Josai Mathematical Monographs vol 3 (2001), pp. 105-113

NOTES ON THE RELATIVE YAMABE INVARIANT

KAZUO AKUTAGAWA

Dedicated to Professor Katsuhiro Shiohama on his sixtieth birthday

Abstract. The Yamabe invariant [K], [S2] of a closed smooth manifold X is a natural differential-topological invariant which arises from a variational problem for the total scalar curvature of Riemannian metrics on X. The relative Yamabe invariant [AB1] of a compact connected smooth manifold W with nonempty boundary is a natural relative version of the Yamabe invariant of X. Hence the relative Yamabe invariant has several fundamental properties analogous to the corresponding ones for the classic Yamabe invariant. In particular, in respect of surgery on X and the interior of W, these two invariants have quite similar properties. In this article, we give those properties.

1. Preliminaries

Let W be a compact connected smooth n-manifold with nonempty boundary ∂W = M (possibly disconnected), and n = dimW ≥ 3. We denote by C(M) and C(W) the spaces of conformal classes on M and W respectively. Let C ∈ C(M) be a conformal class on M and \overline{C} ∈ C(W) a conformal class on W. We say that C is the boundary of \overline{C} if \overline{C}|_{M} = C. We use the notation ∂\overline{C} = C in this case. Let Riem(W) be the space of all Riemannian metrics on W. For each conformal class C ∈ C(M), we define the following subspaces of Riem(W):

Riem_{C}(W) = \{\overline{g} ∈ Riem(W) | ∂[\overline{g}] = C\},
Riem_{C}^{0}(W) = \{\overline{g} ∈ Riem_{C}(W) | H\overline{g} = 0 along M\},

where [\overline{g}] denotes the conformal class of \overline{g} and H\overline{g} the mean curvature along M with respect to \overline{g}. We consider the normalized Einstein-Hilbert functional I : Riem(W) ⟶ ℝ given

2000 Mathematics Subject Classification: Primary 53C; Secondary 57R, 58E.. Partly supported by the Grants-in-Aid for Scientific Research, The Ministry of Education, Science,

Sports and Culture, Japan, No. 11640070.

by

I(\overline{g}) = \frac{∫_{W} R\overline{g} dv\overline{g}}{Vol\overline{g}(W)^{(n−2)/n}} ,

where R\overline{g} and dv\overline{g} are respectively the scalar curvature and the volume element of \overline{g}. As in the case of closed manifolds, we notice the following (cf. [B]):

Proposition 1.1. ([AB1, Theorem 1.1]) The set of critical points of I on the space Riem_{C}^{0} (W) coincides with the set of Einstein metrics \overline{g} on W with ∂[\overline{g}] = C and H\overline{g} = 0.

Remark 1.2. From Claim 3.1 in [AB1], we notice that the set of critical points of I on Riem_{C}(W) is empty for any C ∈ C(M). Moreover, the set of critical points of I on the space \{\overline{g} ∈ Riem_{C}(W) | H\overline{g} is constant along M\} is also empty.

Similarly to the case of closed manifolds (see [Au2]), the functional I is not bounded on Riem_{C}^{0}(W). More precisely, for any W and any C ∈ C(M),

inf_{\overline{g}∈Riem_{C}^{0}(W)} I(\overline{g}) = −∞ , sup_{\overline{g}∈Riem_{C}^{0}(W)} I(\overline{g}) = ∞.

For each conformal class C ∈ C(M), let denote C_{C}(W) the subspace \{\overline{C} ∈ C(W) | ∂\overline{C} = C\} of C(W). For each conformal class \overline{C} ∈ C_{C}(W), we consider the subclass \overline{C}^{0} ⊂ \overline{C} defined by

\overline{C}^{0} = \{\overline{g} ∈ \overline{C} | H\overline{g} = 0 along M\}.

