EXISTENCE RESULTS FOR THE YAMABE PROBLEM ON MANIFOLDS WITH BOUNDARY

FERNANDO C. MARQUES

Abstract. Given a compact Riemannian manifold with umbilic boundary, we prove existence of conformally related metrics of zero scalar curvature and constant mean curvature on the boundary, under hypotheses on the Weyl tensor. In order to carry out the estimates on the Sobolev quotient, we also prove existence of conformal Fermi coordinates.

Contents

1. Introduction 1 2. Some expansion formulas 4 3. Conformal Fermi Coordinates 8 4. Sobolev quotient when n ≥ 8 12 References 21

1. Introduction

The classical Riemann mapping theorem is one of the most famous results in Mathematics. It states that any simply connected, proper domain of the plane is conformally diffeomorphic to a disk. One cannot hope to prove such a theorem in the higher dimensional setting, since the conformal structure becomes much more rigid. For instance, the only bounded open subsets of ℝ^{n}, n ≥ 3, that are conformally diffeomorphic to euclidean balls are the euclidean balls themselves. Nevertheless, as a generalization of the Riemann mapping theorem, we may ask the following question:

Yamabe Problem

Let (M^{n}, g) be an n-dimensional, compact, Riemannian manifold with boundary ∂M, n ≥ 3. Is there a conformally related metric g̃ with zero scalar curvature and boundary ∂M of constant mean curvature?

12000 Mathematics Subject Classification. Primary 53C25; Secondary 58J60.

In analytical terms, the problem we are interested in is equivalent to finding a smooth positive solution to the nonlinear boundary-value problem:

(1.1)  \{ \begin{matrix}Δ_{g}u − \frac{n−2}{4(n−1)}R_{g}u = 0 & in & M, \\ \frac{∂u}{∂η} + \frac{n−2}{2}h_{g}u = \frac{n−2}{2}cu^{\frac{n}{n−2}} & on & ∂M, \end{matrix}

where Δ_{g} denotes the Laplace-Beltrami operator of the metric g, R_{g} is the scalar curvature, h_{g} is the mean curvature of the boundary with respect to g, η is the outward unit normal vector to ∂M and c is a constant. Given a solution u of problem (1.1), the metric g̃ = u^{\frac{4}{n−2}} g has the required properties. Note that the equations (1.1) are specially interesting, given that the nonlinearity resides in the boundary condition.

This problem was studied by J. F. Escobar in [4], where he introduced the Sobolev quotient

Q(M, ∂M) = inf\{Q(ϕ) : ϕ ∈ C^{1}(\overline{M}), ϕ ≢ 0 on ∂M\},

where

Q(ϕ) = \frac{∫_{M}(|∇ϕ|^{2} + \frac{n−2}{4(n−1)}R_{g}ϕ^{2})dv_{g} + \frac{n−2}{2} ∫_{∂M} h_{g}ϕ^{2}dσ_{g}}{(∫_{∂M} |ϕ|^{2 \frac{(n−1)}{(n−2)}} dσ_{g})^{\frac{(n−2)}{(n−1)}}} .

Here dv_{g} and dσ_{g} denote the volume forms of M and ∂M, respectively.

It was proven ([4]) that this Sobolev quotient is conformally invariant, and it always satisfies

Q(M, ∂M) ≤ Q(B^{n}, ∂B^{n}),

where B^{n} denotes the unit ball in ℝ^{n} endowed with the euclidean metric.

Under the hypothesis that Q(M, ∂M) is finite (it could be −∞, as pointed out by J. Zhiren), J. F. Escobar showed that the strict inequality

Q(M, ∂M) < Q(B^{n}, ∂B^{n})

implies the existence of a solution to problem (1.1). In that same paper, using this fact, he establishes existence for many cases:

Theorem 1.1 (Escobar). Suppose one of the following conditions holds: (1) n ≥ 7, and M has a nonumbilic point on ∂M;

(2) n ≥ 6, with M locally conformally flat and ∂M umbilic;

(3) n = 4 or 5, and ∂M umbilic;

(4) n = 3.

If the Sobolev quotient Q(M, ∂M) is finite, then there exists a smooth metric u^{\frac{4}{n−2}} g, u > 0 on \overline{M}, with zero scalar curvature in M and constant mean curvature on ∂M.

For manifolds with umbilic boundary, this last result does not cover the case n ≥ 7, unless the manifold is locally conformally flat.

In this paper we solve the case of manifolds with umbilic boundary, n ≥ 9, under the hypothesis that the Weyl tensor W_{g} does not vanish identically on ∂M. We also prove existence of solutions when n ≥ 8 and the Weyl tensor \overline{W}_{g} of the restriction of the metric to the boundary is not zero at some point P ∈ ∂M.

The exact statement is:

Theorem 1.2. Let (M^{n}, g) be an n-dimensional, compact, Riemannian manifold with umbilic boundary. Assume M^{n} satisfies any of the following two conditions:

(1) n ≥ 8, and \overline{W}_{g}(P) ≠ 0 for some P ∈ ∂M;

(2) n ≥ 9, and W_{g}(P) ≠ 0 for some P ∈ ∂M. Then

Q(M, ∂M) < Q(B^{n}, ∂B^{n}).

Moreover, if the Sobolev quotient Q(M, ∂M) is finite, then there exists a smooth metric u^{\frac{4}{n−2}} g, u > 0 on \overline{M}, with zero scalar curvature and constant mean curvature on ∂M.

This result is analogous to the one in [1], where T. Aubin proves that the Yamabe problem for manifolds without boundary (which consists in finding conformally related metrics of constant scalar curvature) is solvable when the manifold is not locally conformally flat and its dimension is at least six. This theorem together with the solution of the remaining cases by R. Schoen [11] gave an affirmative answer to the Yamabe problem without boundary, which had been studied before by H. Yamabe [13] and N. Trudinger [12].

Our proof is based on explicitly constructing a function ϕ such that

Q(ϕ) < Q(B^{n}, ∂B^{n}).

In order to carry out the estimates, we prove the existence of conformal Fermi coordinates around a given point P ∈ ∂M. More precisely we prove the existence of a conformally related metric for which the volume element in Fermi coordinates is almost Euclidean. They are similar to the conformal normal coordinates [9], useful in the problem without boundary, and they simplify the asymptotic analysis, clarifying, for example, the role played by the Weyl tensor.

Other works concerning conformal deformation on manifolds with boundary include [5], [6], [7] and [8].

Notations. Throughout this paper we will make use of the index notation for tensors, commas denoting covariant differentiation. We will adopt the summation convention whenever confusion is not possible. When dealing with manifolds with boundary, we will use the indices 1 ≤ i, j, k, l, m, p, r, s ≤ n − 1 and 1 ≤ a, b, c, d ≤ n. The greek letters α and β will be multiindices. In Fermi coordinates the letter t will refer to the normal direction.

