Generic Properties of Critical Points of the Weyl Tensor

1 Introduction

Let (M,g) be a connected compact Riemannian manifold of dimension n≥3 without boundary.

The curvature tensor is a map Riem:𝔛⁢(M)3→𝔛⁢(M) defined by

where 𝔛⁢(M) denotes the space of smooth vector fields and [⋅,⋅] is the Lie bracket. The Weyl tensorWeylg is a conformally invariant tensor which is obtained from the full curvature tensor by subtracting out various traces. In a local coordinate system it reads as

where the Riemann tensor is given by

the Schouten curvature tensor is

the Ricci curvature tensorRic is the contraction of Riem, i.e. Ri⁢j=gk⁢l⁢Ri⁢k⁢j⁢l and the scalar curvatureRg is the contraction of Ric, i.e. Rg=gi⁢j⁢Ri⁢j.

Let ℳk be the set of all Ck Riemannian metrics on M with k≥4. Any g∈ℳk determines the Weyl tensor Weylg of (M,g). Set

We assume that M is such that the set

is a non-empty open subset of ℳk. Examples of manifolds with 𝒟k non-empty are given by the product of round spheres, i.e. the product manifold (𝕊n×𝕊m,g𝕊n⊗g𝕊m).

Our goal is to prove that for generic Riemannian metrics g∈𝒟k the critical points of the function 𝒲g are nondegenerate. More precisely, we will prove the following result.

Our result is motivated by the study of compactness of solutions to a linear perturbation of the Yamabe equation. Yamabe [21] asked the question if it is possible to find a metric g~ in the conformal class of g with constant scalar curvature. If g~=u4N-2⁢g, where u is a positive smooth function on M, the problem turns out to be equivalent to find a positive solution to the elliptic problem

for some constant κ. Here ℒg⁢u:=Δg⁢u-n-24⁢(n-1)⁢Rg⁢u is the conformal Laplacian, Δg is the Laplace–Beltrami operator and Rg is the scalar curvature of g. If u solves problem (1.2), the scalar curvature of the metric g~ is 4⁢(n-1)n-2⁢κ. The Yamabe problem has been completely solved by Yamabe [21], Aubin [1], Trudinger [19] and Schoen [15] (see also the proof given by Bahri [2]).

Once the problem is solved, a second question concerns the structure of the set of the metrics conformally equivalent to g with constant scalar curvature or equivalently the set of solutions to the elliptic problem (1.2). The solution is unique in the case of negative scalar curvature and it is unique (up to a constant factor) in the case of zero scalar curvature. The uniqueness is not true anymore in the case of positive scalar curvature, as was proved by Schoen [17, 18] and Pollack [11] who exhibited examples where a large number of high energy solutions of (1.2) with high Morse index exist. Thus it is natural to ask if the set of solutions of (1.2) is compact or not as was raised by Schoen [16]. Obata [10] proved that compactness does not hold in the case of the round sphere (𝕊n,g𝕊n). Indeed, the Yamabe problem (1.2) turns out to be equivalent (via the stereographic projection) to the equation in the Euclidean space

which has infinitely many solutions, the so-called standard bubbles,

Here αn:=[n⁢(n-2)]n-24. The study of compactness in a manifold which is not locally conformally equivalent to the round sphere is a delicate issue. Khuri, Marques and Schoen [7] proved that the compactness holds when the dimension n of the manifold satisfies 3≤n≤24, while it fails when n≥25 thanks to the examples built by Brendle [3] and Brendle and Marques [4]. The proof of compactness strongly relies on proving sharp pointwise estimates at a blow-up point of the solution. In particular, when compactness holds, every sequence of unbounded solutions to (1.2) must blow-up at some points of the manifold which are necessarily isolated and simple, i.e. around each blow-up point ξ0, the solution can be approximated by a standard bubble (see (1.3))

More precisely, let um be a sequence of solutions to problem (1.2). We say that um blows up at a point ξ0∈M if there exists ξm∈M such that ξm→ξ0⁢and⁢um⁢(ξm)→+∞. The point ξ0 is said to be a blow-up point for um. Blow-up points can be classified according to the definitions introduced by Schoen [16]. The point ξ0∈M is an isolated blow-up point for um if there exists ξm∈M such that ξm is a local maximum of um, ξm→ξ0, um⁢(ξm)→+∞, and there exist c>0 and R>0 such that

Moreover, ξ0∈M is an isolated simple blow-up point for um if the function

has exactly one critical point in (0,R).

