2025/03/24 by Wang, Guofang, Zhang, Xuwen
#49Q15 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.19052
In this paper we introduce and study a new class of varifolds in Rn+1 of arbitrary dimensions and co-dimensions, which satisfy a Neumann-type boundary condition characterizing capillarity. The key idea is to introduce a Radon measure on a subspace of the trivial Grassmannian bundle over the supporting hypersurface as a generalized boundary with prescribed angle, which plays a role as a measure-theoretic capillary boundary. We show several structural properties, monotonicity inequality, boundary rectifiability, classification of tangent cones, and integral compactness for such varifolds under reasonable conditions. This Neumann-type boundary condition fits very well in the context of curvature varifold and Brakke flow, which we also discuss.