Generalized boundary rigidity and minimal surface transform
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914116399005696 |
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| author | Busch, Leonard Liimatainen, Tony Salo, Mikko Tzou, Leo |
| author_facet | Busch, Leonard Liimatainen, Tony Salo, Mikko Tzou, Leo |
| contents | We study a generalized boundary rigidity problem, which investigates whether the areas of embedded minimal surfaces can uniquely determine a Riemannian manifold with boundary. We prove that for a conformal perturbation of an analytic metric in dimension $n+1$ ($n \geq 2$), the metric is determined by these volumes under an ampleness condition. Furthermore, we establish Hölder stability for this determination. This result extends earlier works in dimension $2+1$. Instead of relying on reductions to Calderón type problems and complex geometrical optics solutions, we study the linearized forward operator that gives rise to the minimal surface transform, a generalization of the X-ray/Radon transform. We demonstrate that this transform fits into the framework of double fibration transforms and satisfies the Bolker condition in the sense of Guillemin. Under certain assumptions, including a foliation condition, we prove invertibility of this transform on an analytic manifold as well as recovery of the analytic wave front set. The methods developed in this paper offer new tools for addressing the generalized boundary rigidity problem and expand the scope of applications of double fibration transforms. We anticipate that these techniques will also be applicable to other geometric inverse problems. Beyond mathematics, our results have implications for the AdS/CFT correspondence in physics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23366 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized boundary rigidity and minimal surface transform Busch, Leonard Liimatainen, Tony Salo, Mikko Tzou, Leo Analysis of PDEs Mathematical Physics Differential Geometry 35R30, 53A10, 35J25 We study a generalized boundary rigidity problem, which investigates whether the areas of embedded minimal surfaces can uniquely determine a Riemannian manifold with boundary. We prove that for a conformal perturbation of an analytic metric in dimension $n+1$ ($n \geq 2$), the metric is determined by these volumes under an ampleness condition. Furthermore, we establish Hölder stability for this determination. This result extends earlier works in dimension $2+1$. Instead of relying on reductions to Calderón type problems and complex geometrical optics solutions, we study the linearized forward operator that gives rise to the minimal surface transform, a generalization of the X-ray/Radon transform. We demonstrate that this transform fits into the framework of double fibration transforms and satisfies the Bolker condition in the sense of Guillemin. Under certain assumptions, including a foliation condition, we prove invertibility of this transform on an analytic manifold as well as recovery of the analytic wave front set. The methods developed in this paper offer new tools for addressing the generalized boundary rigidity problem and expand the scope of applications of double fibration transforms. We anticipate that these techniques will also be applicable to other geometric inverse problems. Beyond mathematics, our results have implications for the AdS/CFT correspondence in physics. |
| title | Generalized boundary rigidity and minimal surface transform |
| topic | Analysis of PDEs Mathematical Physics Differential Geometry 35R30, 53A10, 35J25 |
| url | https://arxiv.org/abs/2510.23366 |