黎曼流型上的熱核估計

Abstract

在一般的歐氏空間上，我們已經能清楚的把熱核表達出來。但在較為複雜的黎曼曲面上則難以表達。在我的文章中，我將會分為在完備而不緊緻的黎曼流型上以及完備且邊界是凸的黎曼流型上分別討論。在這兩種情況上，藉由得到相同的梯度估計，以此推導出相同的哈拿估計，在由此二估計，推出相同型式的熱核上界。另外，藉由畢社比較定理，估計出在此二種情況中相同型式的熱核下界。最後，利用熱的上界及下界可導出在拉普拉斯運算下的特徵值下界、以及格林函數的估計。

In the common Euclidean spaces, we can explicitly express the form of heat kernel.But in the more complicated Riemannian manifolds, it is hard to express. In my survey, I will discuss in two cases, first, in complete noncompact manifold, second, in compactmanifold with convex boundary. In this two cases, we can get the same form of gradient estimate and Harnack estimate, and by these two estimates, we can get the same form ofheat kernel upper bound in these two cases. Also, by Bishop volume comparison, we can get the same form of heat kernel upper bound in these two cases. With heat kernelestimates, we can estimate the lower bound of eigenvalue of Laplace operator and Green function.