arXiv subjects

Albert Wood

Publications and source records attributed to Albert Wood.

5 recordsLinked to original sources

arXiv2025

Lagrangian Translating Solitons and Special Lagrangians in $\mathbb{C}^m$ with Symmetries

We construct novel families of exact immersed and embedded Lagrangian translating solitons and special Lagrangian submanifolds in $\mathbb{C}^m$ that are invariant under the action of various admissible compact subgroups $G \leq \text{SU}(m-1)$ with cohomogeneity-two. These examples are obtained via an Ansatz generalising a construction of Castro-Lerma in $\mathbb{C}^2$. We give explicit examples of admissible group actions, including a full classification for $G$ simple. We also describe novel Lagrangian translators symmetric with respect to non-compact subgroups of the affine special unitary group $\text{SU}(m)\ltimes \mathbb{C}^m$, including cohomogeneity-one examples.

math.DG↗

arXiv2024

Infinite-Time Singularities of the Lagrangian Mean Curvature Flow

In this paper, we construct solutions of Lagrangian mean curvature flow which exist and are embedded for all time, but form an infinite-time singularity and converge to an immersed special Lagrangian as $t\to\infty$. In particular, the flow decomposes the initial data into a union of special Lagrangians intersecting at one point. This result shows that infinite-time singularities can form in the Thomas--Yau `semi-stable' situation. A precise polynomial blow-up rate of the second fundamental form is also shown. The infinite-time singularity formation is obtained by a perturbation of an approximate family $N^{\varepsilon(t)}$ constructed by gluing in special Lagrangian `Lawlor necks' of size $\varepsilon(t)$, where the dynamics of the neck size $\varepsilon(t)$ are driven by the obstruction for the existence of nearby special Lagrangians to $N^{\varepsilon(t)}$. This is inspired by the work of Brendle and Kapouleas regarding ancient solutions of the Ricci flow.

math.DG↗

arXiv2023

Cohomogeneity-One Lagrangian Mean Curvature Flow

We study mean curvature flow of Lagrangians in $\mathbb{C}^n$ that are cohomogeneity-one with respect to a compact Lie group $G \leq \mathrm{SU}(n)$ acting linearly on $\mathbb{C}^n$. Each such Lagrangian necessarily lies in a level set $μ^{-1}(ξ)$ of the standard moment map $μ\colon \mathbb{C}^n \to \mathfrak{g}^*$, and mean curvature flow preserves this containment. We classify all cohomogeneity-one self-similarly shrinking, expanding and translating solutions to the flow, as well as cohomogeneity-one smooth special Lagrangians lying in $μ^{-1}(0)$. Restricting to the case of almost-calibrated flows in the zero level set $μ^{-1}(0)$, we classify finite-time singularities, explicitly describing the Type I and Type II blowup models. Finally, given any cohomogeneity-one special Lagrangian in $μ^{-1}(0)$, we show it occurs as the Type II blowup model of a Lagrangian MCF singularity. Throughout, we give explicit examples of suitable group actions, including a complete list in the case of $G$ simple. This yields infinitely many new examples of shrinking and expanding solitons for Lagrangian MCF, as well as infinitely many new singularity models.

math.DG↗

arXiv2020

Singularities of Equivariant Lagrangian Mean Curvature Flow

We study almost-calibrated, $O(n)$-equivariant Lagrangian mean curvature flow in $\mathbb{C}^n$, and prove structural theorems about the Type I and Type II blowups of finite-time singularities. In particular, we prove that any Type I blowup of such a flow must be a special Lagrangian pair of transversely intersecting planes, any Type II blowup must be the Lawlor neck with the same asymptotes, and these blowups are independent of the choice of rescaling. We also give a partial classification of when singularities occur in the equivariant case, and examine the intermediate scales between the Type I and Type II models.

math.DG↗

arXiv2019

Lagrangian mean curvature flow with boundary

We introduce Lagrangian mean curvature flow with boundary in Calabi--Yau manifolds by defining a natural mixed Dirichlet-Neumann boundary condition, and prove that under this flow, the Lagrangian condition is preserved. We also study in detail the flow of equivariant Lagrangian discs with boundary on the Lawlor neck and the self-shrinking Clifford torus, and demonstrate long-time existence and convergence of the flow in the first instance and of the rescaled flow in the second.

math.DG↗