這篇文章分成二個部分：第一個部分我們藉由擬埃爾米特流形的Bakry-Emery瑞奇曲率來研究擬埃爾米特流形上與Witten拉普拉斯聯繫在一起的柯西－黎曼熱方程的Perelman’s W-熵公式。第二部分我們建立在海森堡群中Legendrian子流形的基本定理。
第二章，我們導出在（2n+1）維度閉的擬埃爾米特流形與Witten拉普拉斯聯繫在一起的柯西－黎曼熱方程的次梯度估計。對於此次梯度估計的應用，我們得到柯西－黎曼熱方程的Perelman型式的熵公式和與Witten拉普拉斯聯繫在一起的柯西－黎曼熱方程的Perelman型式的熵公式。
第三章，PSH(n)是由n維海森堡群上所有擬埃爾米特變換所形成的李群。我們得到PSH(n)的群表示。除此之外，我們討論這個矩陣李群PSH(n)中的任一元素如何做為在齊次空間PSH(n)/U(n)上的一組標架。因此我們從Maurer-Cantan form可以立即得到活動標架公式。
第四章，我們利用Élie Cartan的活動標架法、李群理論得到海森堡群中的Legendrian子流形的基本定理。我們令Σ是一個n維可定向的曲面，並且做為在海森堡群中的Legendrian子流形。對於任意在Σ裡的totally real point，我們計算Darboux 導數而得到integrability條件。於是我們可以証明對任意的n維黎曼流形如果滿足integrability條件，那麼此黎曼流形就可以局部嵌入到n維的海森堡群裡，做為海森堡群中的Legendrian子流形。

In this thesis, we study Perelman's W-Entropy formula for the CR heat equation associated with the Witten Laplacian on pseudohermitian manifolds via the Bakry-Emery Ricci curvature. In addition we establish the fundamental theorem for Legendrian submanifolds in Heisenberg groups.

In Chapter 2, we derive the subgradient estimate of the CR heat equation associated with the Witten Laplacian on a closed pseudohermitian (2n+1)-manifold. With its application, we obtain Perelman-type entropy formula for
thse CR heat equation and the CR heat equation associated with the Witten Laplacian.

In Chapter 3, we obtain the representation of PSH(n) which is the group of pseudohermitian transformations on n-dimensional Heisenberg groups. Also we discuss how the matrix Lie group PSH(n) interpret as the set of "frames" on the homogeneous space PSH(n)/U(n). Then for the (left-invariant) Maurer-Cartan form, we immediately get the moving frame formula.

In Chapter 4, we use Élie Cartan's method of moving frames, the theory of Lie groups to obtain the fundamental theorem for the Legendrian submanifolds in Hesenberg groups. Let Σ be a n-dimensional oriented surface and f:Σ-->H^n be an embedding as a Legendrian submanifold in H^n. For every totally real point p in Σ, we compute the Darboux derivative of the lifting of f to get the integrability conditions for Σ. Then we show that for any Riemannian manifold which satisfies the integrability conditions can be
locally embedded into H^n as a Legendrian submanifold.

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