We will denote by g the Riemannian metric, by ∇ the covariant derivative and by R_{abcd} the full Riemannian curvature tensor. The Ricci tensor will be denoted by R_{ab}, the scalar curvature by R, the second fundamental form of the boundary by h_{ij} and its trace will be H. The Weyl tensor will be denoted by W. Also det g = det(g_{ab}) and Δ will denote the Laplace-Beltrami operator. Lines under or over an object mean the restriction of the metric to the boundary is involved.

We shall use the definition Sym_{i_{1}...i_{r}}T_{i_{1}...i_{r}} = \frac{1}{r!} ∑_{σ} T_{i_{σ(1)}...i_{σ(r)}}, where σ ranges over all the permutations of the set {1, …, r}.

Acknowledgements. The results in this paper are part of the author’s doctoral thesis [10]. The author would like to express his gratitude to his advisor Prof. José F. Escobar, for his encouragement and support. While the author was at Cornell University, he was fully supported by CNPq-Brazil.

2. Some expansion formulas

In this section we shall prove some expansion formulas that will be useful later.

First we will derive the expansion of a symmetric 2-tensor up to third order in geodesic normal coordinates. We will use this formula in the proof of Lemma 2.2, when computing the expansion of the metric in Fermi coordinates.

Lemma 2.1. Let B_{ij} be a symmetric 2-tensor on M. Then, in geodesic normal coordinates (x_{1}, …, x_{n}) centered at P ∈ M:

B_{ij}(x) = (2.1)

B_{ij}(0) + B_{ij,k}x_{k} + (\frac{1}{2}B_{ij,kl} − \frac{1}{3}Sym_{ij}(R_{iksl}B_{sj}))x_{k}x_{l}

+ (\frac{1}{6}B_{ij,klm} − \frac{1}{3}Sym_{ij}(R_{iksl}B_{sj,m}) − \frac{1}{6}Sym_{ij}(R_{ilsm,k}B_{sj}))x_{k}x_{l}x_{m}

+ O(r^{4}),

where r = |x| and every coefficient above is evaluated at P.

Proof. Let (x_{1}, …, x_{n}) be g-normal coordinates centered at P. We know that Y_{i}(t) = t\frac{∂}{∂x_{i}} is a Jacobi field along the radial geodesic γ(t) = exp_{P} (tT), |T | = 1. Thus ∇_{T}^{2} Y_{i} = R(T, Y_{i})T, and differentiating:

∇_{T}^{3} Y_{i} = (∇_{T} R)(T, Y_{i})T + R(T, ∇_{T} Y_{i})T,
∇_{T}^{4} Y_{i} = (∇_{T}^{2} R)(T, Y_{i})T + 2(∇_{T} R)(T, ∇_{T} Y_{i})T + R(T, ∇_{T}^{2} Y_{i})T,
∇_{T}^{5} Y_{i} = (∇_{T}^{3} R)(T, Y_{i})T + 3(∇_{T}^{2} R)(T, ∇_{T} Y_{i})T + 3∇_{T} R(T, ∇_{T}^{2} Y_{i})T
+ R(T, ∇_{T}^{3} Y_{i})T.

Since Y_{i}(0) = 0 and ∇_{T} Y_{i}(0) = \frac{∂}{∂x_{i}}, we get

∇_{T}^{2} Y_{i}(0) = 0,
∇_{T}^{3} Y_{i}(0) = R(T, \frac{∂}{∂x_{i}})T,
∇_{T}^{4} Y_{i}(0) = 2(∇_{T} R)(T, \frac{∂}{∂x_{i}})T,
∇_{T}^{5} Y_{i}(0) = 3(∇_{T}^{2} R)(T, \frac{∂}{∂x_{i}})T + R(T, R(T, \frac{∂}{∂x_{i}})T)T.

Define f(t) = B(Y_{i}(t), Y_{j}(t)). Since

f^{(r)}(0) = ∑_{p=0}^{r} ∑_{q=0}^{r−p} \binom{r}{p}\binom{r − p}{q}∇_{T}^{r−p−q}B(∇_{T}^{p} Y_{i}, ∇_{T}^{q} Y_{j}),

where f^{(r)} denotes the r-th derivative of f, we compute f(0) = 0, f′(0) = 0 and

f^{(2)}(0) = 2B(\frac{∂}{∂x_{i}} , \frac{∂}{∂x_{j}}),
f^{(3)}(0) = 6∇_{T} B(\frac{∂}{∂x_{i}} , \frac{∂}{∂x_{j}}),
f^{(4)}(0) = 12∇_{T}^{2} B(\frac{∂}{∂x_{i}} , \frac{∂}{∂x_{j}}) + 8Sym_{ij}B(\frac{∂}{∂x_{i}} , R(T, \frac{∂}{∂x_{j}})T),
f^{(5)}(0) = 20∇_{T}^{3} B(\frac{∂}{∂x_{i}} , \frac{∂}{∂x_{j}}) + 40Sym_{ij}∇_{T} B(\frac{∂}{∂x_{i}} , R(T, \frac{∂}{∂x_{j}})T)
+ 20Sym_{ij}B(\frac{∂}{∂x_{i}} , ∇_{T} R(T, \frac{∂}{∂x_{j}})T).

In order to get the expansion formula (2.1), we observe that B_{ij}(tT) = f(t)t^{−2} and write tT = x. □

Suppose (M^{n}, g) is a compact Riemannian manifold with boundary and P ∈ ∂M. Let (x_{1}, …, x_{n−1}) be geodesic normal coordinates on the boundary, centered at P, and denote by T (x) the inward unit vector normal to the boundary at x ∈ ∂M. We say that (x_{1}, …, x_{n−1}, t), for small t ≥ 0, are the Fermi coordinates (centered at P) of the point exp_{x}(tT (x)) ∈ M. It is easy to see that in these coordinates g_{tt} ≡ 1 and g_{ti} ≡ 0. In particular, det(g_{ab}) = det(g_{ij}).

Now we turn to the expansion of the metric (up to fourth order) in such coordinates.

