Large time behavior of the heat equation on complete manifolds with non- negative Ricci curvature
It is shown that if a complete Riemannian manifold M has nonnegative Ricci curvature and if the volume growth of M is of maximum order then \(\lim_{t\to \infty}V(B_ x(\sqrt{t}))H(x,y,t)\) exists, where \(V(B_ x(\sqrt{t}))\) denotes the volume of the geodesic ball centered at x of radius \(\sqrt{t}\) and H(x,y,t) is the heat kernel, i.e. H(x,y,t) solves the heat equation \[ (\Delta -\partial /\partial t)F(x,t)=0\quad on\quad M\times (0,\infty), \] with initial data \(F(x,0)=f(x)\), by setting \(F(x,t)=\int_{M}H(x,y,t)f(y)dy\). An interesting consequence of this result is derived: A nonnegatively Ricci-curved manifold with maximal volume growth must have finite fundamental group. Further, the asymptotic formula is applied to obtain a necessary and sufficient condition for the heat equation with bounded initial data to stabilize as t. Finally, the heat equation is used to study the behavior of bounded superharmonic functions on complete manifolds with nonnegative Ricci curvature.
manifolds with nonnegative Ricci curvature
scientific article; zbMATH DE number 3992498