We call \overline{C}^{0} the normalized conformal class of \overline{C}. For a fixed \overline{g} ∈ \overline{C}, put g̃ = u^{\frac{4}{n−2}} \overline{g} for u ∈ C_{+}^{∞}(W), where C_{+}^{∞}(W) denotes the space of all positive smooth functions on W. Then

Hg̃ = u^{−2/(n−2)}(H\overline{g} + \frac{2(n − 1)}{n − 2}u^{−1} \frac{∂u}{∂ν}), (1.1)

where \frac{∂}{∂ν} denotes the normal derivative with respect to the inward unit normal vector field ν along the boundary M. Hence (1.1) implies the following:

Proposition 1.3. ([E2, Section 1])

(1) \overline{C}^{0} is nonempty for any \overline{C} ∈ C(W).

(2) If \overline{g} ∈ \overline{C}^{0}, then

\overline{C}^{0} = [\overline{g}]^{0} = \{u^{4/(n−2)}\overline{g} ∈ \overline{C} | u ∈ C_{+}^{∞}(W), \frac{∂u}{∂ν} = 0 along M\} .

The relative Yamabe constant Y\overline{C}(W, M; C) of the conformal class \overline{C} ∈ C_{C}(W) (or the pair (\overline{C}, C)) is defined by

Y\overline{C}(W, M; C) = inf_{\overline{g}∈\overline{C}^{0}} I(\overline{g}) = inf_{\overline{g}∈\overline{C}^{0}} \frac{∫_{W} R\overline{g} dv\overline{g}}{Vol\overline{g}(W)^{(n−2)/n}}

(see [Au2], [LP] for the Yamabe constant of a conformal class on a closed manifold). A metric ǧ ∈ \overline{C}^{0} is called a relative Yamabe metric if I(ǧ) = Y\overline{C}(W, M; C), that is, ǧ is a minimizer of I on \overline{C}^{0}. A similar argument to the Yamabe constant given by Aubin [Au1] (cf. [E2]) shows that

(1.2) Y\overline{C}(W, M; C) ≤ Y_{[\overline{h}]}(S_{+}^{n}, S^{n−1}; [h]) = n(n − 1) · Vol\overline{h}(S_{+}^{n})^{2/n} for any (W, M; \overline{C}, C),

where S_{+}^{n} denotes the round n-hemisphere with standard metric \overline{h} of constant curvature 1 and S^{n−1} ⊂ S_{+}^{n} the equator with h = \overline{h}|_{S^{n−1}}. It should be pointed out that from [E3, Proposition 1.4]

inf_{g̃∈\overline{C}} I(g̃) ≤ 0 for any \overline{C} ∈ C(W).

This observation combined with Remark 1.2, Proposition 1.3-(1) and (1.2) may implies that the minimal boundary condition is suitable for a natural relative version of the classic Yamabe constant.

The relative Yamabe invariants of the conformal class C ∈ C(M) (or the triple (W, M; C)) and the pair (W, M) are respectively defined by

Y (W, M; C) = sup_{\overline{C}∈C_{C}(W)} Y\overline{C}(W, M; C) = sup_{\overline{C}∈C_{C}(W)} inf_{\overline{g}∈\overline{C}^{0}} \frac{∫_{W} R\overline{g} dv\overline{g}}{Vol\overline{g}(W)^{(n−2)/n}} ,

Y (W, M) = sup_{C∈C(M)} Y (W, M; C) = sup_{C∈C(M)} sup_{\overline{C}∈C_{C}(W)} inf_{\overline{g}∈\overline{C}^{0}} \frac{∫_{W} R\overline{g} dv\overline{g}}{Vol\overline{g}(W)^{(n−2)/n}} ,

(see [K], [S2], [Le2] for the Yamabe invariant of a closed manifold). The inequality (1.2) guarantees their least upper bound

Y (W, M; C), Y (W, M) ≤ n(n − 1) · Vol\overline{h}(S_{+}^{n})^{2/n},

and hence

Y (S_{+}^{n}, S^{n−1}) = n(n − 1) · Vol\overline{h}(S_{+}^{n})^{2/n}.