Lemma 2.2. In Fermi coordinates (x_{1}, …, x_{n−1}, t) centered at P ∈ ∂M:

g_{ij}(x, t) = (2.2)

δ_{ij} − 2h_{ij}t

−\frac{1}{3}\overline{R}_{ikjl}x_{k}x_{l} − 2h_{ij,\underline{k}}tx_{k} + (−R_{titj} + h_{is}h_{sj})t^{2}

−\frac{1}{6}\overline{R}_{ikjl,\underline{m}}x_{k}x_{l}x_{m} + (−h_{ij,\underline{kl}} + \frac{2}{3}Sym_{ij}(\overline{R}_{iksl}h_{sj}))tx_{k}x_{l}

+ (−R_{titj,\underline{k}} + 2Sym_{ij}(h_{is,k}h_{sj}))t^{2}x_{k}

+ \frac{1}{6}(−2R_{titj,t} + 8Sym_{ij}(h_{is}R_{tstj}))t^{3}

+ (−\frac{1}{20}\overline{R}_{ikjl,\underline{mp}} + \frac{2}{45}\overline{R}_{iksl}\overline{R}_{jmsp})x_{k}x_{l}x_{m}x_{p}

+ (−\frac{1}{3}h_{ij,\underline{klm}} + \frac{2}{3}Sym_{ij}(\overline{R}_{iksl}h_{sj,\underline{m}})

+ \frac{1}{3}Sym_{ij}(\overline{R}_{ilsm,\underline{k}}h_{sj}))tx_{k}x_{l}x_{m}

+ (−\frac{1}{2}R_{titj,\underline{kl}} + \frac{1}{3}Sym_{ij}(\overline{R}_{iksl}R_{tstj}) + \frac{1}{2}(h_{is}h_{sj})_{,\underline{kl}}

− \frac{1}{3}Sym_{ij}(\overline{R}_{ikrl}h_{rs}h_{sj}))t^{2}x_{k}x_{l}

+ \frac{1}{6}(−2R_{titj,t\underline{k}} + 8Sym_{ij}(h_{is,\underline{k}}R_{tstj})

+ 8Sym_{ij}(h_{is}R_{tstj,\underline{k}}))t^{3}x_{k}

+ \frac{1}{24}(−2R_{titj,tt} + 12Sym_{ij}(h_{is}R_{tstj,t})

+ 8R_{tits,t}R_{tstj,t} − 8h_{ir}h_{js}R_{trts})t^{4}
+ O(r^{5}),

where r = |(x, t)| and every coefficient is computed at P. Here underlined indices mean covariant differentiation as tensors on the boundary.

Proof. Let (x_{1}, …, x_{n−1}, t) be Fermi coordinates around P ∈ ∂M. To begin with, we shall differentiate the metric with respect to the normal direction.

Note that \frac{∂}{∂x_{i}} is a Jacobi field along the geodesic γ_{x}(t) = (x, t), x ∈ ∂M. Let us denote by X such a vector field and by T the tangent vector to the geodesic.

Then ∇_{T}^{2} X = R(T, X)T, and by differentiating:

∇_{T}^{3} X = (∇_{T} R)(T, X)T + R(T, ∇_{T} X)T
∇_{T}^{4} X = (∇_{T}^{2} R)(T, X)T + 2(∇_{T} R)(T, ∇_{T} X)T + R(T, ∇_{T}^{2} X)T.

Note that defining v = X(0) ∈ T_{x}(∂M), it follows that ∇_{T} X(0) = −A_{x}v, where A_{x} denotes the selfadjoint linear operator associated with the second fundamental form at x ∈ ∂M, and so

\{ \begin{matrix}∇_{T}^{2} X(0) = R(T, v)T, \\ ∇_{T}^{3} X(0) = (∇_{T} R)(T, v)T − R(T, Av)T, \\ ∇_{T}^{4} X(0) = (∇_{T}^{2} R)(T, v)T − 2(∇_{T} R)(T, Av)T + R(T, R(T, v)T)T. \end{matrix}

By using A(\frac{∂}{∂x_{i}}) = g^{kl}h_{ik} \frac{∂}{∂x_{l}}, we can now differentiate g(\frac{∂}{∂x_{i}} , \frac{∂}{∂x_{j}}) with respect to T:

∂_{t}^{(r)}g_{ij} = ∑_{k=0}^{r} \binom{r}{k}g(∇_{t}^{k} \frac{∂}{∂x_{i}} , ∇_{t}^{r−k} \frac{∂}{∂x_{j}}),

and we get

(2.3)  ∂_{t}g_{ij}(x, 0) = −2h_{ij}(x);
(2.4) ∂_{t}^{(2)}g_{ij}(x, 0) = −2R_{titj} + 2g^{rs}h_{ir}h_{sj};
(2.5) ∂_{t}^{(3)}g_{ij}(x, 0) = −2R_{titj,t} + 8Sym_{ij}(g^{rs}h_{si}R_{tjtr})
(2.6) ∂_{t}^{(4)}g_{ij}(x, 0) = −2R_{titj,tt} + 12Sym_{ij}(g^{rs}h_{si}R_{tjtr,t})
+8(g^{rs}R_{tsti}R_{trtj} − g^{rq}g^{sp}R_{trts}h_{pi}h_{qj}).

In order to get the whole expansion formula (2.2), we just have to consider the right hand sides on equations (2.3),(2.4) and (2.5) as symmetric 2-tensors on the boundary ∂M, and apply Lemma 2.1. □

We will also need the expansion of the inverse g^{ij} under an additional hypothesis on the second fundamental form at P ∈ ∂M.

Lemma 2.3. Suppose \overline{∇}_{α}h_{ij} = 0 at P ∈ ∂M for every |α| ≤ 3. Then, in Fermi coordinates around P,

g^{ij}(x, t) = (2.7)

δ_{ij} + \frac{1}{3}\overline{R}_{ikjl}x_{k}x_{l} + R_{titj}t^{2}

+ \frac{1}{6}\overline{R}_{ikjl,m}x_{k}x_{l}x_{m} + R_{titj,k}t^{2}x_{k} + \frac{1}{3}R_{titj,t}t^{3}

+ (\frac{1}{20}\overline{R}_{ikjl,mp} + \frac{1}{15}\overline{R}_{iksl}\overline{R}_{jmsp})x_{k}x_{l}x_{m}x_{p}

+ (\frac{1}{2}R_{titj,kl} + \frac{1}{3}Sym_{ij}(\overline{R}_{iksl}R_{tstj}))t^{2}x_{k}x_{l}

+ \frac{1}{3}R_{titj,tk}t^{3}x_{k}

+ \frac{1}{12}(R_{titj,tt} + 8R_{tits,t}R_{tstj,t})t^{4} + O(r^{5}),

where r = |(x, t)|, and the curvatures are evaluated at P.

Remark. Because of the hypothesis on the second fundamental form, we do not need to use underlined indices anymore.

Proof. If g = (g_{ij}) we write the expansion

g = I + C_{2} + C_{3} + C_{4} + O(r^{5}),

where I is the identity matrix and C_{i} is homogeneous of degree i. For small r, we have

g^{−1} = ∑_{j=0}^{∞}(−1)^{j}(g − I)^{j},

and so

g^{−1} = I − C_{2} − C_{3} − C_{4} + C_{2}^{2} + O(r^{5}).

The expansion formula (2.7) now follows immediately by applying the equality (2.2). □

3. Conformal Fermi Coordinates

In this section we will prove the existence of special coordinates which will considerably simplify the future computations.

