Variations of Renormalized Volume for Minimal Submanifolds of Poincare-Einstein Manifolds
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2021
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916493245022208 |
|---|---|
| author | Marx-Kuo, Jared |
| author_facet | Marx-Kuo, Jared |
| contents | We investigate the asymptotic expansion and the renormalized volume of minimal submanifolds, $Y^m$ of arbitrary codimension in Poincare-Einstein manifolds, $M^{n+1}$. In particular, we derive formulae for the first and second variations of renormalized volume for $Y^m \subseteq M^{n+1}$ when $m < n + 1$. We apply our formulae to the codimension $1$ and the $M = \mathbb{H}^{n+1}$ case. Furthermore, we prove the existence of an asymptotic description of our minimal submanifold, $Y$, over the boundary cylinder $\partial Y \times \mathbb{R}^+$, and we further derive an $L^2$-inner-product relationship between $u_2$ and $u_{m+1}$ when $M = \mathbb{H}^{n+1}$. Our results apply to a slightly more general class of manifolds, which are conformally compact with a metric that has an even expansion up to high order near the boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_04294 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Variations of Renormalized Volume for Minimal Submanifolds of Poincare-Einstein Manifolds Marx-Kuo, Jared Differential Geometry We investigate the asymptotic expansion and the renormalized volume of minimal submanifolds, $Y^m$ of arbitrary codimension in Poincare-Einstein manifolds, $M^{n+1}$. In particular, we derive formulae for the first and second variations of renormalized volume for $Y^m \subseteq M^{n+1}$ when $m < n + 1$. We apply our formulae to the codimension $1$ and the $M = \mathbb{H}^{n+1}$ case. Furthermore, we prove the existence of an asymptotic description of our minimal submanifold, $Y$, over the boundary cylinder $\partial Y \times \mathbb{R}^+$, and we further derive an $L^2$-inner-product relationship between $u_2$ and $u_{m+1}$ when $M = \mathbb{H}^{n+1}$. Our results apply to a slightly more general class of manifolds, which are conformally compact with a metric that has an even expansion up to high order near the boundary. |
| title | Variations of Renormalized Volume for Minimal Submanifolds of Poincare-Einstein Manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2111.04294 |