In this paper, first we prove the Crandall–Liggett exponential theorem in nonlinear semigroup theory on Hadamard manifolds. This theorem states that a semigroup of contractions can be constructed by the resolvent of a monotone vector field on Hadamard manifolds. Then, we show that the generated semigroup satisfies the evolution equation governed by the monotone vector field. The results of this paper are extensions of the classical results of Crandall and Liggett (Am J Math 93:265–298, 1971) and Brezis and Pazy (Israel J Math 8:367–383, 1970) to Hadamard manifolds. Some examples are also presented in the last part of the paper.

在本文中，我们首先证明了关于 Hadamard 流形的非线性半群理论中的 Crandall-Liggett 指数定理。该定理指出，可以通过对 Hadamard 流形上的单调矢量场进行解算来构造一个半群收缩。然后，我们证明生成的半群满足由单调向量场支配的演化方程。本文的结果是将 Crandall 和 Liggett (Am J Math 93:265–298, 1971) 以及 Brezis 和 Pazy (Israel J Math 8:367–383, 1970) 的经典结果扩展到 Hadamard 流形。本文的最后一部分还提供了一些示例。