A function with nonnegative Laplacian on a closed manifold is locally constant
= A function with nonnegative Laplacian on a closed manifold is locally constant
For the nonnegative <Hodge Laplacian> convention $\Delta f=-\operatorname{div}\operatorname{grad}f$, a smooth function on a <closed manifold> with $\Delta f\ge0$ has $\int\Delta f=0$ and hence $\Delta f=0$. Integration by parts gives $\int|df|^2=0$, so it is constant on each connected component. Global constancy requires connectedness; boundary conditions are needed if a boundary is allowed.