The amplituhedron BCFW triangulation by Chaim Even-Zohar & Tsviqa Lakrec & Ran J. Tessler ISBN 101007/S00222025013161 instant download

The amplituhedron BCFW triangulation by Chaim Even-Zohar & Tsviqa Lakrec & Ran J. Tessler ISBN 101007/S00222025013161 instant download

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 thescale-invariant L2 norm of the second fundamental form. Further, we obtain a graphdescription, in the Lipschitz multi-valued sense, for any stable minimal immersion ofdimension n ≥ 2, that may have a singular set Σ of locally finite Hn−2-measure, andthat is weakly close to a hyperplane. (In fact, if the Hn−2-measure of the singular setvanishes, the conclusion is strengthened to a union of smooth graphs.) This followsdirectly from an ε-regularity theorem, that assumes smallness of the scale-invariantL2 tilt-excess (verified when the hypersurface is weakly close to a hyperplane). Specialising the multi-valued decomposition to the case of embeddings, we recover theSchoen–Simon theorem (Schoen and Simon 34:741–797, 1981). In both ε-regularitytheorems the relevant quantity (respectively, length of the second fundamental formand tilt function) solves a non-linear PDE on the immersed minimal hypersurface.The proof is carried out intrinsically (without linearising the PDE) by implementingan 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 themethod gives explicit constants that quantify the required smallness.)