Conformal gradient vector fields on Riemannian manifolds with boundary

Abstract

Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with smooth boundary. We show that the presence of a nontrivial conformal gradient vector field on $M$, with an appropriate control on the Ricci curvature makes $M$ to be isometric to a hemisphere of ${\mathbb{S}}^n$. We also prove that if an Einstein manifold admits nonzero conformal gradient vector field, then its scalar curvature is positive and it is isometric to a hemisphere of ${\mathbb{S}}^n$. Furthermore, we prove that if $M$ admits a nontrivial conformal vector field and has constant scalar curvature, then the scalar curvature is positive. Finally, a suitable control on the energy of a conformal vector field implies that $M$ is isometric to a hemisphere ${\mathbb{S}}_{+}^n$.

To appear in Colloquium Mathematicum