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In this paper, we study gradient Yamabe solitons realized as warped product manifolds. We apply the maximum principle to find lower bound estimates for both the potential function of the soliton and the scalar curvature of the warped product manifold. By slightly modifying Li–Yau's technique, we can handle the drifted Laplacian and thus find different gradient estimates for the warping function according to the sign of the (constant) scalar curvature of the fiber manifold. We close the article with a theorem stating the nonexistence of gradient Yamabe solitons on top of warped products with a base manifold satisfying certain analytical hypothesis. [ABSTRACT FROM AUTHOR]

In this paper, we take a look at conditions that make a Riemann soliton trivial, compacity being one of them. We also obtain scalar curvature estimates for complete shrinking or steady gradient Riemann solitons whose scalar curvature is bounded from below at infinity. [ABSTRACT FROM AUTHOR]

In this paper, we show that any compact quasi k-Yamabe gradient soliton must have constant σ k -curvature. Moreover, we provide a certain condition for a compact quasi k-Yamabe soliton to be gradient, and for noncompact solitons, we present a geometric rigidity under a decay condition on the norm of the soliton field. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli–Kohn–Nirenberg inequality with the same exponent n (n ≥ 2) , then it has exactly n-dimensional volume growth. As application, we obtain geometric and topological properties of Alexandrov spaces, Riemannian manifolds and Finsler manifolds which support a Caffarelli–Kohn–Nirenberg inequality. [ABSTRACT FROM AUTHOR]