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如果你也在线性代数linearalgebra这个学科遇到相关的难题,请随时添加vx号联系我们的代写客服。我们会为你提供专业的服务。 linearalgebra™长期致力于留学生网课服务,涵盖各个网络学科课程:金融学Finance,经济学Economics,数学Mathematics,会计Accounting,文学Literature,艺术Arts等等。除了网课全程托管外,linearalgebra™也可接受单独网课任务。无论遇到了什么网课困难,都能帮你完美解决! 2. A formula for the critical points of harmonic functionsLet \((M,g)\) be a smooth, compact \(n\)-dimensional Riemannian manifold with smooth boundary \(tial M\). Through all the paper we shall denote by \(\nabla,\Delta,D^{2}\) the gradient, Laplacian and Hessian on \(M\) with respect to the given metric \(g\), respectively, and by \(\bar{\nabla}\) the gradient on \(tial M\) with respect to the induced metric.Given a vector field \(V\) on a smooth manifold \(M\) with an isolated zero \(x\), the _index_ of \(V\) at \(x\), or \(\operatorname{ind}_{x}V\), is defined as the local degree of the map \(\Phi:tial U\to\mathbb{S}^{n-1}\) given by \(\Phi:=\frac{V}{|V|}\). Here we have fixed a system of local coordinates around \(x\) and \(U\) is a small coordinate neighborhood of \(x\) which does not contain other zeros of \(V\).The aim of this section is to prove the following theorem.**Theorem**.: _Let \((M,g)\) be a smooth compact Riemannian manifold with smooth boundary \(tial M\). Let \(u\) be a smooth function on \(M\) which satisfies \(\Delta u=0\) in \(M\), and \(u_{|_{tial M}}=h\). Assume moreover that_1. \(u\) _has isolated critical points in_ \(M\)_;_ 2. \(h\) _has isolated critical points on_ \(tial M\)_;_ 3. \(\nabla u\neq 0\) _on_ \(tial M\)_.__Then_\[\sum_{\{x\in M:\nabla u(x)=0\}}\operatorname{ind}_{x}\nabla u=\chi(M)-\sum_{\{ \xi\intial M:\bar{\nabla}h(\xi)=0,tial_{\mathcal{U}}u(\xi)>0\}} \operatorname{ind}_{\xi}\bar{\nabla}h. \tag{5}\]A crucial role in the proof of .1 is played by the Poincare-Hopf Theorem.**Theorem** (Poincare-Hopf).: _Let \(M\) be an orientable \(n\)-dimensional Riemannian manifold, and let \(V\) be a smooth vector field on \(M\) with isolated zeros \(x_{i}\). If \(M\) has a boundary, assume that \(\langle V,\nu\rangle>0\), where \(\nu\) is the outer unit normal to \(tial M\). Then_\[\sum_{i}\operatorname{ind}_{x_{i}}V=\chi(M).\]**Proof of Theorem** The proof is divided into two step. In the first step we consider the manifold \(M\) and take its _double_, which is a closed manifold. Starting from \(u\), we then define an auxiliary function \(\tilde{u}\) on the double manifold and we relate its critical points to the critical points of \(u\) on \(M\) and of \(h\) on \(tial M\). In the second step we apply the Poincare-Hopf Theorem to the function \(\tilde{u}\)._The double manifold and the auxiliary function._ We consider the _double_ of \(M\), \(\mathscr{D}M=M\cup_{\operatorname{l}d}M\), where \(\operatorname{I}d:tial M\totial M\) is the identity map of \(tial M\): it is obtained from \(M\cup M\) by identifying each boundary point in one copy of \(M\) with the same boundary point in the other. We can give a smooth structure on \(\mathscr{D}M\) using a tubular neighborhood of the boundary (which exists since the boundary is at least \(C^{2}\)) to define smooth charts in a neighborhood of the gluing (see e.g., [8, SS9] for more details). We have