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Large time behavior of the heat equation on complete manifolds with non- negative Ricci curvature. (English) Zbl 0613.58032

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.
Reviewer: M.Fila

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K05 Heat equation
53C20 Global Riemannian geometry, including pinching
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