Minimal Surface Equation and Bernstein Property on RCD spaces
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912268872056832 |
|---|---|
| author | Cucinotta, Alessandro |
| author_facet | Cucinotta, Alessandro |
| contents | We show that if $(X,d,m)$ is an RCD(K,N) space and $u \in W^{1,1}_{loc}(X)$ is a solution of the minimal surface equation, then $u$ is harmonic on its graph (which has a natural metric measure space structure). If K=0 this allows to obtain an Harnack inequality for $u$, which in turn implies the Bernstein property, meaning that any positive solution to the minimal surface equation must be constant. As an application, we obtain oscillation estimates and a Bernstein Theorem for minimal graphs in products $M \times \mathbb{R}$, where $M$ is a smooth manifold (possibly weighted and with boundary) with non-negative Ricci curvature |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02406 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal Surface Equation and Bernstein Property on RCD spaces Cucinotta, Alessandro Differential Geometry Analysis of PDEs We show that if $(X,d,m)$ is an RCD(K,N) space and $u \in W^{1,1}_{loc}(X)$ is a solution of the minimal surface equation, then $u$ is harmonic on its graph (which has a natural metric measure space structure). If K=0 this allows to obtain an Harnack inequality for $u$, which in turn implies the Bernstein property, meaning that any positive solution to the minimal surface equation must be constant. As an application, we obtain oscillation estimates and a Bernstein Theorem for minimal graphs in products $M \times \mathbb{R}$, where $M$ is a smooth manifold (possibly weighted and with boundary) with non-negative Ricci curvature |
| title | Minimal Surface Equation and Bernstein Property on RCD spaces |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2403.02406 |