Proposition 3.1. (Conformal Fermi Coordinates) Let (M^{n}, g) be a compact Riemannian manifold with boundary, and P ∈ ∂M. For any given integer N ≥ 1, there exists a conformally related metric g̃ such that in g̃-Fermi coordinates around P:

det g̃ = 1 + O(r^{N}),

where r = |(x, t)|.

Proof. As in the case of conformal normal coordinates (see [9]), we will acomplish conformal Fermi coordinates by a metric g̃ = e^{2f}g, where f is a polynomial on the variables x_{1}, …, x_{n−1}, t.

Given an integer k ≥ 1, we will say that a tensor T belongs to C_{k} if its components can be written as polynomial expressions on the quantities \overline{∇}_{α}h_{ij} and ∇_{β}R_{abcd}, for |α| ≤ k − 1 and |β| ≤ k − 2. If the degree of f is k, then it is not difficult to see that any tensor in C_{k−1} will not change at P under the conformal deformation.

We will use induction on N. Consider N ≥ 1 and without loss of generality we can suppose

det g = 1 + O(r^{N}).

We want to find a homogeneous polynomial f of degree N such that

det g̃ = 1 + O(r^{N+1}),

with g̃ = e^{2f}g.

Since Fermi coordinates are geodesic normal coordinates when restricted to the boundary, and det(g_{ab}) = det(g_{ij}), we can choose the coefficients of f corresponding to the monomials not involving t just as we do in the case of conformal normal coordinates without boundary. It remains to choose the other coefficients of f, and for that we recall the following formulas (see [5]):

(3.1)  H̃ = e^{−f}(H − (n − 1)f_{,t}),
(3.2)  R̃_{ab} = R_{ab} − (n − 2)f_{,ab} + (n − 2)f_{,a}f_{,b}
−(Δf + (n − 2)|∇f|^{2})g_{ab}.

If we write g = (g_{ij}) = exp A, we get

(3.3)  g = I + A + ∑_{j=2}^{∞} \frac{1}{j!}A^{j},

and also

(3.4)  det g = e^{trA} = 1 + trA + ∑_{j=2}^{∞} \frac{1}{j!}(trA)^{j}.

It is not difficult to see, from the way we derived the expansions of the metric in the last section and from the identities (3.3) and (3.4), that the coefficients of the terms of order N in the expansion of det g̃ are, up to a factor:

(3.5)  Sym_{k_{1}...k_{N−1}}(H̃_{,k_{1}...k_{N−1}} − Ã_{k_{1}...k_{N−1}}),

and

(3.6)  Sym_{l_{1}...l_{r}}(R̃_{tt,t...tl_{1}...l_{r}} − B̃_{l_{1}...l_{r}}^{(r)}),

0 ≤ r ≤ N − 2, where the tensors Ã and B̃^{(r)} are in C_{N−1}. Here we are differentiating R̃_{tt} precisely N − 2 times.

Since, from what we have discussed before, Ã = A and B̃^{(r)} = B^{(r)}, the identities (3.1) and (3.3) imply that all we have to do is to solve:

(3.7)  Sym_{k}f_{,tk_{1}...k_{N−1}} = \frac{1}{n − 1}Sym_{k}(H_{,k_{1}...k_{N−1}} − A_{k_{1}...k_{N−1}}),

and

(3.8) Sym_{l}f_{,t...tl_{1}...l_{r}} = \frac{1}{n − 1}Sym_{l}(R_{tt,t...tl_{1}...l_{r}} − B_{l_{1}...l_{r}}^{(r)} − f_{,t...tiil_{1}...l_{r}}),

0 ≤ r ≤ N − 2. Here R_{tt} is differentiated exactly N − 2 times and f is differentiated N times.

The first set of equations (3.7) is easily accomplished by choosing the appropriates f_{,tk_{1}...k_{N−1}}. Having chosen the coefficients f_{,ik_{1}...k_{N−1}} and f_{,tk_{1}...k_{N−1}}, equality (3.8) determines the coefficients f_{,ttk_{1}...k_{N−2}} and so on, until we reach the last term f_{,tt...t}. □

In the next result we will compute certain geometric quantities at P in conformal Fermi coordinates, when the boundary is assumed umbilic.

Proposition 3.2. Suppose the boundary ∂M is umbilic. Then, in conformal Fermi coordinates around P ∈ ∂M, N ≥ 5,

(1) h_{ij} = h_{ij,k} = h_{ij,kl} = 0;

(2) R_{tt} = R_{tk} = \overline{R}_{kl} = 0;

(3) R_{tt,t} = R_{tt,k} = R_{tk,l} = 0;

(4) Sym_{klm}(\overline{R}_{kl,m}) = 0;

(5) R_{tt,tt} = −2(R_{titj})^{2};

(6) R_{tt,tk} = 0;

(7) Sym_{kl}R_{tt,kl} = 0;

(8) Sym_{klm}H_{,klm} = 0;

(9) Sym_{klmp}(\frac{1}{20} \overline{R}_{kl,mp} + \frac{1}{90} \overline{R}_{rkls}\overline{R}_{rmps}) = 0, where all the quantities above are computed at P ∈ ∂M.

Proof. From the equalities (2.2), (3.3), and (3.4), we obtain (3.9)

det g = 1−2Ht− \frac{1}{3}\overline{R}_{kl}x_{k}x_{l} −2H_{,k}tx_{k} +(−R_{tt} −(h_{ij})^{2} +2H^{2})t^{2} +O(r^{3}).

This already tells us that H = H_{,k} = \overline{R}_{kl} = 0 and R_{tt} = −(h_{ij})^{2}, even when the boundary is not umbilic at P. Since we are assuming umbilicity, we get as a consequence that h_{ij} = h_{ij,k} = 0, and so R_{tt} = 0. From the Codazzi equation

R_{ijkt} = h_{ik,j} − h_{jk,i},

we can also conclude R_{tk} = 0.

Under these conditions we get further:

det g =

1 − \frac{1}{6}\overline{R}_{kl,m}x_{k}x_{l}x_{m} − H_{,kl}tx_{k}x_{l} − R_{tt,k}t^{2}x_{k} − \frac{1}{3}R_{tt,t}t^{3}

+ (−\frac{1}{20}\overline{R}_{kl,mp} − \frac{1}{90}\overline{R}_{iksl}\overline{R}_{imsp})x_{k}x_{l}x_{m}x_{p} − \frac{1}{3}H_{,klm}tx_{k}x_{l}x_{m}

− \frac{1}{2}R_{tt,kl}t^{2}x_{k}x_{l} − \frac{1}{3}R_{tt,tk}t^{3}x_{k}

+ \frac{1}{24}(−2R_{tt,tt} − 4(R_{titj,t})^{2})t^{4} + O(r^{5}).

The result follows immediately, noting that H_{,kl} = 0 implies h_{ij,kl} = 0 because of umbilicity, and another application of Codazzi gives R_{tk,l} = 0.

