Renormalized Area of Hypersurfaces in Hyperbolic Spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918388194869248 |
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| author | Pampano, Alvaro |
| author_facet | Pampano, Alvaro |
| contents | We employ Chen's conformal invariant quantity [8, Theorem 1] in combination with the Chern-Gauss-Bonnet formulas to obtain expressions for the renormalized area of asymptotically minimal hypersurfaces in the $(2n+1)$-dimensional hyperbolic space $\mathbb{H}^{2n+1}$, $n=1,2$. Our results extend Alexakis and Mazzeo's formula for the renormalized area for surfaces in $\mathbb{H}^3$ [1, Proposition 3.1] as well as their relation between the renormalized area of minimal surfaces of $\mathbb{H}^3$ and the Willmore energy of their doubles in $\mathbb{R}^3$ [1, Proposition 8.1] to the non-minimal case and to the higher dimensional case $n=2$.
Moreover, we also generalize our results by considering hypersurfaces in $(2n+1)$-dimensional Poincaré-Einstein spaces and even-dimensional submanifolds of arbitrary codimension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14055 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Renormalized Area of Hypersurfaces in Hyperbolic Spaces Pampano, Alvaro Differential Geometry We employ Chen's conformal invariant quantity [8, Theorem 1] in combination with the Chern-Gauss-Bonnet formulas to obtain expressions for the renormalized area of asymptotically minimal hypersurfaces in the $(2n+1)$-dimensional hyperbolic space $\mathbb{H}^{2n+1}$, $n=1,2$. Our results extend Alexakis and Mazzeo's formula for the renormalized area for surfaces in $\mathbb{H}^3$ [1, Proposition 3.1] as well as their relation between the renormalized area of minimal surfaces of $\mathbb{H}^3$ and the Willmore energy of their doubles in $\mathbb{R}^3$ [1, Proposition 8.1] to the non-minimal case and to the higher dimensional case $n=2$. Moreover, we also generalize our results by considering hypersurfaces in $(2n+1)$-dimensional Poincaré-Einstein spaces and even-dimensional submanifolds of arbitrary codimension. |
| title | Renormalized Area of Hypersurfaces in Hyperbolic Spaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2603.14055 |