that \(tial M\) is a smooth submanifold of \(\mathscr{D}M\). For \(\rho>0\) small, we denote by \(\omega_{\rho}\) a tubular neighborhood of \(tial M\) in \(\mathscr{D}M\), namely \(\omega_{\rho}:=\{x\in\mathscr{D}M:\operatorname{dist}(x,tial M)0\) on \((0,\rho)\). This is achieved e.g. for \(\phi(t)=\frac{2t^{2}}{\rho}-\frac{t^{3}}{\rho^{2}}\). When \(t\in(-\rho,0]\), we define \(\tilde{u}\) by even reflection in the direction of \(\nu\). It is easy to check that \(\tilde{u}\) is \(C^{1}\) on \(\mathscr{D}M\).Let us compute now the gradient of \(\tilde{u}\) in \(\omega_{\rho}\). Since \(\tilde{u}\) is even with respect to reflections through \(tial M\), for simplicity we consider the points \(x\in\omega_{\rho}\) of the form \(x=\Phi(\xi,t)\) for \(t\in(0,\rho)\).We have that \(\nabla\tilde{u}=tial_{t}\tilde{u}\,\nu+\bar{\nabla}_{t}\tilde{u}\), where \(\bar{\nabla}_{t}\) denotes the gradient for the induced metric on the parallel to \(tial M\) at distance \(\phi(t)\) from \(tial M\). We check that\[tial_{t}\tilde{u}(\Phi(\xi,t))=\phi^{\prime}(t)tial_{\nu}u(\Phi(\xi, \phi(t)))\,,\quad\bar{\nabla}_{t}\tilde{u}(\Phi(\xi,t))=\bar{\nabla}\tilde{u}( \Phi(\xi,t))+O(\phi(t)). \tag{7}\]We still denote here by \(\nu\) the natural extension of the normal vector field to \(\omega_{\rho}\) (i.e., \(\nu\) is a unit vector field normal to all parallels to \(tial M\) in \(\omega_{\rho}\)).We look now for critical points of \(\tilde{u}\) in \(\omega_{\rho}\). From the assumption \(iii)\), we deduce that we can choose \(\rho\) small enough such that \(\nabla u\neq 0\) in \(\omega_{\rho}\). Hence, to have both components of the gradient equal to zero, we deduce from (7) that necessarily \(t=0\), hence \(\bar{\nabla}\tilde{u}(\Phi(\xi,0))=0\), which is just \(\bar{\nabla}h=0\). Therefore we conclude that the critical points of \(\tilde{u}\) are given by the union of the interior critical points of \(u\) (on each copy of \(M\) in the double manifold) and of the critical points of \(h\) on \(tial M\)._The Poincare-Hopf Theorem._ We apply .2 to the vector field \(\nabla\tilde{u}\) on the closed manifold \(\mathscr{D}M\). Since by \(i)\) and \(ii)\) both \(u\) and \(h\) have isolated critical points (on \(M\) and \(tial M\), respectively), the function \(\tilde{u}\) also has isolated critical points on \(\mathscr{D}M\) and so\[\begin{split}\chi\left(\mathscr{D}M\right)&=\sum_{\{x \in\omega_{\rho}:\nabla\tilde{u}(x)=0\}}\operatorname{ind}_{x}\nabla\tilde{u}+ \sum_{\{x\in\mathscr{D}M\setminus\overline{\omega}_{\rho}:\nabla\tilde{u}(x)=0 \}}\operatorname{ind}_{x}\nabla\tilde{u} &=\sum_{\{x\in\omega_{\rho}:\nabla\tilde{u}(x)=0\}}\operatorname{ind }_{x}\nabla\tilde{u}+2\sum_{\{x\in M:\nabla u(x)=0\}}\operatorname{ind}_{x} \nabla u.\end{split} \tag{8}\]We have seen that the critical points in \(\omega_{\rho}\) (if \(\rho\) is chosen sufficiently small) actually belong to the submanifold \(tial M\) and they are exactly the same critical points of \(h\). However, their index may change, since each can be either a minimum or a maximum in the normal direction to \(tial M\). Let \(\xi\intial M\) be a critical point for \(h\)图片描述

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