The conformal invariant Y (W, M; C) of C has a clear geometrical meaning in terms of positive scalar curvature (abbreviated as “psc”). Namely, Y (W, M; C) > 0 if and only if any metric g ∈ C can be extended conformally to a psc-metric \overline{g} on W with the minimal boundary condition H\overline{g} = 0. The invariant Y (W, M) is by definition a differential-topological invariant of the pair (W, M), and also Y (W, M) > 0 if and only if there exists a psc-metric \overline{g} on W with the minimal boundary condition.

The rest of the paper is organized as follows. In Section 2, we prove that the relative Yamabe constant of \overline{C} ∈ C(W) coincides with the conformal invariant of \overline{C} defined by Escobar [E2], and we summarize fundamental properties on relative Yamabe constants and relative Yamabe metrics. In Section 3, we also give fundamental properties on relative Yamabe invariants.

2. Relative Yamabe constants

Let \overline{C} ∈ C(W) be a conformal class on W with ∂\overline{C} = C, and \overline{g} a metric in the normalized conformal class \overline{C}^{0}. We define the functional E\overline{g} : L^{1,2}(W) ⟶ ℝ by

E\overline{g}(f) = ∫_{W} (\frac{4(n − 1)}{n − 2}|df|2 ; \overline{g} + R\overline{g}f^{2})dv\overline{g} for f ∈ L^{1,2}(W),

where L^{1,2}(W) denotes the Sobolev space of square-integrable functions on W up to their first weak derivatives. In terms of the functional E\overline{g}, the relative Yamabe constant Y\overline{C}(W, M; C) may be written as the following:

Lemma 2.1.

Y\overline{C}(W, M; C) = inf_{g̃∈\overline{C}^{0}} I(g̃)
(2.1)  = inf_{u∈C_{+}^{∞}(W), \frac{∂u}{∂ν} |_{M} =0} E\overline{g}(u)/∥u∥_{L^{2n/(n−2)}(\overline{g})}^{2}
= inf_{f∈L^{1,2}(W), f≢0} E\overline{g}(f)/∥f∥_{L^{2n/(n−2)}(\overline{g})}^{2}.

Here ∥ · ∥_{L^{2n/(n−2)}(\overline{g})} stands for the L^{2n/(n−2)}-norm on W with respect to \overline{g}.

Remark 2.2. From (2.1), the relative Yamabe constant Y\overline{C}(W, M; C) coincides with the conformal invariant Q(W) = Q(W, \overline{C}) of \overline{C} (up to the positive factor \frac{4(n−1)}{n−2}) defined by Escobar [E2].

Proof of Lemma 2.1. Let g̃ = u^{4/(n−2)}\overline{g} be any metric in \overline{C}^{0}, where u is a positive smooth function on W with \frac{∂u}{∂ν} = 0 along M. Then we have the formula

Rg̃ = u^{−(n+2)/(n−2)}(−\frac{4(n − 1)}{n − 2}△\overline{g}u + R\overline{g}u) on W, (2.2)

where △\overline{g} = \overline{g}^{ij}\overline{∇}_{i}\overline{∇}_{j} denotes the Laplace-Beltrami operator of \overline{g} and \overline{∇} the Levi-Civita connection of \overline{g}. The Neumann boundary condition \frac{∂u}{∂ν} = 0 along M implies

I(g̃) = E\overline{g}(u)/∥u∥_{L^{2n/(n−2)}(\overline{g})}^{2},

and hence

Y\overline{C}(W, M; C) = inf_{u∈C_{+}^{∞}(W), \frac{∂u}{∂ν} |_{M} =0} E\overline{g}(u)/∥u∥_{L^{2n/(n−2)}(\overline{g})}^{2}.

In order to prove the last equality in (2.1), it is enough to prove the following

inf_{u∈C_{+}^{∞}(W), \frac{∂u}{∂ν} |_{M} =0} E\overline{g}(u)/∥u∥_{L^{2n/(n−2)}(\overline{g})}^{2} ≤ inf_{f∈L^{1,2}(W), f≢0} E\overline{g}(f)/∥f∥_{L^{2n/(n−2)}(\overline{g})}^{2}. (2.3)

We notice that

inf_{u∈C_{+}^{∞}(W), \frac{∂u}{∂ν} |_{M} =0} E\overline{g}(u)/∥u∥_{L^{2n/(n−2)}(\overline{g})}^{2}

= inf \{E\overline{g}(f)/∥f∥_{L^{2n/(n−2)}(\overline{g})}^{2} | f ∈ L^{1,2}(W), f is smooth near M and \frac{∂f}{∂ν} |_{M} = 0\} .

