Theorem 1. Any nontrivial complete steady gradient Yamabe solitons with positive scalar curvature are rotationally symmetric.
September 17, 2023
In this paper, we solve the Yamabe soliton version of the Perelman conjecture. We show that any nontrivial complete steady gradient Yamabe solitons with positive scalar curvature are rotationally symmetric.
The Yamabe flow is important for understanding structures in Riemannian geometry. It was introduced by R. Hamilton [1] and further developed by S. Brendle [2], [3]. Yamabe solitons are special solutions of the Yamabe flow, and are expected to be the singularity models. From this perspective, Yamabe solitons have developed rapidly in the last few decades. One of the main problems is the Yamabe soliton version of the Perelman conjecture, that is, any nontrivial complete steady gradient Yamabe solitons with positive scalar curvature are rotationally symmetric. Various researchers have considered the problem under the assumption of locally conformally flat condition. The problem was solved under these assumptions and that the sectional curvature is positive by Daskalopoulos and Sesum [4], and the scalar curvature is positive by Cao, Sun and Zhang [5]. Subsequently, Catino, Mantegazza, and Mazzieri [6] achieved a significant breakthrough by removing the locally conformally flat condition, previously believed impossible to eliminate. They also relaxed the requirement of positive sectional curvature by instead assuming that the Ricci tensor is non-negative everywhere and positive at some point, and applied their result to \(k\)-Yamabe solitons. In [5], [6], they rediscovered Tashiro’s theorem [7] (see also [8]). As a corollary, Cao, Sun and Zhang solved the problem in the 3-dimensional case. However, the higher dimensional case is still open. In this paper, we completely solve the Yamabe soliton version of the Perelman conjecture.
Theorem 1. Any nontrivial complete steady gradient Yamabe solitons with positive scalar curvature are rotationally symmetric.
Remark 2. It is known that any compact gradient Yamabe solitons are trivial \((\)[1], [9]\()\). The original Perelman conjecture [10] is that any \(3\)-dimensional complete noncompact \(\kappa\)-noncollapsed gradient steady Ricci soliton with positive curvature is rotationally symmetric, which was proven by S. Brendle [11]. However, in higher dimensions, it is not well understood \((\)but, see for example [12]\()\).
An \(n\)-dimensional Riemannian manifold \((M^n,g)\) is called a gradient Yamabe soliton if there exists a smooth function \(F\) on \(M\) and a constant \(\lambda\in \mathbb{R}\), such that \(\nabla \nabla F=(R-\lambda)g,\) where \(\nabla\nabla F\) is the Hessian of \(F\), and \(R\) is the scalar curvature on \(M\). If \(F\) is constant, \(M\) is called trivial. If \(\lambda>0\), \(\lambda=0\), or \(\lambda<0\), then the Yamabe soliton is called shrinking, steady, or expanding.
Tashiro’s theorem ([7], see also [5], [6], [8]) is used for proving Theorem 1.
Theorem 3 ([7]). A Riemannian manifold \((M^n,g)\) which satisfies that for any smooth functions \(F\) and \(\varphi\) on \(M\), \(\nabla \nabla F=\varphi g\) is either \((1)\) compact and rotationally symmetric, or \((2)\) rotationally symmetric and equal to the warped product \(([0,\infty),dr^2)\times_{|\nabla F|}(\mathbb{S}^{n-1},{\bar g}_{S}),\) where \(\bar g_{S}\) is the round metric on \(\mathbb{S}^{n-1},\) or \((3)\) the warped product \((\mathbb{R},dr^2)\times_{|\nabla F|} \left(N^{n-1},\bar g\right),\) where the scalar curvature \(\bar R\) of \(N\) satisfies \[\label{key} |\nabla F|^2R=\bar R-(n-1)(n-2)\varphi^2-2(n-1)g(\nabla F,\nabla\varphi).\tag{1}\]
Remark 4. The potential function \(F\) depends only on \(r\), and \(F'(r)>0\) \((\)see, for example, the proof of Theorem \(1.1\) of [8]\()\). The manifold \((M,g,F,\varphi)\) that satisfies the condition \(\nabla \nabla F=\varphi g\) was also studied by Cheeger and Colding [13].
As pointed out in [5], it follows as a corollary of Theorem 3 that any nontrivial complete gradient Yamabe solitons with positive Ricci curvature are rotationally symmetric.
