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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2501.17471 |
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| _version_ | 1866909469296820224 |
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| author | Sharafutdinov, Vladimir A. |
| author_facet | Sharafutdinov, Vladimir A. |
| contents | For a compact Riemannian manifold $(M,g)$ with boundary $\partial M$, the Diri\-chl\-et-to-Neumann operator $Λ_g:C^\infty(\partial M)\longrightarrow C^\infty(\partial M)$ is defined by $Λ_gf=\left.\frac{\partial u}{\partialν}\right|_{\partial M}$, where $ν$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $Δ_gu=0,\ u|_{\partial M}=f$. Let $g_\partial$ be the Riemannian metric on $\partial M$ induced by $g$. The Calderon problem is posed as follows: To what extent is $(M,g)$ determined by the data $(\partial M,g_\partial,Λ_g)$? We prove the uniqueness theorem: A compact connected two-dimensional Riemannian manifold $(M,g)$ with non-empty boundary is determined by the data $(\partial M,g_\partial,Λ_g)$ uniquely up to conformal equivalence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17471 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two-dimensional Calderon problem and flat metrics Sharafutdinov, Vladimir A. Differential Geometry For a compact Riemannian manifold $(M,g)$ with boundary $\partial M$, the Diri\-chl\-et-to-Neumann operator $Λ_g:C^\infty(\partial M)\longrightarrow C^\infty(\partial M)$ is defined by $Λ_gf=\left.\frac{\partial u}{\partialν}\right|_{\partial M}$, where $ν$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $Δ_gu=0,\ u|_{\partial M}=f$. Let $g_\partial$ be the Riemannian metric on $\partial M$ induced by $g$. The Calderon problem is posed as follows: To what extent is $(M,g)$ determined by the data $(\partial M,g_\partial,Λ_g)$? We prove the uniqueness theorem: A compact connected two-dimensional Riemannian manifold $(M,g)$ with non-empty boundary is determined by the data $(\partial M,g_\partial,Λ_g)$ uniquely up to conformal equivalence. |
| title | Two-dimensional Calderon problem and flat metrics |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2501.17471 |