Let r be the distance function to M in W with respect to \overline{g}. Let (x, r) = (x^{1}, ⋯, x^{n−1}, r) denotes a Fermi coordinate system on a tubular neighborhood U_{ε}(M) = {z ∈ W | r(z) < ε} of M, where ε is a small positive constant. For each smooth function f ∈ C^{∞}(W) with f ≢ 0, we define a Lipschitz function f_{ε} by

f_{ε}(z) = \{ \begin{matrix}f(z) & if z ∈ W \ U_{ε}(M) \\ f(x, ε) & if z = (x, r) ∈ U_{ε}(M). \end{matrix}

Note that f_{ε} is smooth near M and \frac{∂f_{ε}}{∂ν} = 0 along M. Then we have

E\overline{g}(f_{ε}) = E\overline{g}(f) − \frac{4(n − 1)}{n − 2} ∫_{U_{ε}(M)}(\frac{∂f}{∂r})^{2} dv\overline{g} + O(ε^{2})

∥f_{ε}∥_{L^{2n/(n−2)}(\overline{g})}^{2} = ∥f∥_{L^{2n/(n−2)}(\overline{g})}^{2} + O(ε^{2}),

and hence

\begin{matrix}E\overline{g}(f_{ε})/∥f_{ε}∥_{L^{2n/(n−2)}(\overline{g})}^{2} \\ = E\overline{g}(f)/∥f∥_{L^{2n/(n−2)}(\overline{g})}^{2} − \frac{4(n − 1)}{(n − 2)} ∫_{U_{ε}(M)}(\frac{∂f}{∂r})^{2} dv\overline{g}/ ∥f∥_{L^{2n/(n−2)}(\overline{g})}^{2} + O(ε^{2}). \end{matrix}

This implies the inequality (2.3).

The Yamabe problem on W related to the relative Yamabe constant Y\overline{C}(W, M; C) was solved by Cherrier [C] and Escobar [E2] under some restrictions (cf. [Au2], [LP], [S1]).

Theorem 2.3.

(1) ([C, Section 6]) There exists a relative Yamabe metric ǧ ∈ \overline{C}^{0} provided

Y\overline{C}(W, M; C) < Y (S_{+}^{n}, S^{n−1}) = n(n − 1) · Vol\overline{h}(S_{+}^{n})^{2/n}.

(2) ([E2, Theorem 6.1]) Assume that W = (W^{n}, \overline{C}) satisfies any of the following three

condition : (i) n = 3, 4, or 5, (ii) W has a nonumbilic point on M = ∂W, (iii) M = (M, C = ∂\overline{C}) is umbilic, and either W is conformally flat or n ≥ 6 and

the Weyl curvature tensor of \overline{C} does not vanish identically on M. Then there exists a relative Yamabe metric ǧ ∈ \overline{C}^{0}.

Standard calculation combined with (2.1), (2.2) also gives the following (cf. [Au2], [LP]):

Proposition 2.4.

(1) ([E2, Lemma 1.1]) For each conformal class \overline{C} ∈ C(W), there exists a metric \overline{g} ∈ \overline{C}^{0} whose scalar curvature does not change sign. The sign is uniquely determined by the conformal structure \overline{C}, and so there are mutually exclusive possibilities : \overline{C} admits a metric of (i) positive, (ii) negative, or (iii) identically zero scalar curvature with the minimal boundary condition.

(2) Each relative Yamabe metric ǧ ∈ \overline{C}^{0} is a metric of constant scalar curvature Rǧ = Y\overline{C}(W, M; C) · Volǧ(W)^{−2/n}.

