Conformal gradient vector fields on Riemannian manifolds with boundary
Volume 159 / 2020
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, causes $M$ to be isometric to a hemisphere of $\mathbb {S}^{n}$. We also prove that if an Einstein manifold with boundary admits a 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 $\mathbb {S}^n_+$.
