S. Brendle in [10] to generalize the test
function estimates of Aubin ([2]) and of Hebey and Vaugon ([26]), and prove
convergence of the Yamabe flow in any dimension.
The basic idea then is to use the Pohozaev Identity as an obstruction tool
in order to rule out the formation of bubbles at blow-up points. The following
is a general version of it in geometric form:
Proposition 1.2 (Pohozaev Identity, [60]). Let (Ωn , g) be a Riemannian domain, n ≥ 3. If X is a vector field on Ω, then
Z
Z
Z
n−2
X(Rg ) dvg + hDg X, Tg i dvg =
Tg (X, ηg ) dσg .
2n Ω
Ω
∂Ω
R
Here Tg = Ricg − ng g is the traceless Ricci tensor, (Dg X)ij = Xi;j + Xj;i −
2
n divg X gij is the conformal Killing operator, and ηg is the outward unit normal
to ∂Ω.
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4
Since the scalar curvature of gν = uνn−2 g is constant, the Pohozaev identity
applied to the geodesic ball Bδ (xν ) = {p ∈ M : r = dg (xν , p) ≤ δ}, endowed
∂
with the Riemannian metric gν and the radial vector field X = r ∂r
, r =
dg (xν , ·), yields
Z
Z
hDgν X, Tgν i dvgν =
(2)
Tgν (X, ηgν ) dσgν .
Bδ
∂Bδ
2
The idea then is to expand both integrals in powers of εν = uν (xν )− n−2 and
compare.
It turns out that the boundary integrals in the identity (2), when appropriately normalized, converge to a quantity which can be bounded above by −m,
where m is the ADM mass of the asymptotically flat and scalar flat metric
4
ĝ = GLn−2 g. Here GL denotes the Green’s function of the conformal Laplacian
with pole at x. The vanishing of the Weyl tensor to order [ n−6
2 ] at x is necessary
in order for the mass of ĝ to be well-defined.
In order to analyze the interior integrals in (2), we let (x1 , . . . , xn ) be normal
coordinates centered at xν . We write the components of the metric g in the form
gij (x) = exp(hij (x)),
and look at the Taylor expansion of hij around the origin:
hij (x) = Hij (x) + O(|x|d+1 ),
where d = [ n−2
2 ]. It is convenient to work in conformal normal coordinates (see
[32]) to simplify the computations. In that case Hij is a matrix whose entries
are polynomials of degree less than or equal to d, and such that
1. Hij (x) = Hji (x),
P
2.
k Hkk (x) = 0,
P
3.
k xk Hik (x) = 0,
for all 1 ≤ i, j ≤ n and x ∈ Rn . Let us denote the vector space of such matrices
by Vn .
The optimal pointwise estimates established in [28] lead to an expansion of
the interior integral of (2) in powers of εν , much like in the work of Aubin [2]. It
turns out that the relevant terms in this expansion are encoded
P in a canonical
quadratic form Pn defined on Vn . If n is odd, for instance, and i,j ∂i ∂j Hij = 0,
the quadratic form is given by
Pn (H, H) =
d
X X
i,j,l s,t=2
cs+t
Z
S1n−1 (0)
1
1
(s)
(t)
(s)
(t)
− ∂j Hij ∂l Hil + ∂l Hij ∂l Hij .
2
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Here H (s) denotes the homegeneous component of H of degree s, and
Z ∞ 2
(s − 1)sk+n−3
ck =
ds
(1 + s2 )n−1
0
for k < n − 2. We refer the reader to the appendix of [28] for a complete
definition of Pn .
The proof of Theorem 1.1 relies on an eigenvalue analysis done in [28]:
Proposition 1.3. The quadratic form Pn , defined on Vn , is positive definite
if n ≤ 24. Moreover, it has negative eigenvalues if n ≥ 25.
Suppose n ≤ 24. The vanishing of Hij at x, which is equivalent to the
vanishing of the Weyl tensor to order [ n−6
2 ], follows from the positivity of Pn
and estimates of the boundary term of (2). Hence the mass of ĝ is well-defined.
Since (M n , g) is not conformally equivalent to the standard sphere, the metric
ĝ is not flat, and therefore m > 0 by the Positive Mass Theorem. It can be seen
that this is in contradiction with the positivity of Pn by letting ν → ∞ in (2).
Therefore we conclude that blowup cannot occur if n ≤ 24.