(3) ([E3, Theorem 4.1]) Let \overline{g} be a metric of nonpositive constant scalar curvature on W with the minimal boundary condition. If \overline{g}′ is another metric of constant scalar curvature in [\overline{g}]^{0}, then \overline{g}′ = c\overline{g} for some positive constant c. Thus this implies that \overline{g} is a relative Yamabe metric.

(4) (cf. [K, Lemma 1.6]) Let \overline{C} ∈ C_{C}(W) be a conformal class of nonpositive Yamabe constant Y\overline{C}(W, M; C). Then, for any \overline{g} ∈ \overline{C}^{0}

(min R\overline{g}) · Vol\overline{g}(W)^{2/n} ≤ Y\overline{C}(W, M; C) ≤ (max R\overline{g}) · Vol\overline{g}(W)^{2/n}.

Similar to the classic Yamabe problem, a metric of positive constant scalar curvature with the minimal boundary condition is not always a relative Yamabe metric. The following uniqueness result on Einstein metrics is a generalization of the corresponding one of Obata [O] to manifolds with boundary (cf. [S2, Proposition 1.4]).

Proposition 2.5. ([E1, Theorem 4.1]) Let ĝ be an Einstein metric of positive scalar curvature on W with totally geodesic boundary. If \overline{g} is another metric of constant scalar curvature in [ĝ]^{0}, then \overline{g} is Einstein. Moreover, if (W^{n}, ĝ) is not conformally equivalent to (S_{+}^{n}, \overline{h}), then \overline{g} = cĝ for some positive constant c. Hence this implies that ĝ is a relative Yamabe metric.

The second variation of I on \overline{C} at a relative Yamabe metric ǧ ∈ \overline{C}^{0} of positive scalar curvature implies the following estimate, which is a characterization of relative Yamabe metrics in the positive case.

Proposition 2.6. (cf. [Au2, Proposition 5.24]) For each relative Yamabe metric ǧ of positive scalar curvature on W, the first nonzero eigenvalue ν_{1}(−△ǧ) of −△ǧ for the Neumann boundary condition can be estimated from below :

ν_{1}(−△ǧ) ≥ \frac{Rǧ}{n − 1}.

The characterization (2.1) of relative Yamabe constants also leads to the following Sobolev inequality on W with a relative Yamabe metric ǧ of positive scalar curvature:

(∫_{W} |f|^{2n/(n−2)} dvǧ)^{(n−2)/n} ≤ \frac{1}{c_{[ǧ]}} ∫_{W} |df|2 ; ǧ dvǧ + \frac{1}{Volǧ(W)^{2/n}} ∫_{W} f^{2} dvǧ (2.4)

for f ∈ L^{1,2}(W), where c_{[ǧ]} = \frac{n−2}{4(n−1)} ·Y_{[ǧ]}(W, M; ∂[ǧ]) > 0. Using the Moser iteration method on the Sobolev inequality (2.4), we obtain

Proposition 2.7. (cf. [Ak1, Section 2]) There exists a positive constant δ_{n} > 0 depending only on n = dim W such that the following holds :

(1) Volǧ(B_{r}(p)) ≥ δ_{n}c_{[ǧ]}^{n/2}r^{n} for p ∈ W and r ≤ c_{[ǧ]}^{−1/2}Volǧ(W)^{1/n},

where B_{r}(p) denotes the geodesic ball of radius r centered at p with respect to ǧ.

diamǧ(W) ≤ 2δ_{n}^{−1}c_{[ǧ]}^{−1/2}Volǧ(W)^{1/n}. (2)

3. Relative Yamabe invariants

Let W be a compact smooth n-manifold (n ≥ 3) with boundary M (possibly M = ∅). We remark that the relative Yamabe invariant Y (W, M) is nothing but the Yamabe invariant Y (W) of the closed manifold W when M is empty. From the characterization (2.1) of relative Yamabe constants, every technique of analyzing Yamabe invariants is available for Y (W, M) on the interior Int(W) of W, under suitable modification. Thus the technique developed by Kobayashi [K] implies the following result, which is a fundamental tool for computations of the relative Yamabe invariant.