□

Since conformal Fermi coordinates are normal on the boundary, we know that (see [9]) \overline{R} = O(r^{2}) and \overline{R}_{,ii} = −\frac{1}{6}|\overline{W} (P)|^{2}. Here \overline{W} denotes the Weyl tensor of the restriction of the metric to the boundary. Also, since R = 2R_{tt} + \overline{R} + ||h_{ij}||^{2} − H^{2}, one gets from the identities (1), (2) and (3) in Proposition 3.2 that R(P) = 0 and R_{,i}(P) = 0. Moreover, from the identity (7) in the same Proposition,

(3.10)  R_{,ii} = −\frac{1}{6}|\overline{W} (P)|^{2}.

We also have that R_{,t}(P) = 0, using the second Bianchi identity and identity (3) in Proposition 3.2.

Using equation (7) in Proposition 3.2 and again the second Bianchi identity, we get that

R_{tk,tk} = R_{tiki,tk} = R_{titi,kk} + R_{tikt,ik} = −R_{titk,ik}.

We also have that

R_{,tt} = 2R_{at,at} = 2R_{ti,it}+2R_{tt,tt} = 2R_{ti,ti}+2(R_{atit}R_{ai}+R_{aiit}R_{ta})−4(R_{titj})^{2}

= 2R_{ti,ti} − 2(R_{titj})^{2},

where we are using the contracted second Bianchi identity, the equalities (2) and (7) in Proposition 3.2, and the fact that, from the Gauss formula, R_{ij} = R_{titj}, since here h_{ij} = 0 and \overline{R}_{ij} = 0 in conformal normal coordinates.

Hence

(3.11)  2R_{titj,ij} = −R_{,tt} − 2(R_{titj})^{2}.

4. Sobolev quotient when n ≥ 8

In this section we will prove an existence theorem for a Yamabe problem on manifolds with boundary.

Let (M^{n}, g) be an n-dimensional, compact, Riemannian manifold with boundary, where n ≥ 3. In what follows, let R_{g} denote the scalar curvature of the metric g, and let h_{g} denote the mean curvature of ∂M.

We are interested in finding positive solutions u to

(4.1)  \{ \begin{matrix}Δ_{g}u − \frac{n−2}{4(n−1)}R_{g}u = 0 & in & M, \\ \frac{∂u}{∂η} + \frac{n−2}{2}h_{g}u = \frac{n−2}{2}cu^{\frac{n}{n−2}} & on & ∂M. \end{matrix}

The operator L_{g} = Δ_{g} − \frac{n−2}{4(n−1)}R_{g} is called the conformal Laplacian of g.

The geometric relevance comes from the fact that if u is such a solution, then the new metric u^{\frac{4}{n−2}} g is scalar-flat and the boundary has constant mean curvature equal to c. For this purpose, we need to introduce the Sobolev quotient of a manifold with boundary.

Given ϕ ∈ H^{1}(M), define

E(ϕ) = ∫_{M}(|∇ϕ|^{2} + \frac{n − 2}{4(n − 1)}R_{g}ϕ^{2})dv_{g} + \frac{n − 2}{2} ∫_{∂M} h_{g}ϕ^{2}dσ,

and also

Q(ϕ) = \frac{E(ϕ)}{(∫_{∂M} |ϕ|^{2 \frac{(n−1)}{(n−2)}} dσ)^{\frac{(n−2)}{(n−1)}}} .

Then the Sobolev quotient Q(M, ∂M) is defined as

Q(M, ∂M) = inf\{Q(ϕ) : ϕ ∈ C^{1}(\overline{M}), ϕ ≢ 0 on ∂M\},

and it is not difficult to see ([4]) that the Sobolev quotient is conformally invariant.

It follows from the work in [4] that, whenever

−∞ < Q(M, ∂M) < Q(B^{n}, ∂B^{n}),

there is a minimizer of Q and this minimizer is automatically a solution to the problem (4.1).

The next theorem gives a criterion for obtaining the second inequality.

Theorem 4.1. Let (M^{n}, g) be an n-dimensional, compact, Riemannian manifold with umbilic boundary. Assume M^{n} satisfies one of the following two conditions:

(1) n ≥ 8, and \overline{W}_{g}(P) ≠ 0 for some P ∈ ∂M;

(2) n ≥ 9, and W_{g}(P) ≠ 0 for some P ∈ ∂M. Then

Q(M, ∂M) < Q(B^{n}, ∂B^{n}).

Furthermore, if the Sobolev quotient Q(M, ∂M) is finite, then there exists a smooth metric u^{\frac{4}{n−2}} g, u > 0 on \overline{M}, with zero scalar curvature and constant mean curvature on ∂M.

Proof. We will construct a function ϕ such that Q(ϕ) < Q(B^{n}, ∂B^{n}). Given ε > 0, define

u_{ε}(x, t) = (\frac{ε}{(ε + t)^{2} + |x|^{2}} )^{\frac{n−2}{2}}

for (x, t) ∈ ℝ_{+}^{n} = {(x, t) : x ∈ ℝ^{n−1}, t > 0}.

We know that u_{ε} is a solution to the boundary-value problem:

(4.2)  \{ \begin{matrix}Δu_{ε} = 0 & in ℝ_{+}^{n}, \\ \frac{∂}{∂η}u_{ε} = (n − 2)uε^{\frac{n}{n−2}} & on ∂ℝ_{+}^{n}. \end{matrix}

Here η denotes the outward normal unit vector.

It was proven by Escobar [3], and independently by Beckner [2], that u_{ε} minimizes the Sobolev quotient on ℝ_{+}^{n}, and one can check

∫_{ℝ_{+}^{n}} |∇u_{ε}|^{2}dxdt = (n − 2) ∫_{∂ℝ_{+}^{n}} u_{ε}^{2 \frac{(n−1)}{n−2}} dx,

and also

(4.3)  Q(B^{n}, ∂B^{n}) = (n − 2)(∫_{∂ℝ_{+}^{n}} u_{ε}^{2 \frac{(n−1)}{n−2}} dx)^{\frac{1}{n−1}} .

Fix a point P ∈ ∂M, and let (x_{1}, …, x_{n−1}, t) be conformal Fermi coordinates with

(4.4)  dv_{g} = (1 + O(|(x, t)|^{N}))dxdt

centered at P, N ≥ 5, as discussed in Proposition 3.1. Since the Sobolev quotient Q(M, ∂M) is a conformal invariant, there is no loss of generality in supposing our initial metric satisfies equality (4.4) , as proven in the last section.

Let ψ(ρ) be a smooth cut-off function satisfying ψ(ρ) = 1 for ρ ≤ ρ_{0}, ψ(ρ) = 0 for ρ ≥ 2ρ_{0}, 0 ≤ ψ ≤ 1 and |ψ′(ρ)| ≤ Cρ_{0}^{−1} if ρ_{0} ≤ ρ ≤ 2ρ_{0}, where ρ_{0} is a small real number to be chosen later. Here ρ = |(x, t)|.