Remark: The Positive Mass Theorem of General Relativity has been established by Schoen and Yau [57] in general for dimensions n ≤ 7. In [65] E. Witten
established it in any dimension for spin manifolds, while the locally conformally
flat case was handled by a special argument in [59]. See [36] for work towards
the general higher dimensional version.
As one of the consequences of compactness, we obtain the following statement about generic metrics (assuming n ≤ 24):
Corollary. Suppose that (M n , g) satisfies the assumptions of Theorem 1.1, and
assume that all critical points in [g] are nondegenerate. Then there are a finite
number of critical points g1 , . . . , gk and we have
1=
k
X
(−1)I(gj ) ,
j=1
where I(gj ) denotes the Morse index of the variational problem with volume
constraint.
1.4. Noncompactness results. One way to understand the noncompactness results is to look closely at the model case of the standard sphere
(S n , g0 ). As was pointed out before, the set of solutions of scalar curvature
n(n − 1) coincides in this particular case with the set of metrics coming from
the action of the conformal group on g0 , and therefore it is noncompact. We
might then be tempted to ask the following question:
Is there a way of perturbing the conformal structure of the standard sphere so
that the noncompactness persists?
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Fernando Codá Marques
It follows from Theorem 1.1 that this is impossible if n ≤ 24, but it turns
out that the answer to this question is yes for all n ≥ 25.
In a surprising paper ([11]), Simon Brendle constructed in 2008 the first examples of C ∞ metrics for which the compactness statement fails. These metrics
were small perturbations of the standard sphere in dimensions greater than or
equal to 52. In a subsequent article ([12]) Brendle and the author were able to
extend these examples to the dimensions 25 ≤ n ≤ 51.
The main theorems of [11] and [12] put together give:
Theorem 1.4. Suppose n ≥ 25. Given any ε > 0, there exists a smooth Riemannian metric g on S n and a sequence of positive functions vν ∈ C ∞ (S n )
(ν ∈ N) with the following properties:
(i) kg − g0 kC [1/ε] (S n ) < ε,
(ii) g is not conformally flat,
(iii) vν is a solution of the Yamabe equation (1) for all ν ∈ N,
4
(iv) Q(vνn−2 g) % Q(S n , g0 ) as ν → ∞,
(v) supS n vν → ∞ as ν → ∞.
The first step of the proof consists in reducing the construction to solving
a finite dimensional variational problem. This follows from a procedure known
as the Lyapunov-Schmidt reduction which we now briefly describe.
Since the standard sphere minus a point is conformally equivalent to the
Euclidean space (Rn , δ) through the stereographic projection, we can translate
the problem to the Euclidean setting. In this setting the solutions of the Yamabe
equation
n+2
(3)
∆ u + n(n − 2)u n−2 = 0
on Rn are the functions
u(ξ,ε) (x) =
ε
ε2 + |x − ξ|2
n−2
2
,
where (ξ, ε) ∈ Rn × (0, ∞). The solutions of the equation (3) can be also seen
as the critical points (at the same energy level) of the functional
Z 2n
|∇ u|2 − (n − 2)2 |u| n−2 dx
Fδ (u) =
Rn
restricted to the space
Z
2n
1,2
n
n
n−2
E = w∈L
(R ) ∩ Wloc (R ) :
|∇w| dx < ∞ .
2
Rn
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819
We consider Riemannian metrics g which are perturbations of the Euclidean
metric with compact support. We write g(x) = exp(h(x)), where h(x) is a tracefree symmetric two-tensor on Rn satisfying h(x) = 0 for |x| ≥ 1, and
|h(x)| + |∂h(x)| + |∂ 2 h(x)| ≤ α
for some small α > 0 and all x ∈ Rn .
Although the linearization of the equation (3) has a kernel, it is possible to
apply the Implicit Function Theorem if we restrict ourselves to the orthogonal
subspace
Z
E(ξ,ε) = w ∈ E :
ϕ(ξ,ε,k) w dx = 0 for k = 0, 1, . . . , n ,
Rn
where
ϕ(ξ,ε,0) (x) =
ε
ε2 + |x − ξ|2
n+2
2
ε2 − |x − ξ|2
ε2 + |x − ξ|2
ϕ(ξ,ε,k) (x) =
ε
ε2 + |x − ξ|2
n+2
2
2ε (xk − ξk )
ε2 + |x − ξ|2
and
for k = 1, . . . , n.