Theorem 3.1. (cf. [K, Theorem 2]) Let W_{1} and W_{2} be compact connected smooth nmanifolds (n ≥ 3) with boundary M_{1} and M_{2} respectively. Then

Y (W_{1}♯W_{2}, M_{1} ⨿ M_{2}) ≥ \begin{cases}−(|Y (W_{1}, M_{1})|^{n/2} + |Y (W_{2}, M_{2})|^{n/2} & )^{2/n} \\ & if Y (W_{1}, M_{1}), Y (W_{2}, M_{2}) ≤ 0, \\ min\{Y (W_{1}, M_{1}), Y (W_{2}, M_{2})\} & otherwise. \end{cases}

Here W_{1}♯W_{2} denotes the connected sum of W_{1} and W_{2}.

Similar to the classic Yamabe invariant, the minimax definition of relative Yamabe invariants is also rather unwieldy for many purposes. However, when the relative Yamabe invariant of a manifold is nonpositive, the following gives a very useful reinterpretation of the invariant.

Proposition 3.2. (cf. [An], [BCG], [Le1]) Let W be a compact connected smooth n-manifold with boundary M. Then

inf_{\overline{g}∈Riem^{0}(W)} ∫_{W} |R\overline{g}|^{n/2}dv\overline{g} = \{ \begin{matrix}0 & if Y (W, M) ≥ 0, \\ |Y (W, M)|^{n/2} & if Y (W, M) ≤ 0. \end{matrix}

Here Riem^{0}(W) = \{\overline{g} ∈ Riem(W) | H\overline{g} = 0 along M\}.

Using Proposition 3.2, we obtain the following surgery theorems corresponding to ones of Petean-Yun [PY] and Petean [P].

Theorem 3.3. (cf. [PY, Theorem 1]) Let W_{1} and W_{2} be compact connected smooth nmanifolds with boundary M_{1} and M_{2} respectively. Suppose that a closed smooth k-manifold S embeds into both Int(W_{1}) and Int(W_{2}) with trivial normal bundle. Assume that k ≤ 3. Let W_{1,2}^{S} be the manifold obtained by gluing W_{1} and W_{2} along S. Then

(i) If Y (W_{1}, M_{1}) ≤ 0 and Y (W_{2}, M_{2}) ≤ 0, then Y (W_{1,2}^{S}, M_{1} ⨿ M_{2}) ≥ −(|Y (W_{1}, M_{1})|^{n/2} + |Y (W_{2}, M_{2})|^{n/2})^{2/n},

(ii) If Y (W_{1}, M_{1}) ≤ 0 and Y (W_{2}, M_{2}) > 0, then Y (W_{1,2}^{S}, M_{1} ⨿ M_{2}) ≥ Y (W_{1}, M_{1}).

Theorem 3.4. (cf. [PY, Corollary 1], [P, Theorem 1]) Let W be a compact connected smooth manifold of dimension n ≥ 4 with boundary M. Let Ŵ be a manifold obtained from W by performing surgery on Int(W) of codimension q (1 ≤ q ≤ n). Suppose that Y (W, M) ≤ 0. Then

(i) If q ≥ 3, then Y (Ŵ, M) ≥ Y (W, M).

(ii) If q is different from 1, 2 and n − 1, then Y (Ŵ, M) = Y (W, M).

Remark 3.5. In [AB1], we developed approximation technique which leads to gluing theorems of the boundary connected sum two manifolds along their boundaries for the relative Yamabe invariant. Using the technique, we showed that there are many examples of manifolds with positive and nonpositive relative Yamabe invariants (see also [Ak2]). In particular, we constructed a family of 4-manifolds with strictly negative relative Yamabe invariant. In [AB2], we are studying the conformal cobordism theory of manifolds with positive conformal classes in terms of the relative Yamabe invariant.

Acknowledgment. The author would like to thank Professor Boris Botvinnik for useful discussions.

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Department of Mathematics Shizuoka University Shizuoka 422-8529 Japan

E-mail Address: smkacta@ipc.shizuoka.ac.jp