The higher dimensional affine Bernstein and Plateau problems

In our previous papers [33, 34], we established the affine Bernstein theorem and proved the interior regularity for the affine Plateau problem in two dimensions. In this paper, we prove the interior regularity for the affine Plateau problem in all dimensions and present a smooth counterexample to the affine Bernstein problem in dimensions ten and higher. We also prove an affine Bernstein theorem in all dimensions under a growth condition.