Gradient Yamabe solitons with conformal vector field. (English) Zbl 1475.53105
The paper under review concerns the theory of Yamabe flow.
By definition, a Riemannian manifold \((M^m, g)\) is called a Yamabe soliton, if it satisfies \[ L_X g = (\lambda - R) g, \] where \(R\) is the scalar curvature, \(L_X\) is the Lie derivative with respect to a vector field \(X\), \(\lambda\) is a constant. If \(X = \nabla f\) for some smooth function \(f\) on \(M^m\), then the Yamabe soliton is referred to as gradient.
Motivated by recent results of R. Sharma [J. Geom. 109, No. 2, Paper No. 33, 7 p. (2018; Zbl 1397.53060)] on Ricci solitons, the authors consider the case where the Yamabe flow \((M^m, g, X, \lambda)\) admits a vector field \(V\) which is conformal so that \(L_V g = 2\sigma g\), where \(\sigma\) is a smooth function.
The main results are the following.
Theorem 1. Let \((M^m, g, X=\nabla f, \lambda)\) be a connected gradient Yamabe soliton with constant scalar curvature \(R\). Suppose it admits a non-homothetic (\(\sigma\not = \mathrm{const}\)) conformal vector field \(V\) such that \(\left[ V , X \right] = 0\). Then the potential function \(f\) is constant and the scalar curvature is equal to \(\lambda\).
Theorem 2. Let \((M^m, g, X=\nabla f, \lambda)\) be a connected gradient Yamabe soliton. Suppose it admits a homothetic (\(\sigma = \mathrm{const}\)) conformal vector field \(V\) leaving the potential vector field \(X\) invariant. Then either \(R= \lambda\) or the vector field \(V\) preserves the potential function \(f\).
Theorem 3. Let \((M^m, g, X=\nabla f, \lambda)\) be a locally conformally flat gradient Yamabe soliton. Suppose it admits a conformal vector field \(V\) leaving the potential vector field \(X\) invariant. Then \((M^m, g)\) has constant curvature, the vector field \(V\) is homothetic.
Theorem 3 is based on the use of previous results from [H.-D. Cao et al., Math. Res. Lett. 19, No. 4, 767–774 (2012, Zbl 1267.53066)] on a specific warped product structure of locally conformally flat Yamabe solitons. Particularly, this allows the authors to determine explicitly the potential function \(f\).
By definition, a Riemannian manifold \((M^m, g)\) is called a Yamabe soliton, if it satisfies \[ L_X g = (\lambda - R) g, \] where \(R\) is the scalar curvature, \(L_X\) is the Lie derivative with respect to a vector field \(X\), \(\lambda\) is a constant. If \(X = \nabla f\) for some smooth function \(f\) on \(M^m\), then the Yamabe soliton is referred to as gradient.
Motivated by recent results of R. Sharma [J. Geom. 109, No. 2, Paper No. 33, 7 p. (2018; Zbl 1397.53060)] on Ricci solitons, the authors consider the case where the Yamabe flow \((M^m, g, X, \lambda)\) admits a vector field \(V\) which is conformal so that \(L_V g = 2\sigma g\), where \(\sigma\) is a smooth function.
The main results are the following.
Theorem 1. Let \((M^m, g, X=\nabla f, \lambda)\) be a connected gradient Yamabe soliton with constant scalar curvature \(R\). Suppose it admits a non-homothetic (\(\sigma\not = \mathrm{const}\)) conformal vector field \(V\) such that \(\left[ V , X \right] = 0\). Then the potential function \(f\) is constant and the scalar curvature is equal to \(\lambda\).
Theorem 2. Let \((M^m, g, X=\nabla f, \lambda)\) be a connected gradient Yamabe soliton. Suppose it admits a homothetic (\(\sigma = \mathrm{const}\)) conformal vector field \(V\) leaving the potential vector field \(X\) invariant. Then either \(R= \lambda\) or the vector field \(V\) preserves the potential function \(f\).
Theorem 3. Let \((M^m, g, X=\nabla f, \lambda)\) be a locally conformally flat gradient Yamabe soliton. Suppose it admits a conformal vector field \(V\) leaving the potential vector field \(X\) invariant. Then \((M^m, g)\) has constant curvature, the vector field \(V\) is homothetic.
Theorem 3 is based on the use of previous results from [H.-D. Cao et al., Math. Res. Lett. 19, No. 4, 767–774 (2012, Zbl 1267.53066)] on a specific warped product structure of locally conformally flat Yamabe solitons. Particularly, this allows the authors to determine explicitly the potential function \(f\).
Reviewer: Vasyl Gorkaviy (Kharkov)
MSC:
| 53E20 | Ricci flows |
| 53C18 | Conformal structures on manifolds |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
Keywords:
Yamabe flow; gradient Yamabe soliton; conformal vector field; scalar curvature; locally conformally flat Riemannian manifoldReferences:
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