arXiv subjects

Myles Workman

Publications and source records attributed to Myles Workman.

5 recordsLinked to original sources

arXiv2026

Parabolic rectifiability of the Brakke flow

We prove that the support of the canonical space-time measure for a Brakke flow is a parabolic $(k+2)$-rectifiable set. As a consequence, we obtain that at almost all points along the flow, with respect to this canonical space-time measure, there exists a unique, static, planar tangent flow, and that various notions of density for the flow agree at these points. Moreover, following on from our previous work `The space-time-Grassmann measure of the Brakke flow', we continue to develop the approach to the Brakke flow as a space-time-Grassmann measure. We prove that the standard notion of convergence for Brakke flows, coming from the compactness theorem of Ilmanen (7.1 of `Elliptic regularization and partial regularity for motion by mean curvature'), is equivalent to the convergence of these space-time-Grassmann Radon measures. This gives an alternate notion of varifold convergence to the one exhibited in 7.1(ii) of `Elliptic regularization and partial regularity for motion by mean curvature'.

math.DG↗

arXiv2025

The space-time-Grassmann measure of the Brakke flow

For a $k$-dimensional Brakke flow on an open subset $U \subset \mathbf{R}^{n}$, over an open time interval $J$, we prove the existence of a canonical space-time-Grassmann measure $\lambda$, over $J \times \mathbf{G}_{k} (U)$, 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.

math.DG↗

arXiv2024

Gradient flow of phase transitions with fixed contact angle

We study the gradient flow of the Allen-Cahn equation with fixed boundary contact angle in Euclidean domains for initial data with bounded energy. Under general assumptions, we establish both interior and boundary convergence properties for the solutions and associated energy measures. Under various boundary non-concentration assumptions, we show that, for almost every time, the associated limiting varifolds satisfy generalised contact angle conditions and have bounded first variation, as well as deducing that the trace of the limit of the solutions coincides with the limit of their traces. Moreover, we derive an Ilmanen type monotonicity formula, for initial data with bounded energy, valid for the associated energy measures up to the boundary.

math.AP↗

arXiv2023

Upper Semicontinuity of Index Plus Nullity for Minimal and CMC Hypersurfaces

We consider a sequence of bubble converging minimal hypersurfaces, or H-CMC hypersurfaces, in compact Riemannian manifolds without boundary, of dimension 4, 5, 6 or 7, and prove upper semicontinuity of index plus nullity, for such a bubble converging sequence. This complements the previously known lower semicontinuity of index obtained by Buzano--Sharp, and Bourni--Sharp--Tinaglia. The strategy of our proof is to analyse a weighted eigenvalue problem along our sequence of hypersurfaces. This strategy is inspired by the recent work of Da Lio--Gianocca--Rivi\`{e}re. A key aspect of our proof is making use of a Lorentz--Sobolev inequality to study the behaviour of these weighted eigenfunctions on the neck regions along the sequence, as well as the index and nullity of our non-compact bubbles.

math.DG↗

arXiv2022

Embeddedness of Min-Max CMC Hypersurfaces in Manifolds with Positive Ricci Curvature

We prove that on a compact Riemannian manifold of dimension $3$ or higher, with positive Ricci curvature, the Allen--Cahn min-max scheme (implemented by the first author and N. Wickramasekera in 2020), with prescribing function taken to be a non-zero constant $\lambda$, produces an embedded hypersurface of constant mean curvature $\lambda$ ($\lambda$-CMC). More precisely, we prove that the interface arising from said min-max contains no even-multiplicity minimal hypersurface and no quasi-embedded points (both of these occurrences are in principle possible in the conclusions of the aforementioned work by the first author and N. Wickramasekera). The immediate geometric corollary is the existence (in ambient manifolds as above) of embedded, closed $\lambda$-CMC hypersurfaces (with Morse index $1$) for any prescribed non-zero constant $\lambda$, with the expected singular set when the ambient dimension is $8$ or higher.

math.DG↗