arXiv subjects
Publications and source records attributed to David Hoffman.
We construct new expanders for mean curvature flow that are smoothly asymptotic to cones arising from certain shrinkers. For each such cone, we prove the existence of expanders of arbitrarily large genus. Thus, for a fixed incoming shrinker, the genus of the outgoing expander can be chosen much larger than the genus before the singularity, contrary to Ilmanen's genus-reduction conjecture.
We prove existence for many examples of shrinkers by producing compact, smoothly embedded surfaces that, under mean curvature flow, develop singularities at which the shrinkers occur as blowups.
We introduce generative interpretation, a new approach to estimating contractual meaning using large language models. As AI triumphalism is the order of the day, we proceed by way of grounded case studies, each illustrating the capabilities of these novel tools in distinct ways. Taking well-known contracts opinions, and sourcing the actual agreements that they adjudicated, we show that AI models can help factfinders ascertain ordinary meaning in context, quantify ambiguity, and fill gaps in parties' agreements. We also illustrate how models can calculate the probative value of individual pieces of extrinsic evidence. After offering best practices for the use of these models given their limitations, we consider their implications for judicial practice and contract theory. Using LLMs permits courts to estimate what the parties intended cheaply and accurately, and as such generative interpretation unsettles the current interpretative stalemate. Their use responds to efficiency-minded textualists and justice-oriented contextualists, who argue about whether parties will prefer cost and certainty or accuracy and fairness. Parties--and courts--would prefer a middle path, in which adjudicators strive to predict what the contract really meant, admitting just enough context to approximate reality while avoiding unguided and biased assimilation of evidence. As generative interpretation offers this possibility, we argue it can become the new workhorse of contractual interpretation.
We construct a family of complete, properly embedded, annular translators $M$ such that $M$ lies in a slab and is invariant under reflections in the vertical coordinate planes. Each translator in the family is asymptotic as $z\to -\infty$ to four vertical planes $\{y= \pm b\}$ and $\{y= \pm B\}$, where $0 0$, there is a translator in the family with inner width $b$ and with necksize $s$. (We also show that there are no translators with inner width $<\pi/2$ having the properties of the examples we construct.)
For a class of functions (called minimal Rad\'o functions) that arise naturally in minimal surface theory, we bound the number of interior critical points (counting multiplicity) in terms of the boundary data and the Euler characteristic of the domain of the function.
We prove existence and uniqueness for a two-parameter family of translators for mean curvature flow. We get additional examples by taking limits at the boundary of the parameter space. Some of the translators resemble well-known minimal surfaces (Scherk's doubly periodic minimal surfaces, helicoids), but others have no minimal surface analogs. A one-parameter subfamily of the examples (the pitchforks) have finite topology and quadratic area growth, and thus might arise as blowups at singularities of initially smooth, closed surfaces flowing by mean curvature flow.
We construct a one-parameter family of singly periodic translating solutions to mean curvature flow that converge as the period tends to $0$ to the union of a grim reaper surface and a plane that bisects it lengthwise. The surfaces are semigraphical: they are properly embedded, and, after removing a discrete collection of vertical lines, they are graphs. We also provide a nearly complete classification of semigraphical translators.
The purpose of these notes is to provide an introduction to those who want to learn more about translating solitons for the mean curvature flow in $\mathbb{R}^3$, particularly those which are complete graphs over domains in $\mathbb{R}^2$. In this paper we describe a full classification of complete translating graphs in $\mathbb{R}^3$.
In this paper we provide a full classification of complete translating graphs in $\mathbf{R}^3$. We also construct two $(n-1)$-parameter families of new examples of translating graphs in $\mathbf{R}^{n+1}$.
We develop a theory of "minimal $\theta$-graphs" and characterize the behavior of limit laminations of such surfaces, including an understanding of their limit leaves and their curvature blow-up sets. We use this to prove that it is possible to realize families of catenoids in euclidean space as limit leaves of sequences of embedded minimal disks, even when there is no curvature blow-up. Our methods work in a more general Riemannian setting, including hyperbolic space. This allows us to establish the existence of a complete, simply connected, minimal surface in hyperbolic space that is not properly embedded.
For every genus $g$, we prove that $S^2 \times R$ contains complete, properly embedded, genus-$g$ minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the $S^2$ tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in $R^3$ that are helicoidal at infinity. We prove that helicoidal surfaces in $R^3$ of every prescribed genus occur as such limits of examples in $S^2\times R$.
We prove that for each positive integer g, there exists a complete minimal surface of genus g that is properly embedded in three-dimensional euclidean space and that is asymptotic to the helicoid.
For every genus g, we prove that S^2 x R contains complete, properly embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the S^2 tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in Euclidean 3-space R^3 that are helicoidal at infinity. In a companion paper, we prove that helicoidal surfaces in R^3 of every prescribed genus occur as such limits of examples in S^2 x R.
For any prescribed closed subset of a line segment in Euclidean 3-space, we construct a sequence of minimal disks that are properly embedded in an open solid cylinder around the segment and that have curvatures blowing up precisely at the points of the closed set.
We prove that if a complete, properly embedded, finite-topology minimal surface in S^2 x R contains a line, then its ends are asymptotic to helicoids, and that if the surface is an annulus, it must be a helicoid.
We generalize the following result of White: Suppose $N$ is a compact, strictly convex domain in $\RR^3$ with smooth boundary. Let $Σ$ be a compact 2-manifold with boundary. Then a generic smooth curve $Γ\cong \partialΣ$ in $\partial N$ bounds an odd or even number of embedded minimal surfaces diffeomorphic to $Σ$ according to whether $Σ$ is or is not a union of disks. First, we prove that the parity theorem holds for any compact riemannian 3-manifold $N$ such that $N$ is strictly mean convex, $N$ is homeomorphic to a ball, $\partial N$ is smooth, and $N$ contains no closed minimal surfaces. We then further relax the hypotheses by allowing $N$ to be weakly mean convex and to have piecewise smooth boundary. We extend the parity theorem yet further by showing that, under an additional hypothesis, it remains true for minimal surfaces with prescribed symmetries. The parity theorems are used in an essential way to prove the existence of embedded genus-$g$ helicoids in $\SS^2\times \RR$. We give a very brief outline of this application. (The full argument will appear elsewhere.)
We prove: a properly embedded, genus-one minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into two connected components that lie on either side of the helicoid. We prove an analogous result for periodic helicoid-like surfaces. We also give a simple condition guaranteeing that an immersed minimal surface with finite genus and bounded curvature is asymptotic to a helicoid at infinity.
We prove by variational means the existence of a complete, properly embedded, genus-one minimal surface in R^3 that is asymptotic to a helicoid at infinity. We also prove existence of surfaces that are asymptotic to a helicoid away from the helicoid's axis, but that have infinitely many handles arranged periodically along the axis. Finally, we prove some new properties of such helicoid-like surfaces.