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On Ricci solitons whose potential is convex

In this paper we consider the Ricci curvature of a Ricci soliton. In particular, we have showed that a complete gradient Ricci soliton with non-negative Ricci curvature possessing a non-constant convex potential function having finite weighted Dirichlet integral satisfying an integral condition is Ricci flat and also it isometrically splits a line. We have also proved that a gradient Ricci soliton with non-constant concave potential function and bounded Ricci curvature is non-shrinking and hence the scalar curvature has at most one critical point.


Introduction and preliminaries
In 1982, Hamilton [8] introduced the concept of Ricci flow. The Ricci flow is defined by an evolution equation for metrics on the Riemannian manifold (M, g 0 ): g(t) = −2Ric, g(0) = g 0 .
A complete Riemannian manifold (M, g) of dimension n ≥ 2 with Riemannian metric g is called a Ricci soliton if there exists a vector field X satisfying where λ is a constant and £ denotes the Lie derivative. The vector field X is called potential vector field. The Ricci solitons are self-similar solutions to the Ricci flow. Ricci solitons are natural generalization of Einstein metrics, which have been significantly studied in differential geometry and geometric analysis. A Ricci soliton is an Einstein metric if the vector field X is zero or Killing. Throughout the paper by M we mean an n-dimensional, n ≥ 2, complete Riemannian manifold endowed with Riemannian metric g. Let C ∞ (M) be the ring of smooth functions on M. If X is the gradient of some function u ∈ C ∞ (M), such a manifold is called a gradient Ricci soliton, and then (1) reduces to the form where ∇ 2 u is the Hessian of u and the function u is called potential function. The Ricci soliton (M, g, X, λ) is called shrinking, steady and expanding according as λ > 0, λ = 0 and λ < 0, respectively. Each type of Ricci solitons determines some unique topology of the manifold.
For example, if the scalar curvature of a complete gradient shrinking Ricci soliton is bounded, then the manifold has finite topological type [5]. Munteanu and Wang proved that an ndimensional gradient shrinking Ricci soliton with non-negative sectional curvature and positive Ricci curvature must be compact [11], for more results see [10,12]. Perelman [13] proved that a compact Ricci soliton is always gradient Ricci soliton. For the detailed treatment on Ricci solitons and their interaction to Ricci flow, we refer to [2,4]. A smooth function ϕ : M → R is said to be convex [15,16] if for any p ∈ M and for any vector v ∈ T p M gradϕ, v p ≤ ϕ(exp p v) − ϕ(p).
If ϕ is convex, then −ϕ is called concave.
The paper is arranged as follows: In the first section, we have proved that a complete noncompact gradient Ricci soliton with non-negative Ricci curvature possessing a non-constant convex function with finite weighted Dirichlet integral satisfying an integral condition is Ricci flat and also it isometrically splits a line. We have also deduced a corollary relating to the Ricci soliton and harmonic function. In the last section, we have proved that if in a complete gradient Ricci soliton, the potential function is a non-constant concave function with bounded Ricci curvature then the scalar curvature possesses at most one critical point, see Theorem 3.3.

Ricci soliton and Ricci flat manifold
Lemma 2.1. Let (M, g) be a complete Riemannian manifold with non-negative Ricci curvature.
If u ∈ C ∞ (M) is a non-constant convex function with finite weighted Dirichlet integral, i.e., and also satisfies the relation is an open ball with center p and radius r, then the hessian of u vanishes in M.
Since u is harmonic, so ∆u = 0. Therefore, the above equation reduces to Combining ϕ 2 r with (6) and then integrating we obtain Using integration by parts we get which implies that ∇ 2 u = 0.  Proof. In view of Lemma 2.1, it follows that hessian of u vanishes. Again, Lemma 2.2 implies that u is an affine function. Therefore, using Theorem 2.3, we conclude that M is isometric to the Riemannian product N × R, where N is a totally geodesic submanifold of M.  Proof. The proof is same as that of Theorem 2.5 except the part where we have proved the harmonicity of the function u and hence we omit. Proof. Since u is a non-trivial concave function in M, the function −u is non-constant convex and it implies that the manifold M is non-compact. Let us consider a length minimizing normal geodesic γ : [0, t 0 ] → M for some arbitrary large t 0 > 0. Take p = γ(0) and X(t) = γ ′ (t) for t > 0. Then X is the unit tangent vector along γ. Now integrating (2) along γ, we get
[7] Let (M, g) be a steady gradient Ricci soliton with positive Ricci curvature.
Then there is atmost one critical point of R. Proof. Since λ ≥ 0, by using Theorem 3.1 we can prove that λ = 0. Therefore, M is a steady Ricci soliton. Now using Lemma 3.2, the result easily follows.