在古典微分幾何裡，我們藉由高斯曲率和平均曲率來分類3維歐式空間裡的曲面。根據高斯-博內定理，我們知道高斯曲率的曲面積分是一個拓撲不變量。因此我們會好奇平均曲率的曲面積分是否也會有類似的結果。在1965年，Willmore提出相關的猜想。這篇論文主要目的是驗證Willmore 猜想在其中的一個主曲率是常數的曲面下是成立的。

In classical differential geometry, we use the Gaussian curvature and the mean curvature to classify surfaces in Euclidean 3-space. According to the Gauss-Bonnet theorem, we know that the surface integral of the Gaussian curvature is a topological invariant. It is an interesting problem whether the surface integral of the mean curvature has a similar result. In 1965, Willmore proposed related conjecture. The main purpose of this paper is to verify that Willmore Conjecture holds in the case of a surface with a constant principal curvature.

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