Set ϕ = ψu_{ε}. Denote by B_{ρ}^{+} = {(x, t) : |x|^{2} + t^{2} < ρ^{2}, t ≥ 0}. Let us first estimate E(ϕ) on A_{ρ_{0}}^{+} = B_{2ρ_{0}}^{+} \ B_{ρ_{0}}^{+}. First observe

|∇ϕ|_{g}^{2} ≤ C|∇ϕ|^{2} ≤ C(|∇ψ|^{2}u_{ε}^{2} + ψ^{2}|∇u_{ε}|^{2}).

Then

∫_{A_{ρ_{0}}^{+}} |∇ψ|^{2}u_{ε}^{2}dxdt ≤ Cρ_{0}^{−2} ∫_{A_{ρ_{0}}^{+}} u_{ε}^{2}dxdt

≤ Cρ_{0}^{−2}ε^{n−2} ∫_{A_{ρ_{0}}^{+}}(\frac{1}{t^{2} + |x|^{2}} )^{n−2}dxdt

= Cρ_{0}^{−2}ε^{n−2} ∫_{ρ_{0}}^{2ρ_{0}} ρ^{3−n}dρ ≤ Cε^{n−2}ρ_{0}^{2−n}.

Since

|∇u_{ε}|^{2} = (n − 2)^{2} \frac{ε^{n−2}}{((ε + t)^{2} + |x|^{2})^{n−1}} ,

we also have

∫_{A_{ρ_{0}}^{+}} ψ^{2}|∇u_{ε}|^{2}dxdt ≤ Cε^{n−2} ∫_{ρ_{0}}^{2ρ_{0}} ρ^{1−n}dρ ≤ Cε^{n−2}ρ_{0}^{2−n}.

Now

∫_{A_{ρ_{0}}^{+}} R_{g}ϕ^{2}dxdt ≤ C ∫_{A_{ρ_{0}}^{+}} u_{ε}^{2}dxdt ≤ Cε^{n−2}ρ_{0}^{4−n},

and

∫_{A_{ρ_{0}}^{+}∩∂M} h_{g}ϕ^{2}dx ≤ C ∫_{A_{ρ_{0}}^{+}∩∂M} u_{ε}^{2}dx ≤ Cε^{n−2}ρ_{0}^{3−n}.

We conclude that E_{M\B_{ρ_{0}}^{+}}(ϕ) ≤ Cε^{n−2}ρ_{0}^{2−n}. Now let us turn our attention to the quantity E_{B_{ρ_{0}}^{+}}(ϕ). Fist observe that, if dv_{g} = (1 + O(|(x, t)|)^{N})dxdt, then

∫_{B_{ρ_{0}}^{+}} |∇ϕ|_{g}^{2}dv_{g} ≤ ∫_{B_{ρ_{0}}^{+}} |∇ϕ|_{g}^{2}dxdt + C ∫_{B_{ρ_{0}}^{+}} |∇ϕ|_{g}^{2}|(x, t)|^{N}dxdt
(4.5)  ≤ ∫_{B_{ρ_{0}}^{+}} |∇ϕ|_{g}^{2}dxdt + Cρ_{0}^{N−5}ε^{5},

for n ≥ 8, where, in the last inequality, we are performing the change of variables (Lemma 3.4 in [4]):

∫_{B_{ρ_{0}}^{+}} \frac{t^{k}x_{i}^{m}dxdt}{((ε + t)^{2} + |x|^{2})^{n−l}} = ε^{k+m+2l−n} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{k}y_{i}^{m}dyds}{((1 + s)^{2} + |y|^{2})^{n−l}} ,

1 ≤ i ≤ n − 1.

In what follows, 1 ≤ i, j ≤ n − 1.

∫_{B_{ρ_{0}}^{+}} |∇ϕ|_{g}^{2}dxdt = ∫_{B_{ρ_{0}}^{+}} |∇u_{ε}|^{2}dxdt + ∫_{B_{ρ_{0}}^{+}}(g^{ij} − δ^{ij})(u_{ε})_{i}(u_{ε})_{j}dxdt

= ∫_{B_{ρ_{0}}^{+}} |∇u_{ε}|^{2}dxdt

+ ∫_{B_{ρ_{0}}^{+}}(\frac{1}{24}g_{,tttt}^{ij}t^{4} + \frac{1}{4}g_{,ttkl}^{ij}t^{2}x_{k}x_{l} + \frac{1}{24}g_{,klmp}^{ij}x_{k}x_{l}x_{m}x_{p})(u_{ε})_{i}(u_{ε})_{j}dxdt

+ O(ε^{5}).

By symmetry arguments, we only have to consider the fourth order terms in the expansion of g^{ij}. This follows from Lemma 2.3 and Proposition 3.2.

The expansion formula in Lemma 2.3 and the equality (5) in Proposition 3.2 imply that

A : = \frac{1}{24} ∫_{B_{ρ_{0}}^{+}} g_{,tttt}^{ij}t^{4}(u_{ε})_{i}(u_{ε})_{j}dxdt

= \frac{1}{24} \frac{(n − 2)^{2}}{n − 1} ε^{n−2}(2R_{tt,tt} + 16(R_{titj})^{2}) ∫_{B_{ρ_{0}}^{+}} \frac{t^{4}|x|^{2}}{((ε + t)^{2} + |x|^{2})^{n}} dxdt

= \frac{1}{2} \frac{(n − 2)^{2}}{n − 1} ε^{4}(R_{titj})^{2} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{4}|y|^{2}}{((1 + s)^{2} + |y|^{2})^{n}} dyds.

Also

B : = \frac{1}{4} ∫_{B_{ρ_{0}}^{+}} g_{,ttkl}^{ij}t^{2}x_{k}x_{l}(u_{ε})_{i}(u_{ε})_{j}dxdt

= \frac{(n − 2)^{2}}{4} ε^{n−2}g_{,ttkl}^{ij} ∫_{B_{ρ_{0}}^{+}} \frac{t^{2}x_{k}x_{l}x_{i}x_{j}}{((ε + t)^{2} + |x|^{2})^{n}} dxdt

= \frac{(n − 2)^{2}}{4} ε^{n−2}\{∑_{i}(2R_{titi,ii}) ∫_{B_{ρ_{0}}^{+}} \frac{t^{2}x_{i}^{4}}{((ε + t)^{2} + |x|^{2})^{n}} dxdt

+ ∑_{i≠j}(2R_{titi,jj} + 4R_{titj,ij}) ∫_{B_{ρ_{0}}^{+}} \frac{t^{2}x_{i}^{2}x_{j}^{2}}{((ε + t)^{2} + |x|^{2})^{n}} dxdt\},

where we are using that \overline{R}_{iksl} = −\overline{R}_{kisl} in order to get rid of the term involving Sym_{ij}(\overline{R}_{iksl}R_{tstj}).