Proof of Theorem 1. To show rotational symmetry of \(M\), we only have to consider \((3)\) of Theorem 3. By the soliton equation, Remark 4 and 1 , one has \[\label{key2} \rho'\rho^2+(n-1)(n-2)\rho'^2+2(n-1)\rho\rho''=\bar R,\tag{2}\] where \(\rho=F'\). Differentiating both sides of 2 , one has \[\label{key3} \rho''\rho^2+2\rho\rho'^2+2(n-1)^2\rho'\rho''+2(n-1)\rho\rho'''=0.\tag{3}\] Since the left hand side of \(\eqref{key2}\) depends only on \(r\), the scalar curvature \(\bar R\) of \(N\) is constant. Positivity of the scalar curvature shows that \(\bar R>0\) (which was shown in [5]). In fact, if \(\bar R\) is nonpositive, then \(\rho''\) is nonpositive, hence the positive function \(\rho\) is concave. Therefore, \(\rho\) is constant, which cannot occur. Since \(\rho'>0\), \(\rho\) is monotone increasing. Furthermore, one can show that \(\rho\) goes to infinity. Assume that \(\rho\) is bounded from above, that is, we assume that \(F'\nearrow c\) as \(r\nearrow+\infty\) for some positive constant \(c\). If \(\rho''>0\) on \(\mathbb{R}\), then the bounded function \(\rho\) is convex, which is a contradiction. Assume that there exists \(\tilde{r}\in\mathbb{R}\) such that \(\rho''(\tilde{r})=0\). By 3 , \(\rho'''<0\) at \(\tilde{r}\). Therefore, by iterating the same argument, \(\rho'\) is weakly monotone decreasing on \((\tilde{r},+\infty)\). Hence, \(\rho'\searrow0\) as \(r\nearrow+\infty\). By 2 , one has \(\rho''\rightarrow \frac{\bar R}{2(n-1)c}>0\) as \(r\nearrow+\infty\), which cannot occur, because \(\rho<c\).
The equation 2 is an autonomous second order equation and can be made into a first order equation by using \(\rho\) as a new independent variable. If \(\rho'=G(\rho)\), then \(\rho''=\dot{G} G,\) and one has \[\label{1st} G\rho^2+(n-1)(n-2)G^2+2(n-1)\rho\dot{G}G=\bar R.\tag{4}\] By differentiating the equation, one has \[\label{2nd} \dot{G}\rho^2+2\rho G+2(n-1)^2\dot{G} G+2(n-1)\rho\ddot G G+2(n-1)\rho \dot{G}^2=0.\tag{5}\] Assume that \(\dot{G}>0\) at some point \(\rho_0\in (0,+\infty)\), that is, \(\dot{G}>0\) on some open interval \(\Omega=(\rho_1,\rho_2)(\ni\rho_0)\). If \(\Omega=(\rho_1,+\infty)\), by 4 , \(G\rho^2+(n-1)(n-2)G^2<\bar R\) on \((\rho_1,+\infty).\) However, the left hand side goes to infinity as \(\rho\nearrow +\infty\), which cannot occur. Thus, one can assume that \(\dot{G}=0\) at \(\rho_2\). Then, by \(\eqref{2nd}\), \(2(n-1)\rho_2\ddot GG+2\rho_2 G=0\) at \(\rho_2\). Hence, \(\ddot G<0\) at \(\rho_2\), and \(G\) is monotone decreasing on \((\rho_2,\rho_3)\) for some \(\rho_3\). Iterating the same argument, one can extend \(\rho_3\) to \(+\infty\), that is, \(G\) is monotone decreasing on \((\rho_2,+\infty)\). Hence, if there exists such an open interval \(\Omega\), it must be \((0,\rho_2)\) and \(\rho_2\) is the maximum point of \(G\). Let \(p\in\mathbb{R}\) be the maximum point of \(\rho'\). Since \(\rho'>0\), there exists a point \(q\in (-\infty, p)\) such that \(\rho''(q)>0\) and \(\rho'''(q)=0\). However, by 2 , one has \(\rho''\rho^2+2\rho \rho'^2+2(n-1)^2\rho'\rho''=0\) at \(q\), which cannot occur. We finally obtain \(\Omega=\emptyset.\)
Therefore, one has \(0\geq\dot{G}=\frac{\rho''(r)}{\rho'(r)}\) for every \(r\in\mathbb{R}\), and \(\rho''\leq 0\) on \(\mathbb{R}\). Since the positive smooth function \(\rho\) is concave on \(\mathbb{R}\), \(\rho\) is constant, which cannot occur. ◻
By the similar argument, we also get the similar result for shrinking solitons:
Theorem 5. Any nontrivial complete shrinking gradient Yamabe solitons with \(R>\lambda\) are rotationally symmetric.
Remark 6. The assumption \(R>\lambda\) is optimal. In fact, for any \(b\in\mathbb{R}\) and any Riemannian manifold \((N,\bar{g}_N)\) with constant positive scalar curvature \(\bar R\), \((\mathbb{R}\times N,dr^2+\frac{\bar R}{\lambda}\,\bar{g}_N,\sqrt{\frac{\bar R}{\lambda}}\,r+b)\) is a nontrivial shrinking gradient Yamabe soliton with \(R=\lambda\).
Acknowledgements.
The author would like to express his gratitude to Ken Shirakawa for valuable discussions.
Data availability statement
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Conflict of interest
There is no conflict of interest in the manuscript.
The author is partially supported by the Grant-in-Aid for Scientific Research (C), No.23K03107, Japan Society for the Promotion of Science.↩︎