No abstract

We study generalizations of Reifenberg's Theorem for measures in R n under assumptions on the Jones' β-numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which holds for general measures without density assumptions, give effective measure bounds on µ away from a closed k-rectifiable set with bounded Hausdorff measure. We show examples to see the sharpness of our results. Under further density assumptions one can translate this into a global measure bound and k-rectifiable structure for µ. Applications include quantitative Reifenberg theorems on sets and discrete measures, as well as upper Ahlfor's regularity estimates on measures which satisfy β-number estimates on all scales. CONTENTS NICK EDELEN, AARON NABER, AND DANIELE VALTORTA 6.3. Tilting control in good balls 36 6.4. Properties of the approximating manifolds 40 6.5. Packing estimate 42 6.6. Bounding excess and Finishing the Proof 43 7. Bad tree 44 8. Finishing the proof 47 8.1. Measure and packing decomposition 49 8.2. Structure of C ′ 0 53 9. Packing/Measure estimates and density 54 9.1. Discretizing a measure 55 9.2. Proofs of Theorems 60 10. Rectifiability 62 11. Remaining Theorems 66 11.1. Proof of the statements in the introduction 67 11.2. Applications 68 References 70

Abstract. We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for freeboundary Brakke flows, define a Gaussian monotonicity formula valid at all points, and use this to adapt the local regularity theorem of White [23] to the free-boundary setting. Using Ilmanen's elliptic regularization procedure [10], we prove existence of free-boundary Brakke flows.

In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow [34][35][36] to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish such a bound via the maximum principle for a triple-approximation scheme, which combines ideas from Edelen [8], Haslhofer-Hershkovits [14], and Volkmann [33]. Other important new ingredients are a Bernstein-type theorem and a sheeting theorem for low entropy free boundary flows in a halfslab, which allow us to rule out multiplicity 2 (half-)planes as possible tangent flows and, for mean convex domains, as possible limit flows.

We adapt the method of Simon [25] to prove a C 1,α -regularity theorem for minimal varifolds which resemble a cone C 2 0 over an equiangular geodesic net. For varifold classes admitting a "no-hole" condition on the singular set, we additionally establish C 1,α -regularity near the cone C 2 0 × R m . Combined with work of Allard [3], Simon [25], Taylor [26],, our result implies a C 1,α -structure for the top three strata of minimizing clusters and size-minimizing currents, and a Lipschitz structure on the (n − 3)-stratum.