As a consequence we can find an (n + 1)-dimensional family of approximate
solutions:
Proposition 1.5. Let α > 0 be sufficiently small, depending only on the dimension. Given (ξ, ε) ∈ Rn × (0, ∞), there exists a function v(ξ,ε) ∈ E such that
v(ξ,ε) − u(ξ,ε) ∈ E(ξ,ε) and
Z 4
n−2
Rg v(ξ,ε) ψ − n(n − 2) |v(ξ,ε) | n−2 v(ξ,ε) ψ = 0
h∇v(ξ,ε) , ∇ψig +
4(n − 1)
Rn
for all test functions ψ ∈ E(ξ,ε) .
The problem reduces to finding critical points of the finite-dimensional functional Fg : Rn × (0, ∞) → R, given by
Z 2n
n−2
2
|∇v(ξ,ε) |2g +
Fg (ξ, ε) =
Rg v(ξ,ε)
− (n − 2)2 |v(ξ,ε) | n−2 dx.
4(n − 1)
Rn
We have that (ξ, ε) ∈ Rn × (0, ∞) is a critical point of Fg if and only if v(ξ,ε)
is a nonnegative weak solution (therefore smooth by a result of Trudinger [63])
of the Yamabe equation
∆g v(ξ,ε) −
n+2
n−2
n−2
= 0.
Rg v(ξ,ε) + n(n − 2)v(ξ,ε)
4(n − 1)
The positivity of v(ξ,ε) then follows from the Maximum PrincipleRand the fact
that v(ξ,ε) and u(ξ,ε) are close to each other in the norm kwkE = Rn |dw|2 dx.
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Fernando Codá Marques
The construction of the counterexample relies on a gluing procedure based
on some local model metrics. The model metrics g(x) = exp(h(x)) are such
that
X
hik (x) = µ λ2m f (λ−2 |x|2 )
Wipkq xp xq
p,q
for |x| ≤ ρ, where µ, λ, ρ are positive constants satisfying µ ≤ 1 and λ ≤ ρ ≤ 1,
f is a polynomial, and W : Rn × Rn × Rn × Rn → R is a nontrivial multi-linear
form which satisfies all the algebraic properties of the Weyl tensor. It is also
necessary that
n−2
.
2 deg(f ) + 2 <
2
These choices make it possible to approximate the energy function Fg (ξ, ε)
at appropriate scales by an auxiliary function F (ξ, ε), ξ ∈ Rn , ε ∈ (0, ∞), and
we are left with the algebraic problem of finding a polynomial f such that
F (ξ, ε) has a strict local minimum at (0, 1).
Notice that
X
µ λ2m f (λ−2 |x|2 )
Wipkq xp xq
p,q
belongs to Vn (as defined in the previous section), and it turns out that the
algebraic problem of finding f can be solved when the quadratic form Pn has
negative eigenvalues. It is proven in [11] that f can be chosen of degree 1 for
all n ≥ 52, and in [12] that it can be chosen of degree 3 for all 25 ≤ n ≤ 51.
The counterexamples g(x) = exp(h(x)) are obtained by gluing infinite copies
of the local models supported in small disjoint balls placed along the x1 -axis.
The N -th ball has radius 1/(2N 2 ) and is centered at yN = ( N1 , 0, . . . , 0) ∈ Rn ,
N ∈ N. If η : R → R is a smooth cutoff function such that η(s) = 1 for s ≤ 1
and η(s) = 0 for s ≥ 2, the two-tensor h(x) is given by
hik (x) =
∞
X
1
η(4N 2 |x − yN |) 2−(m+ 8 )N f (2N |x − yN |2 ) Hik (x − yN ),
N =N0
P
where yN = ( N1 , 0, . . . , 0) ∈ Rn , m = deg(f ), Hik (x) = p,q Wipkq xp xq , and
N0 is sufficiently large.
Since the metric g is in local model form in each of the infinitely many
balls B1/(2N 2 ) (yN ), we can apply the Lyapunov-Schmidt reduction infinitely
many times to obtain a sequence vν of solutions to the Yamabe equation as in
Theorem 1.4.
We should notice that even though the Weyl tensor of these counterexamples
vanishes to all orders at the blow-up point (0 ∈ Rn ), recent work of the author
([39]) shows that they can be perturbed to provide counterexamples to the Weyl
Vanishing Conjecture as well.
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821
2. The Connectedness Problem
In this section we will discuss a connectedness result for the space of positive
scalar curvature metrics on an orientable compact 3-manifold. We refer the
reader to [50] for a nice survey on related questions.
In 1916 H. Weyl proved the following connectedness result:
Theorem 2.1 ([64]). Let g be a metric of positive scalar curvature on the twosphere S 2 . There exists a continuous path of metrics µ ∈ [0, 1] → gµ on S 2 , of
positive scalar curvature, such that g0 = g and g1 has constant curvature.
