一個關於傑勒西單調性公式的評注

Abstract

廣義相對論中的霍金準局部能量為 S. W. Hawking 在1968年提出的概念，其在逆平均曲率流下的單調性以隱晦的方式初見於一篇1973年的文章，文章作者為 Robert Geroch ，因此該性質一般稱作傑勒西單調性公式。標誌著非負的時變率，這個公式在近代許多幾何流、數學相對論的文獻中被明白揭示並證明，其中一篇文獻是 Gerhard Huisken 與 Alexander Polden 在1996年完成的工作，此二人證明公式的手法為取得幾個演化方程後再求能量的時變率。在這份評注中，我們將詳述 Huisken 與 Polden 如何在那篇1996年的文章中證明傑勒西單調性公式。

The Hawking quasi-local energy in general relativity is a notion proposed by S. W. Hawking in 1968. Its monotonicity under inverse mean curvature flow was first suggested in a 1973 article authored by Robert Geroch, commonly known as the Geroch monotonicity formula. As a non-negative time derivative, this formula is explicitly stated and proved in many of the modern references on mathematical relativity and geometric flows, including an article composed by Gerhard Huisken and Alexander Polden in 1996. Huisken and Polden proved the formula by taking the time derivative of the energy function after some evolution equations were developed. In this note, we shall present a detailed exposition of how Huisken and Polden prove the Geroch monotonicity formula in the 1996 article.