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Background:

Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of thickness $\epsilon>0.$ Let $\lambda_i(T_\epsilon,g)$ denote the Neumann eigenvalues with respect to the Laplacian $\Delta_g$ and $\lambda_i(\gamma,g_{\text{eucl}})$ denote Neumann eigenvalues on the interval $[0,\pi]$ (here $g_{\text{eucl}}$ denotes the euclidean metric on the interval $[0,\pi]$).

Question:

Is it true that each eigenvalue $\lambda_i(T_\epsilon,g)\to \lambda_i(\gamma,g_{\text{eucl}})$ as $\epsilon\to 0?$

  • Is there an explicit form in which I can write the induced metric on $T_{\epsilon}?$ I read about Fermi normal coordinates but I am not sure how to write the metric in fermi normal coordinates around a geodesic on the sphere.
  • Is it true that each eigenvalue $\lambda_i(T_\epsilon,g)\to \lambda_i(\gamma,g_{\text{eucl}})$ as $\epsilon\to 0?$

Background:

Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of thickness $\epsilon>0.$ Let $\lambda_i(T_\epsilon,g)$ denote the Neumann eigenvalues with respect to the Laplacian $\Delta_g$ and $\lambda_i(\gamma,g_{\text{eucl}})$ denote Neumann eigenvalues on the interval $[0,\pi]$ (here $g_{\text{eucl}}$ denotes the euclidean metric on the interval $[0,\pi]$).

Question:

Is it true that each eigenvalue $\lambda_i(T_\epsilon,g)\to \lambda_i(\gamma,g_{\text{eucl}})$ as $\epsilon\to 0?$

Background:

Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of thickness $\epsilon>0.$ Let $\lambda_i(T_\epsilon,g)$ denote the Neumann eigenvalues with respect to the Laplacian $\Delta_g$ and $\lambda_i(\gamma,g_{\text{eucl}})$ denote Neumann eigenvalues on the interval $[0,\pi]$ (here $g_{\text{eucl}}$ denotes the euclidean metric on the interval $[0,\pi]$).

Question:

  • Is there an explicit form in which I can write the induced metric on $T_{\epsilon}?$ I read about Fermi normal coordinates but I am not sure how to write the metric in fermi normal coordinates around a geodesic on the sphere.
  • Is it true that each eigenvalue $\lambda_i(T_\epsilon,g)\to \lambda_i(\gamma,g_{\text{eucl}})$ as $\epsilon\to 0?$
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Do the eigenvalues Convergence of a thin strip around a semicircle converge to the respectivemetric and eigenvalues ofon a semicircle?tubular neighbourhood

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Student
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Background:

Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of thickness $\epsilon>0.$ Let $\lambda_i(T_\epsilon,g)$ denote the Neumann eigenvalues with respect to the Laplacian $\Delta_g$ and $\lambda_i(\gamma,g_{\text{eucl}})$ denote Neumann eigenvalues on the interval $[0,\pi]$ (here $g_{\text{eucl}}$ denotes the euclidean metric on the interval $[0,\pi]$).

Question:

Is it true that each eigenvalue $\lambda_i(T_\epsilon,g)\to \lambda_i(\gamma,g_{\text{eucl}})$ as $\epsilon\to 0?$

Background:

Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of thickness $\epsilon>0.$ Let $\lambda_i(T_\epsilon,g)$ denote the Neumann eigenvalues with respect to the Laplacian $\Delta_g$ and $\lambda_i(\gamma,g_{\text{eucl}})$ denote Neumann eigenvalues on the interval $[0,\pi]$ (here $g_{\text{eucl}}$ denotes the euclidean metric on the interval $[0,\pi]$).

Question:

Is it true that $\lambda_i(T_\epsilon,g)\to \lambda_i(\gamma,g_{\text{eucl}})$ as $\epsilon\to 0?$

Background:

Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of thickness $\epsilon>0.$ Let $\lambda_i(T_\epsilon,g)$ denote the Neumann eigenvalues with respect to the Laplacian $\Delta_g$ and $\lambda_i(\gamma,g_{\text{eucl}})$ denote Neumann eigenvalues on the interval $[0,\pi]$ (here $g_{\text{eucl}}$ denotes the euclidean metric on the interval $[0,\pi]$).

Question:

Is it true that each eigenvalue $\lambda_i(T_\epsilon,g)\to \lambda_i(\gamma,g_{\text{eucl}})$ as $\epsilon\to 0?$

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