2023/10/02 by Costante Bellettini, Bellettini, Costante · 6 citations
Computer Science · Mathematics · #35J15 #35J60 #49Q05 #49Q20 #53A10 #53C42 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2310.01340
openalex publication_date 2023/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give an alternative proof of the Schoen--Simon--Yau curvature estimates and associated Bernstein-type theorems (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 L2 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≥ 2, that may have a singular set Σ of locally finite Hn-2-measure, and that is weakly close to a hyperplane. (In fact, if Hn-2(Σ)=0, 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 L2 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 (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.)