Weyl’s proof is a nice application of the uniformization theorem. It is based
on the existence of a constant curvature metric g in the conformal class of g.
We can choose g so that Rg = 2. If g = e2f g, we define gµ = e2µf g. The
transformation law for the scalar curvature in two dimensions gives
Rgµ = e−2µf Rg − 2µ ∆g f
=
(1 − µ)Rg e−2µf + 2µe2(1−µ)f .
It is clear that Rgµ > 0 for all µ ∈ [0, 1], if Rg > 0. The space of metrics of
positive scalar curvature on S 2 is in fact contractible (see [51]).
There are several positive curvature conditions that are satisfied by the standard sphere in higher dimensions (positive scalar curvature, Ricci curvature,
sectional curvature, etc). Each one of them leads to a different connectedness
problem. Since there is no general uniformization theorem, other tools have to
be developed.
In his famous 1982 paper R. Hamilton ([23]) introduced the equation
∂g
= −2Ricg ,
∂t
known as the Ricci flow, and proved the existence of short time solutions
with arbitrary compact Riemannian manifolds as initial data. In [23], Hamilton
proved that the Ricci flow preserves positive scalar curvature in any dimension,
and that positive Ricci curvature (Ric > 0) and positive sectional curvature
(sec > 0) are both preserved in dimension three. He also proved a convergence
result: if g(t) denotes a solution to the Ricci flow on a compact 3-manifold M
such that g(0) has positive Ricci curvature, then the flow becomes extinct at
finite time T > 0, and the volume one rescalings g̃(t) of g(t) converge to a constant curvature metric as t → T . From that he could conclude that any compact
3-manifold of positive Ricci curvature is diffeomorphic to a spherical quotient
S 3 /Γ. It also follows from the method that Weyl’s connectedness result extends
to three dimensions under the conditions Ric > 0 or sec > 0.
We want to address the connectedness question in dimension three under
the weaker condition of positive scalar curvature R > 0. In order to state the
main result, let us introduce some notation. If M is a compact manifold, we will
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Fernando Codá Marques
denote by R+ (M ) the set of Riemannian metrics g on M with positive scalar
curvature Rg . The associated moduli space is the quotient R+ (M )/Diff(M ) of
R+ (M ) under the standard action of the group of diffeomorphisms Diff(M ).
Unless otherwise specified, the space of metrics on a given manifold will be
endowed with the C ∞ topology.
In [40], we prove the following connectedness theorem:
Theorem 2.2. Suppose that M 3 is a compact orientable 3-manifold such that
R+ (M ) 6= ∅. Then the moduli space R+ (M )/Diff(M ) is path-connected.
As a corollary we obtain:
Corollary. Let g be a metric of positive scalar curvature on the three-sphere
S 3 . There exists a continuous path of metrics µ ∈ [0, 1] → gµ on S 3 , of positive
scalar curvature, such that g0 = g and g1 has constant sectional curvature.
Remark: Since the set Diff + (S 3 ) of orientation-preserving diffeomorphisms of
the 3-sphere is path-connected (J. Cerf, [15]), we have that the total space
R+ (S 3 ) is path-connected.
We should point out that the results for scalar curvature in higher dimensions are quite different. This was first noticed by N. Hitchin ([27]) in 1974. He
considered some index-theoretic invariants associated to the Dirac operator of
spin geometry, and proved that the spaces R+ (S 8k ) and R+ (S 8k+1 ) are disconnected for each k ≥ 1. In 1988 R. Carr ([14]) proved that the space R+ (S 4k−1 )
has infinitely many connected components for each k ≥ 2, extending the 7dimensional case (k = 2) established earlier by Gromov and Lawson in 1983 (see
Theorem 4.47 of [22]). This result was improved by M. Kreck and S. Stolz ([31])
in 1993, where they show that even the moduli space R+ (S 4k−1 )/Diff(S 4k−1 )
has infinitely many connected components for k ≥ 2. This means that on those
spheres there are infinitely many nonequivalent metrics of positive scalar curvature which are exotic in the sense that they do not come from deformations of
the standard metric. The same statement holds true for any nontrivial spherical
quotient of dimension greater than or equal to five, as proved by B. Botvinnik
and P. Gilkey in [7]. The surgery arguments used in these proofs break down
in the three-dimensional case.
The evolution equation for the scalar curvature under the Ricci flow is
∂Rg
= ∆Rg + 2|Ricg |2 .