Using

∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}y_{i}^{4}}{((1 + s)^{2} + |y|^{2})^{n}} dyds = 3 ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}y_{i}^{2}y_{j}^{2}}{((1 + s)^{2} + |y|^{2})^{n}} dyds,

and equality (7) in Proposition 3.2 we get

B = \frac{(n − 2)^{2}}{3} ε^{4}(R_{titj,ij}) ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}y_{i}^{4}}{((1 + s)^{2} + |y|^{2})^{n}} dyds

= \frac{(n − 2)^{2}}{(n − 1)(n + 1)}ε^{4}(R_{titj,ij}) ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}|y|^{4}}{((1 + s)^{2} + |y|^{2})^{n}} dyds.

Now, again from Lemma 2.3,

C : = \frac{(n − 2)^{2}}{24} ε^{n−2}g_{,klmp}^{ij} ∫_{B_{ρ_{0}}^{+}} \frac{x_{i}x_{j}x_{k}x_{l}x_{m}x_{p}}{((ε + t)^{2} + |x|^{2})^{n}} dxdt

= \frac{(n − 2)^{2}}{24} ε^{n−2}Sym_{klmp}(\frac{1}{20}\overline{R}_{ikjl,mp}

+\frac{1}{15}\overline{R}_{iklr}\overline{R}_{jmpr}) ∫_{B_{ρ_{0}}^{+}} \frac{x_{i}x_{j}x_{k}x_{l}x_{m}x_{p}}{((ε + t)^{2} + |x|^{2})^{n}} dxdt = 0

by symmetry of the curvature tensor.

Now, using 2R_{titj,ij} = −R_{,tt} − 2(R_{titj})^{2} (see eq. (3.11)),

∫_{B_{ρ_{0}}^{+}} |∇ϕ|_{g}^{2}dxdt =

∫_{B_{ρ_{0}}^{+}} |∇u_{ε}|^{2}dxdt + \frac{1}{2} \frac{(n − 2)^{2}}{n − 1} ε^{4}(R_{titj})^{2} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{4}|y|^{2}}{((1 + s)^{2} + |y|^{2})^{n}} dyds

−\frac{(n − 2)^{2}}{2(n − 1)(n + 1)}ε^{4}(R_{,tt} + 2(R_{titj})^{2}) ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}|y|^{4}}{((1 + s)^{2} + |y|^{2})^{n}} dyds

+E_{1},

where E_{1} = O(ε^{5}), if n ≥ 8.

Now

∫_{B_{ρ_{0}}^{+}} Ru_{ε}^{2}dv = \frac{1}{2} ∫_{B_{ρ_{0}}^{+}}(R_{,ij}x_{i}x_{j} + R_{,tt}t^{2})u_{ε}^{2}dxdt + E_{2},

where E_{2} = O(ε^{5}), if n ≥ 8.

And, using eq. (3.10),

\frac{1}{2} ∫_{B_{ρ_{0}}^{+}}(R_{,ij}x_{i}x_{j} + R_{,tt}t^{2})u_{ε}^{2}dxdt

= −\frac{1}{12(n − 1)}|\overline{W} (P)|^{2}ε^{4} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{|y|^{2}}{((1 + s)^{2} + |y|^{2})^{n−2}} dyds

+\frac{1}{2}R_{,tt}ε^{4} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}}{((1 + s)^{2} + |y|^{2})^{n−2}} dyds.

Then

\frac{(n − 2)}{4(n − 1)} ∫_{B_{ρ_{0}}^{+}} Ru_{ε}^{2}dv

= −\frac{(n − 2)}{48(n − 1)^{2}} |\overline{W} (P)|^{2}ε^{4} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{|y|^{2}}{((1 + s)^{2} + |y|^{2})^{n−2}} dyds

+\frac{(n − 2)}{8(n − 1)}R_{,tt}ε^{4} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}}{((1 + s)^{2} + |y|^{2})^{n−2}} dyds + cE_{2}

Now the integral involving the mean curvature term can be estimated as follows

∫_{B_{ρ_{0}}^{n−1}} hϕ^{2}dσ = \frac{1}{2} ∫_{B_{ρ_{0}}^{n−1}} h_{,\underline{rs}}x^{r}x^{s}u_{ε}^{2}dx + E_{3} = E_{3},

where E_{3} = O(ε^{5}), if n ≥ 8. The last equality follows from the fact that H_{,kl}(P) = 0 and by using symmetry in the third order term.

Note that from the properties (4.2) and the identity (4.3), we can get

∫_{B_{ρ_{0}}^{+}} |∇u_{ε}|^{2}dxdt ≤ Q(B^{n}, ∂B^{n}) (∫_{B_{ρ_{0}}^{n−1}} u_{ε}^{2 \frac{(n−1)}{n−2}} dx)^{\frac{n−2}{n−1}} . (4.6)

Here we are using that \frac{∂u_{ε}}{∂η} < 0 on ∂B_{ρ_{0}}^{+} ∩ ℝ_{+}^{n}, where η denotes the outward unit normal vector.

18 FERNANDO C. MARQUES

Thus E(ϕ) ≤ Q(B^{n}, ∂B^{n})(∫_{∂M} ϕ^{\frac{2(n−1)}{(n−2)}} dσ)^{\frac{n−2}{n−1}}

+\frac{1}{2} \frac{(n − 2)^{2}}{n − 1} ε^{4}(R_{titj})^{2} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{4}|y|^{2}}{((1 + s)^{2} + |y|^{2})^{n}} dyds

−\frac{(n − 2)^{2}}{2(n − 1)(n + 1)}ε^{4}(R_{,tt} + 2(R_{titj})^{2}) ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}|y|^{4}}{((1 + s)^{2} + |y|^{2})^{n}} dyds

−\frac{(n − 2)}{48(n − 1)^{2}} |\overline{W} (P)|^{2}ε^{4} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{|y|^{2}}{((1 + s)^{2} + |y|^{2})^{n−2}} dyds

+\frac{(n − 2)}{8(n − 1)}R_{,tt}ε^{4} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}}{((1 + s)^{2} + |y|^{2})^{n−2}} dyds + E,

where E = O(ε^{5}), if n ≥ 8.

A computation shows that

∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}}{((1 + s)^{2} + |y|^{2})^{n−2}} dyds (4.7)

= 4\frac{n − 2}{n + 1} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}|y|^{4}}{((1 + s)^{2} + |y|^{2})^{n}} dyds + O(ρ_{0}^{6−n}ε^{n−6}).

This last equality allows us to forget the terms involving R_{,tt}. Also

∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{4}|y|^{2}}{((1 + s)^{2} + |y|^{2})^{n}} dyds = (4.8)

\frac{12}{(n − 2)(n + 1)} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}|y|^{4}}{((1 + s)^{2} + |y|^{2})^{n}} dyds + O(ρ_{0}^{6−n}ε^{n−6}).

