On Hypersurfaces of Space Forms That Are Gradient Yamabe Solitons

Abstract

This paper studies nontrivial Yamabe gradient solitons occurring as complete immersed hypersurfaces with constant scalar curvature in space forms $\left(\overline{M}^n(C), \overline{g}\right)$. For solitons with non-vanishing gradients, we establish a structural characterization of the soliton by analyzing a regular level set $\Sigma$ of the soliton function. In particular, imposing a suitable condition on the traceless part of the second fundamental form, $\Phi^{\Sigma}_{\overline{M}}$ of $\Sigma$ as a submanifold of the space form, we prove that the soliton either decomposes as a Riemannian product of $\mathbb{R}$ and a totally umbilical submanifold of the space form, or satisfies a sharp lower bound on $\sup\limits_\Sigma\left|\Phi^{\Sigma}_{\overline{M}}\right|$. Furthermore, we show that the lower bound is attained if, and only if, the soliton is isometric to a Riemannian product of $\mathbb{R}$ and a parallel submanifold of the space form.