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Abstract |
Many gradient estimates in differential geometry can be naturally treated by stochastic methods involving Brownian motion on a Riemannian manifold. In this talk, we discuss Hamilton's gradient estimate of bounding the gradient of the logarithm of a positive harmonic function in terms of its supremum. We will see how naturally this form of gradient estimates follows from Ito's formula and extend it to manifolds with convex oundary by considering reflecting Brownian motion. Furthermore, we will show that in fact Hamilton's gradient estimate can be embedded as the limiting case of a family of gradient estimates which can be treated just as easily by the same stochastic method. |
