## 标准的椭圆理论:一个能量不等式

Proposition 1. 假设$u$是方程
$$0=\Delta u-\frac{1}{2}x\cdot \nabla u.$$
的光滑解, 则我们有如下的能量不等式:
$$\int_{|x|< r}e^{-\frac{|x|^2}{4}}|\nabla u|^2\rd x\leq\frac{c}{r^2}\int_{r< |x|< 2r}e^{-\frac{|x|^2}{4}}u^2\rd x,\quad\forall r >0.$$
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## 几何分析中的变分问题与方法

这是丁伟岳院士的一个Talk, 原文链接在其主页上有: 几何分析中的变分问题与方法.

1. 历史的回顾:1960年以前 变分法有很长的历史, 如果从欧拉和拉格朗日提出以他们的名字命名的变分方程算起, 至今己有250年的历史. 在开始的时候, 变分法的创立和应用主要是围绕物理学(力学, 光学, 天文学等等 )中的各种变分问题. 比如, 与拉普拉斯方程相联系 的Dirichlet原理就是在研究引力或电场的位势时提出的.
变分法对于几何的应用在早期主要是对曲面上的测地线和欧氏空间中给定边界的极小曲面(Plateau问题 )的研究. 但在很长时期内仅限于一些特殊情形, 没有重要进展.
直到上世纪早期, 为了研究曲面上的测地线的个 数, Morse(20-30年代)和俄国数学家(40年代)分别建立了Morse和 Ljusternik-Schnirelman理论. 其中, Morse理论不仅对变分问题的解的个数估计有许多应用而且在流形的拓扑问题有重要应用. (阅读全文 …)

好的.

## A Proof of Trigonometric Formulas in the Plane of Constant Curvature

Abstract.  In this paper, we solve the geodesics equation in the geodesic polar coordinates of a two dimensional Riemannian manifolds of constant sectional curvature. The relation between edges and angles of geodesic triangle has obtained and as a result the trigonometric formulae has been derived, that is the law of sines, the law of cosines.
1. Induced Connection Along a Mapping Suppose $M$ and $N$ be two smooth manifolds, and $\phi\mathpunct{:}N\to M$ is a smooth mapping. A vector field $X$ along $\phi$ is an assignment which corresponding each $x\in N$ to a vector $X(x)\in T_{\phi(x)}M$. In particular, for any vector field $V$ on $N$, $\phi_\ast V$ may not be a vector field on $M$, but it is a vector field along $\phi$. Clearly, the collection of vector fields along $\phi$ is a vector space, with the natural defined addtion and scalar multiplication.
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## The Peter-Weyl theorem

From Tao’s Blog: The Peter-Weyl theorem, and non-abelian Fourier analysis on compact groups.

Let $G$ be a compact group. (Throughout this post, all topological groups are assumed to be Hausdorff.) Then $G$ has a number of unitary representations, i.e. continuous homomorphisms $\rho: G \rightarrow U(H)$ to the group $U(H)$ of unitary operators on a Hilbert space $H$, equipped with the strong operator topology. In particular, one has the left-regular representation $\tau: G \rightarrow U(L^2(G))$, where we equip $G$ with its normalised Haar measure $\mu$ (and the Borel $\sigma$-algebra) to form the Hilbert space $L^2(G)$, and $\tau$ is the translation operation
$$\tau(g) f(x) := f(g^{-1} x).$$ (阅读全文 …)

## Weak Convergence in Sobolev Spaces

Suppose $\Omega\subset\R^n$ and donote $W^{1,p}:=W^{1,p}(\Omega)$ be the sobolev space for some $1< p< +\infty$. Recall that $f_i\in W^{1,p}$ convergent weakly to $f\in W^{1,p}$, if for any $\phi$ in the dual space of $W^{1,p}$, we have $\inner{f_i,\phi}\to\inner{f,\phi}$, denote as $f_i\weakto f$. This is distinguished by strongly convergence, as we use the dual normal instead of $W^{1,p}$ normal.
Proposition 1.  If $f_i\weakto f$ in $W^{1,p}$, then $f_i\to f$ in $L^p$.
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## Overview of moduli spaces,review of G-bundles and connections

All the contents are from this wikisite, which is aimed to have a E-version lecture notes of the seminar given by Prof. Mrowka. All rights are reserved by the original wikisite, any reprint should be indicate this.
Main Contributor:Christian

These lecture notes are based on notes from the 18.999 geometry seminar class taught by Tomasz Mrowka and from the IAS/Park city Mathematic series book Gauge theory and the topology of four-manifolds 1. Moduli spaces A moduli space can be viewed as a geometric object which classifies the solutions of some problem. For example, there is the moduli space of $n$-tuples of points in $I=[0,1]$,that is, points in $I^{n}$ modulo symmetric transformations:$I^{n}/ Sym_{n}$. Given a complex vector space $V$ of dimension $n$, we can look at the space of endomorphisms of $V$ modulo isomorphisms:
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