Motivated by the previous observations, we are interested in studying the compactness of solutions to a linear perturbation of the Yamabe problem

where the first eigenvalue of -ℒg is positive and ϵ is a small parameter. In particular, we address the questions of the existence of solutions blowing-up at a point ξ0 as ϵ goes to zero and the nature of the blow-up point ξ0, namely ξ0 can be either a clustering blow-up point (a non-isolated blow-up point) or a towering blow-up point (a non-isolated simple blow-up point).

Problem (1.4) does not have any blowing-up solutions when ϵ<0 as proved by Druet [5] in the low-dimensional case, i.e. n=3,4,5 (except when the manifold is conformally equivalent to the round sphere). In the high-dimensional case, i.e. n≥6, the problem is still open. The situation is completely different when ϵ>0. Indeed, if n=3, no blowing-up solutions exist as proved by Li–Zhu [8], while if n≥4 blowing-up solutions do exist as showed by Esposito, Pistoia and Vetois [6]. In particular, if the dimension n≥7 and the manifold is not locally conformally flat, in [6] blowing-up solutions are built whose blow-up point is a nondegenerate critical point ξ0 of the function 𝒲g defined in (1.1) with 𝒲g⁢(ξ0)≠0. Moreover, ξ0 turns out to be a clustering blow-up point as soon as it is a nondegenerate minimum point of 𝒲g with 𝒲g⁢(ξ0)≠0 as proved by Pistoia and Vaira [12]. We strongly believe that ξ0 is also a towering blow-up point, as suggested by a recent result obtained by Morabito, Pistoia and Vaira [9] under the additional assumption that M is symmetric with respect to the point ξ0. We also point out that if ξ0 is either a clustering or a towering blow-up point, an arbitrary large number of solutions blowing-up at ξ0 do exist as was shown in [12] and in [9]. Therefore, the existence and multiplicity of blowing-up solutions to problem (1.4) strictly rely on the existence of nondegenerate critical points of the function 𝒲g defined in (1.1) having a corresponding non-vanishing critical value.

The proof of Theorem 1.1 relies on a transversality argument (see Theorem 2.1) and it is carried out in Section 2. Section 3 contains the proof of the crucial transversality condition.

2 Formulation of the Problem and Proof of the Main Result

We denote by 𝒮k the space of all Ck symmetric covariant 2-tensors on M. It is a Banach space equipped with the norm ∥⋅∥k defined in the following way: We fix a finite covering {Vα}α∈L of M such that the closure of Vα is contained in Uα, where {Uα,ψα} is the open coordinate neighborhood. If h∈𝒮k, we denote by hi⁢j the components of h with respect to the coordinates (x1,…,xn) on Vα. We define

The set ℳk of all Ck Riemannian metrics on M is an open set of 𝒮k.

In the following, we will assume k≥4.

Given g^∈ℳk, it is possible to define an atlas on M whose charts are (Bg^⁢(ξ,R),φ-1), where φ:B⁢(0,R)→Bg^⁢(ξ,R). Here Bg^⁢(ξ,R)⊂M is the ball centered at ξ with radius R given by the metric g^ and B⁢(0,R)⊂ℝn is the ball centered at 0 with radius R in the Euclidean space ℝn. Let ℬρ:={h∈𝒮k:∥h∥k<ρ} be the ball centered at 0 with radius ρ in 𝒮k.

For any ξ∈M and h∈ℬρ, with ρ small enough so that g^+h∈ℳk, we consider the Weyl curvature tensor Wg^+h⁢(ξ):=Weylg^+h⁢(ξ) of (M,g^+h) at the point ξ∈M whose components are Wa⁢b⁢c⁢dg^+h⁢(ξ). We introduce the function

Here and in the following, we use the Einstein summation convention, i.e. when an index variable appears twice in a single term, once in an upper (superscript) and/or in a lower (subscript) position, it implies that we are summing over all of its possible values.

Given ξ0∈M and the chart (Bg^⁢(ξ0,R),φ-1), we set

Now, we introduce the C1-map F:ℬρ×B⁢(0,R)⊂𝒮k×ℝn→ℝn defined by

We shall apply to the map F an abstract transversality theorem (see [13, 14, 20]) which we recall from [14, Theorem 1.1].

By Theorem 2.1 we obtain the following result, which is crucial for deducing Theorem 1.1.

3 The Transversality Condition