Three-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor
- PDF / 976,218 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 53 Downloads / 166 Views
Three‑dimensional complete gradient Yamabe solitons with divergence‑free Cotton tensor Shun Maeta1 Received: 8 January 2020 / Accepted: 15 June 2020 © Springer Nature B.V. 2020
Abstract In this paper, we classify three-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor. We also give some classifications of complete gradient Yamabe solitons with nonpositively curved Ricci curvature in the direction of the gradient of the potential function. Keywords Yamabe solitons · Cotton tensor · Bach tensor · Scalar curvature Mathematics Subject Classification 53C21 · 53C25 · 53C20
1 Introduction A Riemannian manifold (M n , g) is called a gradient Yamabe soliton if there exist a smooth function F on M and a constant 𝜌 ∈ ℝ , such that (1)
(R − 𝜌)g = ∇∇F,
where R is the scalar curvature on M and ∇∇F is the Hessian of F. If 𝜌 > 0 , 𝜌 = 0 , or 𝜌 < 0 , then the Yamabe soliton is called shrinking, steady or expanding. If the potential function F is constant, then the Yamabe soliton is called trivial. It is known that any compact Yamabe soliton is trivial (see, for example, [7, 12]). Yamabe solitons are special solutions of the Yamabe flow which was introduced by Hamilton [10]. The Yamabe soliton equation (1) is similar to the equation of Ricci solitons. Ricci solitons are special solutions of the Ricci flow which was also introduced by Hamilton [11] (see also [8]). As is well known, by using the Ricci flow, Perelman [16–18] proved Thurston’s geometrization conjecture [19] and Poincaré conjecture. In the first paper of Perelman, he mentioned that “any 3-dimensional complete noncompact 𝜅-noncollapsed gradient steady Ricci soliton with The author is partially supported by the Grant-in-Aid for Young Scientists, Nos. 15K17542 and 19K14534 Japan Society for the Promotion of Science, and JSPS Overseas Research Fellowships 20172019 No. 70. * Shun Maeta [email protected]; [email protected]‑u.ac.jp 1
Department of Mathematics, Shimane University, Nishikawatsu 1060, Matsue 690‑8504, Japan
13
Vol.:(0123456789)
Annals of Global Analysis and Geometry
positive curvature is rotationally symmetric, namely Bryant soliton.” In [5], Cao et al. gave a partial answer to the conjecture. Finally, Brendle proved the conjecture [2]. In this paper, we consider the similar problem. More precisely, we consider the following problem.
Problem 1 Classify nontrivial nonflat complete three-dimensional gradient Yamabe solitons.
Daskalopoulos and Sesum [9] showed that “all locally conformally flat complete gradient Yamabe solitons with positive sectional curvature have to be rotationally symmetric.” The proof was inspired by Cao and Chen’s paper [3] (For quasi Yamabe solitons, by using the similar idea, Huang and Li [13] showed the similar problem). Furthermore, they constructed some examples of rotationally symmetric gradient Yamabe solitons on ℝn with positive sectional curvature. Recently, Cao et al. relaxed the assumption and showed that any nontrivial nonflat complete and locally conformally flat gradient Yama
Data Loading...
