2025/04/14 by Agnoletto, Marcos, Hoyos, Julio C. Correa, da Silva, Márcio Fabiano +1
#28A75 #49Q15 #49Q20 #53A10 #58A25 #58C35 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2504.10758
In this paper, we prove Allard's Interior ε-Regularity Theorem for m-dimensional varifolds with generalized mean curvature in Lploc, for p ∈ ℝ such that p>m, in Alexandrov spaces of dimension n with double-sided bounded intrinsic sectional curvature. We first give an intrinsic proof of this theorem in the case of varifolds in Riemannian manifolds of dimension n whose metric tensor is at least of class C2, without using Nash's Isometric Embedding Theorem. This approach provides explicitly computable constants that depend only on n, m, the injectivity radius and bounds on the sectional curvature, which is essential for proving our main theorem, as we establish it through a density argument in the topological space of Riemannian manifolds with positive lower bounds on the injectivity radius and double-sided bounds on sectional curvature, equipped with the C1,α topology, for every α∈ ]0,1[ (in fact, it is enough with the W2,q topology for some suitable q large enough).