We prove a flatness result for entire nonlocal minimal graphs having some partial derivatives bounded from either above or below. This result generalizes fractional versions of classical theorems due to Bernstein and Moser. Our arguments rely on a general splitting result for blow-downs of nonlocal minimal graphs. Employing similar ideas, we establish that entire nonlocal minimal graphs bounded on one side by a cone are affine. Moreover, we show that entire graphs having constant nonlocal mean curvature are minimal, thus extending a celebrated result of Chern on classical CMC graphs.

中文翻译：

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非局部极小图的 Bernstein-Moser 型结果

我们证明了整个非局部极小图的平坦度结果，这些图具有一些从上方或下方有界的偏导数。这个结果概括了 Bernstein 和 Moser 的经典定理的分数版本。我们的论点依赖于对非局部极小图的破坏的一般分裂结果。使用类似的想法，我们确定在一侧以锥体为界的整个非局部极小图是仿射的。此外，我们表明具有恒定非局部平均曲率的整个图是最小的，从而扩展了陈在经典 CMC 图上的著名结果。