Extensions of Schoen–Simon–Yau and Schoen–Simon theorems via iteration à la De Giorgi

Abstract We give an alternative proof of the Schoen–Simon–Yau curvature estimates and associated Bernstein-type theorems (Schoen et al. in Acta Math. 134:275–288, 1975), and extend the original result by including the case of 6-dimensional (stable minimal) immersions. The key step is an ε-regularity theorem, that assumes smallness of the scale-invariant $L^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> norm of the second fundamental form. Further, we obtain a graph description, in the Lipschitz multi-valued sense, for any stable minimal immersion of dimension $n\geq 2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:math> , that may have a singular set $\Sigma $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> of locally finite $\mathcal{H}^{n-2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> -measure, and that is weakly close to a hyperplane. (In fact, if the $\mathcal{H}^{n-2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> -measure of the singular set vanishes, the conclusion is strengthened to a union of smooth graphs.) This follows directly from an ε-regularity theorem, that assumes smallness of the scale-invariant $L^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> tilt-excess (verified when the hypersurface is weakly close to a hyperplane). Specialising the multi-valued decomposition to the case of embeddings, we recover the Schoen–Simon theorem (Schoen and Simon 34:741–797, 1981). In both ε-regularity theorems the relevant quantity (respectively, length of the second fundamental form and tilt function) solves a non-linear PDE on the immersed minimal hypersurface. The proof is carried out intrinsically (without linearising the PDE) by implementing an iteration method à la De Giorgi (from the linear De Giorgi–Nash–Moser theory). Stability implies estimates (intrinsic weak Caccioppoli inequalities) that make the iteration effective despite the non-linear framework. (In both ε-regularity theorems the method gives explicit constants that quantify the required smallness.)