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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2101.07475 |
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| _version_ | 1866929391003500544 |
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| author | Dey, Akashdeep |
| author_facet | Dey, Akashdeep |
| contents | The semi-linear, elliptic PDE $AC_{\varepsilon}(u):=-\varepsilon^2Δu+W'(u)=0$ is called the Allen-Cahn equation. In this article we will prove the existence of finite energy solution to the Allen-Cahn equation on certain complete, non-compact manifolds. More precisely, suppose $M^{n+1}$ (with $n+1\geq 3$) is a complete Riemannian manifold of finite volume. Then there exists $\varepsilon_0>0$, depending on the ambient Riemannian metric, such that for all $0<\varepsilon\leq\varepsilon_0$, there exists $\mathfrak{u}_{\varepsilon}:M\rightarrow (-1,1)$ satisfying $AC_{\varepsilon}(\mathfrak{u}_{\varepsilon})=0$ with the energy $E_{\varepsilon}(\mathfrak{u}_{\varepsilon})<\infty$ and the Morse index $\text{Ind}(\mathfrak{u}_{\varepsilon})\leq 1$. Moreover, $0<\liminf_{\varepsilon\rightarrow 0}E_{\varepsilon}(\mathfrak{u}_{\varepsilon})\leq\limsup_{\varepsilon\rightarrow 0}E_{\varepsilon}(\mathfrak{u}_{\varepsilon})<\infty.$ Our result is motivated by the theorem of Chambers-Liokumovich and Song, which says that $M$ contains a complete minimal hypersurface $Σ$ with $0<\mathcal{H}^n(Σ)<\infty.$ This theorem can be recovered from our result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_07475 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Allen-Cahn equation on the complete Riemannian manifolds of finite volume Dey, Akashdeep Differential Geometry The semi-linear, elliptic PDE $AC_{\varepsilon}(u):=-\varepsilon^2Δu+W'(u)=0$ is called the Allen-Cahn equation. In this article we will prove the existence of finite energy solution to the Allen-Cahn equation on certain complete, non-compact manifolds. More precisely, suppose $M^{n+1}$ (with $n+1\geq 3$) is a complete Riemannian manifold of finite volume. Then there exists $\varepsilon_0>0$, depending on the ambient Riemannian metric, such that for all $0<\varepsilon\leq\varepsilon_0$, there exists $\mathfrak{u}_{\varepsilon}:M\rightarrow (-1,1)$ satisfying $AC_{\varepsilon}(\mathfrak{u}_{\varepsilon})=0$ with the energy $E_{\varepsilon}(\mathfrak{u}_{\varepsilon})<\infty$ and the Morse index $\text{Ind}(\mathfrak{u}_{\varepsilon})\leq 1$. Moreover, $0<\liminf_{\varepsilon\rightarrow 0}E_{\varepsilon}(\mathfrak{u}_{\varepsilon})\leq\limsup_{\varepsilon\rightarrow 0}E_{\varepsilon}(\mathfrak{u}_{\varepsilon})<\infty.$ Our result is motivated by the theorem of Chambers-Liokumovich and Song, which says that $M$ contains a complete minimal hypersurface $Σ$ with $0<\mathcal{H}^n(Σ)<\infty.$ This theorem can be recovered from our result. |
| title | The Allen-Cahn equation on the complete Riemannian manifolds of finite volume |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2101.07475 |