RCD空间上的极小曲面方程与Bernstein性质
Minimal Surface Equation and Bernstein Property on RCD spaces
AI总结：
本文在RCD(K,N)空间中研究极小曲面方程，证明其解在图上调和，并在K=0时利用Harnack不等式推出Bernstein性质，进而给出乘积空间中极小图的振荡估计与Bernstein定理。
摘要：
我们证明，若$(X,d,m)$是一个RCD(K,N)空间，且$u \in W^{1,1}_{loc}(X)$是极小曲面方程的一个解，则$u$在其图（该图具有自然的度量测度空间结构）上是调和的。当K=0时，这允许我们得到关于$u$的Harnack不等式，进而推出Bernstein性质，即极小曲面方程的任何正解必为常数。作为一个应用，我们得到了乘积空间$M \times \mathbb{R}$中极小图的振荡估计和一个Bernstein定理，其中$M$是一个具有非负Ricci曲率的光滑流形（可能带权且带边）。
英文摘要：
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
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