Equalities (4.7) and (4.8) follow from the fact that, if

I_{1} = ∫_{ℝ_{+}^{n}} \frac{s^{2}}{((1 + s)^{2} + |y|^{2})^{n−2}} dyds,

I_{2} = ∫_{ℝ_{+}^{n}} \frac{s^{2}|y|^{4}}{((1 + s)^{2} + |y|^{2})^{n}} dyds,

and

I_{3} = ∫_{ℝ_{+}^{n}} \frac{s^{4}|y|^{2}}{((1 + s)^{2} + |y|^{2})^{n}} dyds,

EXISTENCE RESULTS FOR THE YAMABE PROBLEM 19

we have I_{1} = 4\frac{n−2}{n+1}I_{2} and I_{3} = \frac{12}{(n−2)(n+1)}I_{2}. In order to see this, one can use Fubini’s theorem, the change of variables y = (1 + s)x and polar coordinates to obtain:

I_{1} = σ_{n−2} ∫_{0}^{∞} \frac{s^{2}}{(1 + s)^{n−3}} ds ∫_{0}^{∞} \frac{r^{n−2}}{(1 + r^{2})^{n−2}} dr
= \frac{2σ_{n−2}}{(n − 4)(n − 5)(n − 6)} ∫_{0}^{∞} \frac{r^{n−2}}{(1 + r^{2})^{n−2}} dr,
I_{2} = σ_{n−2} ∫_{0}^{∞} \frac{s^{2}}{(1 + s)^{n−3}} ds ∫_{0}^{∞} \frac{r^{n+2}}{(1 + r^{2})^{n}} dr
= \frac{2σ_{n−2}}{(n − 4)(n − 5)(n − 6)} ∫_{0}^{∞} \frac{r^{n+2}}{(1 + r^{2})^{n}} dr,

and

I_{3} = σ_{n−2} ∫_{0}^{∞} \frac{s^{4}}{(1 + s)^{n−1}} ds ∫_{0}^{∞} \frac{r^{n}}{(1 + r^{2})^{n}} dr
= \frac{24σ_{n−2}}{(n − 2)(n − 3)(n − 4)(n − 5)(n − 6)} ∫_{0}^{∞} \frac{r^{n}}{(1 + r^{2})^{n}} dr.

Integration by parts yields

∫_{0}^{∞} \frac{r^{n−2}}{(1 + r^{2})^{n−2}} dr = \frac{2(n − 2)}{n − 1} ∫_{0}^{∞} \frac{r^{n}}{(1 + r^{2})^{n−1}} dr.

and also

(4.9)  ∫_{0}^{∞} \frac{r^{n}}{(1 + r^{2})^{n−1}} dr = \frac{2(n − 1)}{n + 1} ∫_{0}^{∞} \frac{r^{n+2}}{(1 + r^{2})^{n}} dr,

so

∫_{0}^{∞} \frac{r^{n−2}}{(1 + r^{2})^{n−2}} dr = \frac{4(n − 2)}{n + 1} ∫_{0}^{∞} \frac{r^{n+2}}{(1 + r^{2})^{n}} dr.

Now identity (4.9) and the equality

∫_{0}^{∞} \frac{r^{n}}{(1 + r^{2})^{n−1}} dr = ∫_{0}^{∞} \frac{r^{n}}{(1 + r^{2})^{n}} dr + ∫_{0}^{∞} \frac{r^{n+2}}{(1 + r^{2})^{n}} dr

immediately imply

∫_{0}^{∞} \frac{r^{n}}{(1 + r^{2})^{n}} dr = \frac{n − 3}{n + 1} ∫_{0}^{∞} \frac{r^{n+2}}{(1 + r^{2})^{n}} dr,

finishing the proof of the equalities (4.7) and (4.8).

Therefore

E(ϕ) ≤ Q(B^{n}, ∂B^{n})(∫_{∂M} ϕ^{\frac{2(n−1)}{(n−2)}} dσ)^{\frac{n−2}{n−1}} (4.10)

−\frac{(n − 2)}{48(n − 1)^{2}} |\overline{W} (P)|^{2}ε^{4} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{|y|^{2}}{((1 + s)^{2} + |y|^{2})^{n−2}} dyds

−\frac{(n − 2)(n − 8)}{(n − 1)(n + 1)}ε^{4}(R_{titj})^{2} ∫_{B+ ; \frac{ρ_{0}}{ε}} \frac{s^{2}|y|^{4}}{((1 + s)^{2} + |y|^{2})^{n}} dyds + O(ε^{5}).

Recall

W_{abcd} = R_{abcd} − \frac{1}{n−2}(R_{ac}g_{bd} − R_{ad}g_{bc} + R_{bd}g_{ac} − R_{bc}g_{ad}) (4.11)

+\frac{R}{(n−2)(n−1)}(g_{ac}g_{bd} − g_{ad}g_{bc}).

It is clear, from expansion (4.10), that if n ≥ 8 and \overline{W} (P) ≠ 0, then Q(ϕ) < Q(B^{n}, ∂B^{n}) for small ε > 0.

To get the result for n ≥ 9 we assert that, under our hypotheses, \overline{W}_{ijkl} = 0 and R_{titj} = 0 if and only if W_{abcd} = 0, these quantities being computed at P. First, W_{tttt} = W_{ttti} = W_{ttij} = 0 by the symmetries of the Weyl tensor. Second, using definition (4.11), parts (1) and (2) in Proposition 3.2, and the Codazzi equation we obtain W_{tijk} = 0.

Again from definition (4.11), we can compute

(4.12)  W_{titj} = \frac{n − 3}{n − 2}R_{titj},

and also

(4.13) W_{ijkl} = \overline{W}_{ijkl} − \frac{1}{n − 2}(R_{titk}g_{jl} − R_{titl}g_{jk} + R_{tjtl}g_{ik} − R_{tjtk}g_{il}),

where we are using that R_{tt} = 0, R = 0 and the facts that R_{ij} = R_{titj} and R_{ijkl} = \overline{W}_{ijkl}. Note that

R_{ij} = R_{titj} + R_{kikj} = R_{titj} + \overline{R}_{kikj} = R_{titj} + \overline{R}_{ij} = R_{titj},

where the second equality follows from the Gauss equation and the fact that h_{ij} = 0, and the last one follows from the fact that \overline{R}_{ij} = 0. The same argument applies to show that R_{ijkl} = \overline{W}_{ijkl}. The identities (4.12) and (4.13) finish the proof of the assertion.

Thus it is clear that if W (P) ≠ 0 then, for small ε,

Q(ϕ) < Q(B^{n}, ∂B^{n}),

when n ≥ 9.

The result follows immediately. □

Fernando C. Marques, Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460-320, Rio de Janeiro - RJ, Brazil

Email address : coda@impa.br

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