∂t
It follows from the parabolic maximum principle that if Rg(0) ≥ R0 > 0, then
min Rg(t) ≥
M
1
R0
1
,
− n2 t
and the flow must end in finite time.
In two dimensions Hamilton ([24]) proved a convergence result: if g has
positive scalar curvature (or Gauss curvature) on S 2 , then the solution to the
Scalar Curvature, Conformal Geometry, and the Ricci Flow
823
normalized Ricci flow with initial condition (S 2 , g) converges to a constant
curvature metric. (See [17] for an extension to arbitrary g). It is interesting to
note that his arguments were made independent of uniformization by Chen, Lu
and Tian in [16]. This gives a Ricci flow proof of Weyl’s theorem.
The great difficulty in studying the scalar curvature connectedness problem
in dimensions greater than two is that the condition of Rg > 0 is too weak to imply convergence results. For instance, the condition of positive scalar curvature
is stable under connected sums if n ≥ 3. There is a construction of Gromov and
Lawson ([21]) that starts with two compact Riemannian manifolds (M1n , g1 )
and (M2n , g2 ), with positive scalar curvature, and replaces the union of two
small balls Bδ (p1 ) ⊂ M1 and Bδ (p2 ) ⊂ M2 with a small neck-like region N .
The result is a metric g1 #g2 of positive scalar curvature on the connected sum
M1 #M2 that coincides with the original metrics g1 and g2 outside N . Therefore, unlike in the case of positive Ricci curvature, neck-pinching singularities
can occur under the Ricci flow.
In order to deal with this kind of situation Hamilton introduced in [25], in
the context of four-manifolds with positive isotropic curvature, a discontinuous
evolution process known as Ricci flow with surgery. The Ricci flow with surgery,
with (M, g0 ) as initial condition, can be thought of as a sequence of standard
Ricci flows (Mi , gi (t)), each defined for t ∈ [ti , ti+1 ) and becoming singular at
t = ti+1 , where 0 = t0 < t1 < · · · < ti < ti+1 < · · · < ∞ is a discrete set,
M0 = M , and g0 (0) = g0 . The initial condition (Mi , gi (ti )) for each of these
Ricci flows is a compact Riemannian manifold obtained from the preceding
Ricci flow (Mi−1 , gi−1 (t))t∈[ti−1 ,ti ) by a specific process called surgery, which
depends on some choice of parameters. Entire components with uniformly large
curvature are discarded at each ti . The flow becomes extinct in finite time T > 0
if T = tj+1 for some j ≥ 0 and Mj+1 = ∅.
In three dimensions the existence of a Ricci flow with surgery and the study
of its properties were accomplished by G. Perelman in a series of three papers
[46], [47], [48]. It follows from Perelman’s breakthroughs that the surgeries
needed are of the simplest type, restricted to almost cylindrical regions. He is
able to prove, through a backwards induction argument, that if the Ricci flow
with surgery of an orientable compact Riemannian 3-manifold becomes extinct
in finite time, then the manifold is diffeomorphic to a connected sum of spherical
space forms and finitely many copies of S 2 × S 1 . Since this is the case if the
fundamental group is trivial, a proof of the Poincaré Conjecture is obtained
as an application. There is a different argument for the finite extinction time
result that uses minimal surfaces, due to T. Colding and B. Minicozzi (see [18]).
Another application is the topological classification by Perelman of the orientable compact 3-manifolds which admit metrics of positive scalar curvature
(see [58] and [22] for earlier results with different methods). Since the surgeries
only increase scalar curvature, the associated Ricci flows with surgery have
to become extinct in finite time. Therefore the assumption of Theorem 2.2 is
824
Fernando Codá Marques
equivalent to asking that M is diffeomorphic to a connected sum of spherical
space forms and finitely many copies of S 2 × S 1 .
In order to explain the strategy to prove Theorem 2.2 let us introduce the
concept of a canonical metric. Let h be the metric on the unit sphere S 3 induced
by the standard inclusion S 3 ⊂ R4 . A canonical metric is any metric obtained
from the 3-sphere (S 3 , h) by attaching to it finitely many constant curvature
spherical quotients (through the Gromov-Lawson procedure), and adding to it
finitely many handles (Gromov-Lawson connected sums of S 3 to itself). The
resulting manifold M is diffeomorphic to
S 3 #(S 3 /Γ1 )# . . . #(S 3 /Γk )#(S 2 × S 1 )# . . . #(S 2 × S 1 ),
where Γ1 , . . . , Γk are finite subgroups of SO(4) acting freely on S 3 . The resulting metric ĝ is locally conformally flat and has positive scalar curvature.
