Let $M$ be a rotationally symmetric Riemannian manifold, and $\Delta$ be the Laplace-Beltrami operator on $M$. We establish a necessary and sufficient condition for the constant function 1 to be a semismall perturbation of $-\Delta +1$ on $M$, and give optimal sufficient conditions for uniqueness of nonnegative solutions of the Cauchy problem to the heat equation. As an application, we determine the structure of all nonnegative solutions to the heat equation on $M\times(0,T)$.
"Nonnegative solutions of the heat equation on rotationally symmetric Riemannian manifolds and semismall perturbations." Rev. Mat. Iberoamericana 27 (3) 885 - 907, September, 2011.