The space-time-Grassmann measure of the Brakke flow

2025/12/22 by Yu Liu, Liu, Yu Tong, Myles Workman +1
Mathematics · Economics, Econometrics and Finance · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Stochastic processes and financial applications

paper · doi:10.48550/arxiv.2512.19227

Abstract

For a k-dimensional Brakke flow on an open subset U ⊂ Rⁿ, over an open time interval J, we prove the existence of a canonical space-time-Grassmann measure λ, over J × G_k (U), and give a characterisation of the flow with respect to the space-time weight of this measure. This results in a new definition of the Brakke flow, as that of a space-time measure which satisfies the Brakke inequality in a distributional sense. Each such space-time measure corresponds to a class of equivalent (classical) Brakke flows, thus yielding an equivalence between the classical definitions of the Brakke flow, and this new definition. Moreover, we prove that the mean curvature vector, density, and tangent map along the flow, are all measurable with respect to this space-time weight measure.

Citations

• Approximate mean curvature flows of a general varifold, and their limit spacetime Brakke flow
• A priori bounds for geodesic diameter. Part II. Fine connectedness properties of varifolds
• A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness
• Equality of the usual definitions of Brakke flow

Cited by

• Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

Related

• Equality of the usual definitions of Brakke flow
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• Approximate mean curvature flows of a general varifold, and their limit spacetime Brakke flow
• On the avoidance principle for codimension 1 space-time Brakke flows