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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! 3. A generic formula for the critical points of Steklov eigenfunctionsIn this section we apply .1 to the case of Steklov eigenfunctions and prove that for generic metrics on a given manifold, formula (2) holds, along with its consequences (formulas (3) and (4)).**Theorem**.: _Let \(M\) be a smooth compact \(n\)-dimensional manifold with smooth boundary \(tial M\) and let \(G\) be the set of smooth Riemannian metrics on \(M\). Then for a residual (hence dense) subset of \(G\) of smooth metrics we have that non-constant Steklov eigenfunctions \(u\) satisfy_\[\sum_{\{x\in M:\nabla u(x)=0\}}\mathrm{ind}_{x}\nabla u=\chi(M)-\sum_{\{\xi\in tial M:\bar{\nabla}u(\xi)=0,u(\xi)>0\}}\mathrm{ind}_{\xi}\bar{\nabla}u. \tag{13}\]_Moreover, if_\[\mathscr{P}:=\{\xi\intial M:u(\xi)>0\}\]_we have_\[\sum_{\{x\in M:\nabla u(x)=0\}}\mathrm{ind}_{x}\nabla u=\chi(M)-\chi(\mathscr{ P}). \tag{14}\]Before proving .1, we discuss a few consequences.Assume that \(\mathscr{P}\) is the union of \(\ell\) disjoint connected components \(\mathscr{P}_{i}\). Then by (14) we get\[\sum_{\{x\in M:\nabla u(x)=0\}}\mathrm{ind}_{x}\nabla u=\chi(M)-\sum_{i=1}^{ \ell}\chi(\mathscr{P}_{i}). \tag{15}\]Since \(u\) and its restriction on \(tial M\) are generically Morse (see [13] and Theorem A.1), we have from (13) that\[\sum_{\{x\in M:\nabla u(x)=0\}}(-1)^{\mathfrak{m}(x)}=\chi(M)-\sum_{\{\xi\in tial M:\bar{\nabla}u(\xi)=0,\,u(\xi)>0\}}(-1)^{\mathfrak{m}(\xi)}.\]If \(n=2\) (i.e., for Riemannian surfaces), each \(\mathscr{P}_{i}\) is either an interval (hence \(\chi(\mathscr{P}_{i})=1\)) or a simple closed curve (hence \(\chi(\mathscr{P}_{i})=0\)). Therefore we deduce that, generically\[\sum_{\{x\in M:\nabla u(x)=0\}}\mathrm{ind}_{x}\nabla u=\chi(M)-\sum_{j=1}^{L} \ell_{j}, \tag{16}\]where \(L\) is the number of connected components \(\Gamma_{j}\) of \(tial M\) where \(u\) changes sign and \(\ell_{j}\) is the number of times that \(u\) changes sign on \(\Gamma_{j}\). Moreover, we also have\[\sharp\{\text{critical points of }u\}= \quad\sharp\{\text{maxima of }u\text{ on }tial M\text{ with }u(\xi)>0\}\] \[-\sharp\{\text{minima of }u\text{ on }tial M\text{ with }u(\xi)>0\}\] \[-\chi(M)\] \[= \sum_{j=1}^{L}\ell_{j}-\chi(M).\]**Proof of Theorem**.: We apply .1 to a non-constant Steklov eigenfunction \(u\). To do so, we verify that hypotheses \(i)\), \(ii)\) and \(iii)\) of .1 are satisfied by \(u\) and \(h=u_{|_{tial M}}\) for a generic metric \(g\) on \(M\). In [13] it is proved that, generically, restrictions of Steklov eigenfunctions on \(tial M\) are Morse functions, hence \(ii)\) is verified. In [13] it is also proved that generically there are no singular zeros for \(u_{|_{tial\Omega}}\), which means that if \(u(\xi)=0\) for some \(\xi\intial M\), then \(\bar{\nabla}u(\xi)\neq 0\). Since \(u(\xi)=0\) if and only if \(tial_{\nu}(\xi)=0\) by the Steklov condition, we deduce that if \(tial_{\nu}u(\xi)=0\), then \(\bar{\nabla}u(\xi)\neq 0\). Hence also \(iii)\) is generically satisfied. On the other hand, in [13] generic properties of Steklov eigenfunctions in \(M\) are not discussed. We prove in Appendix A (see Theorem A.1) that \(u\) is generically a Morse function in \(M\), which implies \(i)\). Therefore .1 applies, and since \(tial_{\nu}u(\xi)>0\) for \(\xi\intial\Omega\) if and only if \(u(\xi)>0\) by the Steklov condition, we deduce (13) from (5).In order to prove (14), we use Poincare-Hopf Theorem on \(\mathscr{P}\), which is the union of \(n-1\)-dimensional connected manifolds \(\mathscr{P}_{i}\), possibly with boundary. If \(\mathscr{P}_{i}\) has boundary (which is a zero level set of \(u_{|_{tial M}}\)), from \(iii)\) we deduce that generically \(\bar{\nabla}u\neq 0\) on \(tial\mathscr{P}_{i}\), which implies that \(\langle\bar{\nabla}u,N\rangle>0\), where \(N\) is the outer unit normal to \(\mathscr{P}_{i}\). Hence we can apply .2 and get that\[\sum_{\{\xi\intial M:\bar{\nabla}u(\xi)=0,u(\xi)>0\}}\operatorname{ind}_{ \xi}\bar{\nabla}u=\sum_{i=1}^{\ell}\chi(\mathscr{P}_{i})=\chi(\mathscr{P}). \tag{17}\]Now (14) follows from (13) and (17). 图片描述

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