On isometry groups of gradient Ricci solitons

A Appendix

In this Appendix we consider the case of each level set of a GRS is Euclidean, which was mentioned in the proof of Theorem 1.1. We first adapt the gradient Ricci soliton equation (1.1) to the cohomogeneity one setting, essentially using the methodology and notation of [16].

Let G be a Lie group acting isometrically on a Riemannian manifold ( M , g ) . The action is of cohomogeneity one if the orbit space M / G is one-dimensional. In this case, we choose a unit speed geodesic γ ⁢ ( t ) that intersects all principal orbits perpendicularly. Then, it is possible to define a G-equivariant diffeomorphism Φ : I × P ↦ M 0 given by Φ ⁢ ( t , h ⁢ K ) = h ⋅ γ ⁢ ( t ) . Here, M 0 ⊂ M is an open dense subset, I is an open interval; P = G / K where K is the isotropy group along γ ⁢ ( t ) . Then the pullback metric is of the form

where g t is a one-parameter family of G-invariant metrics on P. We let L denote the shape operator L ⁢ ( X ) = ∇ X ⁡ N , where N = Φ * ⁢ ( ∂ t ) is a unit normal vector field. We will consider L t = L ∣ Φ ( t × P ) to be a one-parameter family of endormorphisms on TP via identification T ⁢ ( Φ ⁢ ( t × P ) ) = T ⁢ P . Following [16], we have ∂ t ⁡ g = 2 ⁢ g t ∘ L t , that is for X , Y ∈ T ⁢ P ,

From Gauss, Codazzi, and Riccati equations, we find that the Ricci curvature of ( M 0 , g ) is totally determined by the geometry of the shape operator and how it evolves. Moreover, if the function f is invariant by the group action, then the gradient Ricci soliton equation (1.1) is reduced to

(A.1)

where Ric t denotes the Ricci curvature of ( P , g t ) , δ ⁢ L = ∑ i ∇ e i ⁡ L ⁢ ( e i ) for an orthonormal basis and tr ⁡ T = tr g t ⁡ T t .

Now, we consider a GRS ( M n , g , f ) ( n ≥ 3 ) with local Euclidean level sets:

where each function h i is smooth. From (A.2) and note that ∂ t ⁡ g t = 2 ⁢ g t ∘ L t , we get

Since the shape operator L satisfies the Riccati equation [19, p. 117], the sectional curvature of the 2-plane section spanned by e i = ∂ i h i and N is given by

Using the Gauss equation [37, Theorem 3.2.4], we see that the sectional curvature of the 2-plane section spanned by e i and e j is given by

The Ricci curvature is then given by Ric ⁡ ( N , N ) = - ∑ i h i ′′ h i , and

From these results, we imply that the Scalar curvature is given by

where A := ∑ i h i ′ h i , B := ∑ i ( h i ′ h i ) 2 . Thus, generically, the Weyl tensor is NOT vanishing.

Plugging the above results in (A.1), we conclude that

Let u 0 := f ′ and u i := h i ′ h i . The above system can be written as follows:

This is a system of first order ODEs and the Picard–Lindelöf theorem yields local existence and uniqueness.