Ben Lambert

• University of Science and Arts of Oklahoma

Consider a convex cone in three-dimensional Minkowski space which either contains the light cone or is contained in it. This work considers mean curvature flow of a proper spacelike strictly mean convex disc in the cone which is graphical with respect to its rays. Its boundary is required to have constant intersection angle with the boundary of the...

We demonstrate that the property of being Alexandrov immersed is preserved along mean curvature flow. Furthermore, we demonstrate that mean curvature flow techniques for mean convex embedded flows such as noncollapsing and gradient estimates also hold in this setting. We also indicate the necessary modifications to the work of Brendle–Huisken to al...

We obtain estimates on nonlocal quantities appearing in the volume preserving mean curvature flow (VPMCF) in the closed, Euclidean setting. As a result we demonstrate that blowups of finite time type I singularities of VPMCF are ancient solutions to mean curvature flow (MCF), prove that monotonicity methods may always be applied at these finite tim...

Consider a convex cone in three-dimensional Minkowski space which either contains the lightcone or is contained in it. This work considers mean curvature flow of a proper spacelike strictly mean convex disc in the cone which is graphical with respect to its rays. Its boundary is required to have constant intersection angle with the boundary of the...

We study the isometric spacelike embedding problem in scaled de Sitter space, and obtain Weyl-type estimates and the corresponding closedness in the space of embeddings.

We obtain estimates on nonlocal quantities appearing in the Volume Preserving Mean Curvature Flow (VPMCF) in the closed, Euclidean setting. As a result we demonstrate that blowups of finite time singularities of VPMCF are ancient solutions to Mean Curvature Flow (MCF), prove that monotonicity methods may always be applied at finite times and obtain...

Ecology is rich in theories that aim to explain why natural communities have as many species as they do. Neutral theory, for example, supposes that a community’s diversity depends on the rate at which it gains species by immigration or speciation and loses them to ecological drift [1–5]. Classical niche theory, by contrast, supposes that diversity...

We demonstrate that the property of being Alexandrov immersed is preserved along mean curvature flow. Furthermore, we demonstrate that mean curvature flow techniques for mean convex embedded flows such as noncollapsing and gradient estimates also hold in this setting. We also indicate the necessary modifications to the work of Brendle--Huisken to a...

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 t...

We study the isometric spacelike embedding problem in scaled de Sitter space, and obtain Weyl-type estimates and the corresponding closedness in the space of embeddings.

We use a locally constrained mean curvature flow to prove the isoperimetric inequality for spacelike domains in generalized Robertson–Walker spaces satisfying the null convergence condition.

Ancient solutions of Lagrangian mean curvature flow in C^n naturally arise as Type II blow-ups. In this extended note we give structural and classification results for such ancient solutions in terms of their blow-down and, motivated by the Thomas-Yau Conjecture, focus on the almost calibrated case. In particular, we classify Type II blow-ups of al...

We prove the existence of compact spacelike hypersurfaces with prescribed k -curvature in de Sitter space, where the prescription function depends on both space and the tilt function.

Neutral models of evolution assume the absence of natural selection. Formerly confined to ecology and evolutionary biology, neutral models are spreading. In recent years they’ve been applied to explaining the diversity of baby names, scientific citations, cryptocurrencies, pot decorations, literary lexica, tumour variants and much more besides. Her...

Here we investigate the evolutionary dynamics of several kinds of modern cultural artefacts—pop music, novels, the clinical literature and cars—as well as a collection of organic populations. In contrast to the general belief that modern culture evolves very quickly, we show that rates of modern cultural evolution are comparable to those of many an...

Sometimes the normal course of events is disrupted by a particularly swift and profound change. Historians have often referred to such changes as “revolutions”, and, though they have identified many of them, they have rarely supported their claims with statistical evidence. Here, we present a method to identify revolutions based on a measure of mul...

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...

We use a locally constrained mean curvature flow to prove the isoperimetric inequality for spacelike domains in generalized Robertson-Walker spaces satisfying the null convergence condition.

We prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space \(\mathbb {R}^{n,m}\), which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. As an application, we prove long-time existence and convergence of the \({{\,\mathrm{G}\,}}_2...

We prove existence of compact spacelike hypersurfaces with prescribed k - curvature in de Sitter space, where the prescription function depends on both space and the tilt function.

Ancient solutions of Lagrangian mean curvature flow in C^n naturally arise as Type II blow-ups. In this extended note we give structural and classification results for such ancient solutions in terms of their blow-down and, motivated by the Thomas-Yau Conjecture, focus on the almost calibrated case. In particular, we classify Type II blow-ups of al...

We prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space $\mathbb{R}^{n,m}$, which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. As an application, we prove long-time existence and convergence of the $G_2$-Laplacian flow in c...