The Calderón problem on Riemannian surfaces and of minimal surfaces
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917703672922112 |
|---|---|
| author | Cârstea, Cătălin I. Liimatainen, Tony Tzou, Leo |
| author_facet | Cârstea, Cătălin I. Liimatainen, Tony Tzou, Leo |
| contents | In this paper we prove two results. The first shows that the Dirichlet-Neumann map of the operator $Δ_g+q$ on a Riemannian surface can determine its topological, differential, and metric structure. Earlier work of this type assumes a priori that the surface is a planar domain [36] or that the geometry is a priori known [29]. We will then apply this result to study a geometric inverse problem for determining minimal surfaces embedded in $3$-dimensional Riemannian manifolds. In particular we will show that knowledge of the volumes of embedded minimal surfaces determine not only their topological and differential structure but also their Riemannian structure as an embedded hypersurface. Such geometric inverse problems are partially inspired by the physical models proposed by the AdS/CFT correspondence.
The crucial ingredient in removing the planar domain assumption is the determination of the boundary trace of holomorphic functions from knowledge of the Dirichlet-Neumann map of $Δ_g +q$. This requires a new type of argument involving Carleman estimates and construction of CGO whose phase functions are not Morse as in the case of [29]. We anticipate that these techniques could be of use for studying other inverse problems in geometry and PDE. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16944 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Calderón problem on Riemannian surfaces and of minimal surfaces Cârstea, Cătălin I. Liimatainen, Tony Tzou, Leo Analysis of PDEs Mathematical Physics Differential Geometry 35R30, 53A10, 35J25 In this paper we prove two results. The first shows that the Dirichlet-Neumann map of the operator $Δ_g+q$ on a Riemannian surface can determine its topological, differential, and metric structure. Earlier work of this type assumes a priori that the surface is a planar domain [36] or that the geometry is a priori known [29]. We will then apply this result to study a geometric inverse problem for determining minimal surfaces embedded in $3$-dimensional Riemannian manifolds. In particular we will show that knowledge of the volumes of embedded minimal surfaces determine not only their topological and differential structure but also their Riemannian structure as an embedded hypersurface. Such geometric inverse problems are partially inspired by the physical models proposed by the AdS/CFT correspondence. The crucial ingredient in removing the planar domain assumption is the determination of the boundary trace of holomorphic functions from knowledge of the Dirichlet-Neumann map of $Δ_g +q$. This requires a new type of argument involving Carleman estimates and construction of CGO whose phase functions are not Morse as in the case of [29]. We anticipate that these techniques could be of use for studying other inverse problems in geometry and PDE. |
| title | The Calderón problem on Riemannian surfaces and of minimal surfaces |
| topic | Analysis of PDEs Mathematical Physics Differential Geometry 35R30, 53A10, 35J25 |
| url | https://arxiv.org/abs/2406.16944 |