Two canonical metrics on M are in the same path-connected component of the
moduli space R+ (M )/Diff(M ).
Given a metric g0 in R+ (M ), the strategy is to use the Ricci flow with
surgery (Mi3 , gi (t))t∈[ti ,ti+1 ) with initial condition g0 (0) = g0 to construct a
continuous path in R+ (M ) that starts at g0 and ends at a canonical metric.
As in the proof of the Poincaré Conjecture we use backwards induction on
the set of singular times ti . We need a combination of the heat flow technique
(Hamilton’s convergence result [23]) and the conformal method (as in Weyl’s
proof) to deform the entire components that are discarded along the flow,
including those at the extinction time. These components have known topology:
S 3 , RP 3 , RP 3 #RP 3 , or S 2 ×S 1 . We also use the connected sum construction of
Gromov and Lawson to undo the surgeries, making sure the final deformation
is continuous despite the fact that the Ricci flow with surgery is a discontinuous
process in its nature. A key observation for the induction is that a GromovLawson connected sum of finitely many canonical metrics is in the same pathconnected component (in the space of positive scalar curvature metrics) of a
single canonical component. This follows from the conformal method.
The work of Perelman on the description of singularities (the existence of
canonical neighborhoods, for example) under Hamilton’s Ricci flow is fundamental ([46], [47], and [48]). We refer the reader to [13], [30], and [41] for some
detailed presentations of the arguments due to Perelman. See also [5], [6] and
[42].
2.1. Some applications to General Relativity. It turns out that
we can use the previous results to study the topology of certain spaces of metrics
which are relevant in General Relativity. In this section the spaces are always
endowed with the topology associated to some natural weighted Hölder norm
Cβk,α (see [40] for more details).
Scalar Curvature, Conformal Geometry, and the Ricci Flow
825
We say that (g, h) is an asymptotically flat initial data set on R3 if g is a
Riemannian metric and h is a symmetric (0, 2)-tensor on R3 such that
|gij − δij |(x) + |x||∂ gij |(x) + |x|2 |∂ 2 gij |(x) = O(1/|x|),
|hij |(x) = O(1/|x|2 ),
as x → ∞.
The full set of solutions to the vacuum Einstein constraint equations is the
set M of all asymptotically flat initial data sets (g, h) defined on R3 such that
a) Rg + (trg h)2 − |h|2 = 0,
b) ∇i hi j − ∇j (trg h) = 0.
It goes back to the work of Choquet-Bruhat that the equations above are the
precise conditions one needs in order to solve the Cauchy problem for Einstein
equations: find a spacetime V (4-dimensional Lorentzian manifold) satisfying
the vacuum Einstein equations RicV = 0 (zero Ricci curvature) and an embedded hypersurface M 3 ⊂ V such that the induced metric on M is g and the
second fundamental form of M is h. We refer the reader to [3] for a nice survey
on the constraint equations.
Question: Is the space M path-connected?
These metrics no longer have nonnegative scalar curvature, so it would be
interesting to find methods to study their deformations.
There is an important special case in which Rg ≥ 0. Let M0 be the set of
all asymptotically flat initial data sets (g, h) on R3 such that
a) trg h = 0,
b) Rg = |h|2 ,
c) (divg h)j := ∇i hi j = 0.
In [40] we prove
Theorem 2.3. The set M0 is path-connected.
The idea is to first connect an initial data (g0 , h0 ) ∈ M0 into data of the form
(ĝ, 0) with ĝ scalar-flat, through the conformal method (Lichnerowicz equation).
We can then assume, by a perturbation argument, that ĝ can be conformally
compactified, i.e., ĝ is a blow-up G4x g of a positive scalar curvature metric g on
S 3 . Here Gx denotes the Green’s function associated to the conformal Laplacian
Lg = ∆g − 18 Rg of g, with pole at x ∈ S 3 . By deforming g, the connectedness of
R+ (S 3 ) can be used to construct a continuous path of asymptotically flat and
scalar-flat metrics on R3 connecting G4x g to the flat metric. Along the way we
also prove that the space of asymptotically flat metrics on R3 of nonnegative
scalar curvature is path-connected. This space was studied previously by B.
Smith and G. Weinstein in [62], where they established connectedness of the
subspace of metrics that admit a quasi-convex global foliation.
826
Fernando Codá Marques
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