On Ricci solitons whose potential is convex
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On Ricci solitons whose potential is convex CHANDAN KUMAR MONDAL and ABSOS ALI SHAIKH∗ Department of Mathematics, University of Burdwan, Golapbag, Burdwan 713 104, India * Corresponding Author E-mail: [email protected]; [email protected]; [email protected]
MS received 22 August 2019; revised 2 April 2020; accepted 2 April 2020 Abstract. 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. Keywords. Ricci soliton; scalar curvature; Ricci flat; convex function; critical point; Riemannian manifold. Mathematics Subject Classification.
53C20, 53C21, 53C44.
1. 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, g0 ): ∂ g(t) = −2 Ric, g(0) = g0 . ∂t 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 1 (1) Ric + £ X g = λg, 2 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 ∇ 2 u + Ric = λg,
(2)
© Indian Academy of Sciences 0123456789().: V,-vol
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Proc. Indian Acad. Sci. (Math. Sci.)
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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 [11] proved that an n-dimensional gradient shrinking Ricci soliton with non-negative sectional curvature and positive Ricci curvature must be compact (for more results, see [10,12]). Perelman [13] proved that a compact Ricci soliton is always gradient Ricci soliton. For the detai
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