变分原理与几何学有自然的联系。测地线、极小曲面、Einstein度量和调和映射都可以刻划为某一泛函的临界点。通常，这类变分问题很难用常规方法解决。为了克服这一困难，几何学家们创造了几何发展方程的方法。几何发展方程的基本思想是将给定的初始几何结构沿着某种流加以形变，以获得一个极限结构。典型的几何发展方程包括Ricci流、平均曲率流和调和映射流。这些几何流的研究广泛借用了诸如自相似解等来自非线性抛物方程的方法。今天，几何发展方程已大量应用于许多几何问题和拓扑问题当中。对于平均曲率流，本文考虑了自相似浸入的情形，证明了平均曲率孤立子方程的图像解若导数有界，则光滑收敛于某个具有特殊性质的函数。事实上，令$F(x) = (x,f(x)), x \in \mathbb{R}^{n}, f(x) \in \mathbb{R}^{k}$ 为平均曲率孤立子方程$\vec{H}(x) + F^{\bot}(x) = 0$的图像解。设$\sup_{\mathbb{R}^{n}}|Df(x)| \le C_{0} < + \infty$，则存在唯一光滑函数$f_{\infty}: \mathbb{R}^{n}\to \mathbb{R}^k$，满足$f_{\infty}(x) = \lim_{\lambda \to \infty}f_{\lambda}(x)$且对任意实数$r \neq 0$，$f_{\infty}(r x)=r f_{\infty}(x)$，其中$f_{\lambda}(x) = \lambda^{-1}f(\lambda x)$.单调性公式在几何发展方程中占据了重要的地位。本文考虑了完备非紧黎曼流形上的$L^2$ $1-$形式在有界曲率Ricci流下的发展。本文证明了在此流形上具有紧支集的光滑$1$-形式的$L^2$范数沿有界曲率Ricci流是单调非增的。对$L^{\infty}$范数我们也证明了相应的单调性。之后我们利用具有紧支集的$1$-形式的$L^{\infty}$范数研究了$S^1\times \mathbb{R}^{n-1}$上的Ricci流奇异性模型。Li-Yau型梯度估计在热方程、Ricci流和平均曲率流中有很多应用。本文推广了这一估计，将其应用于Ricci曲率有下界的黎曼流形上某些半线性热方程的研究。本文证明了对方程$u_{t} = \Delta u + \lambda u \log u, \; \lambda > 0$的整体正解，对$|\nabla u|^{2}/u^{2} - \alpha u_{t}/u + \alpha \lambda \log u$有Li-Yau型梯度估计，其中$\alpha > 1$。本文还证明了对方程$u_{t} = \Delta u + u^{p}, \; p > 1$在有限时间$T$ blow up的正解，若当$t \to T^{-}$时$u$满足某一增长率上界，则对$|\nabla u|^{2}/u^{2} - \alpha u_{t}/u + \alphau^{p-1}$有Li-Yau型梯度估计，其中$\alpha > 1$。

Variational principles arise naturally in geometry. Geodesics, minimal surfaces, Einstein metrics, or harmonic maps can be characterized as critical points of certain functionals. Since these variational problems rarely can be attacked by standard methods, geometers invented geometric evolution equations to overcome the difficulty. The basic idea of geometric evolution equations is to deform a given geometric structure along a flow to get a limit structure. Typical geometric evolution equations include the Ricci flow, the mean curvature flow, and the harmonic map flow. In studying these geometric flows, methods such as self-similar solutions borrowed from nonlinear parabolic equations are widely used. Today geometric evolution eqations have many applications in geometric and topological problems.For the mean curvature flow, we consider self-similar immersions and show that a graph solution to the mean curvature soliton equation, provided it has bounded derivative, converges smoothly to a function which has some special properties. In fact, let $F(x) = (x,f(x)), x \in \mathbb{R}^{n}, f(x) \in \mathbb{R}^{k}$ be a graph solution to the mean curvature soliton equation $\vec{H}(x) + F^{\bot}(x) = 0$. Assume $\sup_{\mathbb{R}^{n}}|Df(x)| \le C_{0} < + \infty$. Then there exists a unique smooth function $f_{\infty}: \mathbb{R}^{n}\to \mathbb{R}^k$ such that $f_{\infty}(x) = \lim_{\lambda \to \infty}f_{\lambda}(x)$ and $f_{\infty}(r x)=r f_{\infty}(x)$ for any real number $r \neq 0$, where $f_{\lambda}(x) = \lambda^{-1}f(\lambda x)$.Monotonicity formulae play an important r\^{o}le in studying geometric evolution equations. We consider the evolution of $L^2$ one forms under Ricci flow with bounded curvature on a noncompact Riemannian manifold. We show on such a manifold that the $L^2$ norm of a smooth one form with compact support is nonincreasing along the Ricci flow with bounded curvature. We show the $L^{\infty}$ norm also has monotonicity property. Then we use $L^{\infty}$ cohomology of one forms with compact support to study the singularity model for the Ricci flow on $S^1\times \mathbb{R}^{n-1}$.Li-Yau type gradient estimates have many applications in the heat equation, the Ricci flow, and the mean curvature flow. We generalize this estimate to some semilinear heat equations on a Riemannian manifold with a Ricci curvature lower bound. We prove that for a global positive solution to the equation $u_{t} = \Delta u + \lambda u \log u, \; \lambda > 0$, a Li-Yau type gradient estimates holds for $|\nabla u|^{2}/u^{2} - \alpha u_{t}/u + \alpha \lambda \log u$ with some $\alpha > 1$. We also prove that for a positive solution to the equation $u_{t} = \Delta u + u^{p}, \; p > 1$ which blows up at finite time $T$, a Li-Yau type gradient estimates holds for $|\nabla u|^{2}/u^{2} - \alpha u_{t}/u + \alpha u^{p-1}$ with some $\alpha > 1$ under the condition that $u$ has some growth rate bound as $t \to T^{-}$.