arXiv subjects
Publications and source records attributed to Brian White.
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.
This paper proves that, in mean curvature flow of a compact surface in a complete $3$-manifold with Ricci curvature bounded below, the genus of the regular set is a decreasing function of time as long as the only singularities are given by shrinking sphere and shrinking cylinder tangent flows. The paper also proves some local versions of that fact.
Carbon matching aims to improve corporate carbon accounting by tracking emissions rather than energy consumption and production. We present a mathematical derivation of carbon matching using marginal emission rates, where the unit of matching is tons of carbon emitted. We present analysis and open source notebooks showing how marginal emissions can be calculated on simulated electric bus networks. Importantly, we prove mathematically that distinct emissions rates can be assigned to all aspects of the electric grid - including transmission, storage, generation, and consumption - completely allocating electric grid emissions. We show that carbon matching is an accurate carbon accounting framework that can inspire ambitious and impactful action. This research fills a gap by blending carbon accounting expertise and power systems modeling to consider the effectiveness of alternative methodologies for allocating electric system emissions.
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.
Large Language Models (LLMs) have shown strong general capabilities in many applications. However, how to make them reliable tools for some specific tasks such as automated short answer grading (ASAG) remains a challenge. We present SteLLA (Structured Grading System Using LLMs with RAG) in which a) Retrieval Augmented Generation (RAG) approach is used to empower LLMs specifically on the ASAG task by extracting structured information from the highly relevant and reliable external knowledge based on the instructor-provided reference answer and rubric, b) an LLM performs a structured and question-answering-based evaluation of student answers to provide analytical grades and feedback. A real-world dataset that contains students' answers in an exam was collected from a college-level Biology course. Experiments show that our proposed system can achieve substantial agreement with the human grader while providing break-down grades and feedback on all the knowledge points examined in the problem. A qualitative and error analysis of the feedback generated by GPT4 shows that GPT4 is good at capturing facts while may be prone to inferring too much implication from the given text in the grading task which provides insights into the usage of LLMs in the ASAG system.
We complete the classification of semigraphical translators for mean curvature flow in $\mathbb{R}^3$ that was initiated by Hoffman-Mart\'in-White. Specifically, we show that there is no solution to the translator equation on the upper half-plane with alternating positive and negative infinite boundary values, and we prove the uniqueness of pitchfork and helicoid translators. The proofs use Morse-Rad\'o theory for translators and an angular maximum principle.
In the theory of flat chains with coefficients in a normed abelian group, we give a simple necessary and sufficient condition on a group element $g$ in order for the following fundamental regularity principle to hold: if a mass-minimizing chain is, in a ball disjoint from the boundary, sufficiently weakly close to a multiplicity $g$ disk, then, in a smaller ball, it is a $C^{1,\alpha}$ perturbation with multiplicity $g$ of that disk.
For each $g\ge 3$, we prove existence of a compact, connected, smoothly embedded, genus-$g$ surface $M_g$ with the following property: under mean curvature flow, there is exactly one singular point at the first singular time, and the tangent flow at the singularity is given by a shrinker with genus $(g-1)$ and with two ends. Furthermore, we show that if $g$ is sufficiently large, then $M_g$ fattens at the first singular time. As $g\to\infty$, the shrinker converges to a multiplicity $2$ plane.
Consider a pair of smooth, possibly noncompact, properly immersed hypersurfaces moving by mean curvature flow, or, more generally, a pair of weak set flows. We prove that if the ambient space is Euclidean space and if the distance between the two surfaces is initially nonzero, then the surfaces remain disjoint at all subsequent times. We prove the same result when the ambient space is a complete Riemannian manifold of nonzero injectivity radius, provided the curvature tensor (of the ambient space) and all its derivatives are bounded.
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.)
Vegetation structure mapping is critical for understanding the global carbon cycle and monitoring nature-based approaches to climate adaptation and mitigation. Repeated measurements of these data allow for the observation of deforestation or degradation of existing forests, natural forest regeneration, and the implementation of sustainable agricultural practices like agroforestry. Assessments of tree canopy height and crown projected area at a high spatial resolution are also important for monitoring carbon fluxes and assessing tree-based land uses, since forest structures can be highly spatially heterogeneous, especially in agroforestry systems. Very high resolution satellite imagery (less than one meter (1m) Ground Sample Distance) makes it possible to extract information at the tree level while allowing monitoring at a very large scale. This paper presents the first high-resolution canopy height map concurrently produced for multiple sub-national jurisdictions. Specifically, we produce very high resolution canopy height maps for the states of California and Sao Paulo, a significant improvement in resolution over the ten meter (10m) resolution of previous Sentinel / GEDI based worldwide maps of canopy height. The maps are generated by the extraction of features from a self-supervised model trained on Maxar imagery from 2017 to 2020, and the training of a dense prediction decoder against aerial lidar maps. We also introduce a post-processing step using a convolutional network trained on GEDI observations. We evaluate the proposed maps with set-aside validation lidar data as well as by comparing with other remotely sensed maps and field-collected data, and find our model produces an average Mean Absolute Error (MAE) of 2.8 meters and Mean Error (ME) of 0.6 meters.
For a manifold-with-boundary moving by mean curvature flow, the entropy at a later time is bounded by the entropy at an earlier time plus a boundary term. This paper controls the boundary term in a geometrically natural way. In particular, it shows (under mild hypotheses)that the entropy of a compact translator is less than or equal to the entropy of the boundary plus the maximal cone density of the boundary.
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.
For hypersurfaces moving by standard mean curvature flow with boundary, we show that if a tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is an initially smooth surface in 3-space that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a nonempty open set of such initial surfaces. Let k be the largest number with the following property: if M is a minimal surface in 3-space bounded by a smooth simple closed curve of total curvature less than k, then M is a disk. Examples show that $k<4\pi$. In this paper, we use mean curvature flow to show that $k >3\pi$. We get a slightly larger lower bound for orientable surfaces.
In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) can be promoted to smoothness and symmetry in the interior. The key feature, in contrast with the classical formulation of the moving plane principle, is that smoothness is a conclusion rather than an assumption. We implement our moving plane method in the setting of compact varifolds with smooth boundary and in the setting of complete varifolds. A key ingredient is a Hopf lemma for stationary and CMC-varifolds. Our Hopf lemma provides a new tool to establish smoothness of varifolds, and works in arbitrary dimensions and without any stability assumptions. As applications of our new moving plane method, we prove varifold uniqueness results for the catenoid, spherical caps, and Delaunay surfaces that are inspired by classical uniqueness results by Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold version of Alexandrov's Theorem for compact CMC-varifolds in hyperbolic space.
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 prove that every stationary polyhedral varifold minimizes area in the following senses: (1) its area cannot be decreased by a one-to-one Lipschitz ambient deformation that coincides with the identity outside of a compact set, and (2) it is the varifold associated to a mass-minimizing flat chain with coefficients in a certain metric abelian group. NOTE: After this paper was posted, I learned that (1) and (2) were already proved by Choe and Morgan, respectively. Thus this paper is an exposition of their results.
Suppose that $N$ is a smooth manifold with a smooth Riemannian metric $g_0$, and that $Γ$ is a smooth submanifold of $N$. This paper proves that for a generic (in the sense of Baire category) smooth metric $g$ conformal to $g_0$, if $F$ is any simple $g$-minimal immersion of a closed manifold into N, then $F$ is transverse to $Γ$ and $F$ is self-transverse. The theorem remains true with "transverse" and "self-transverse" replaced by "strongly transverse" and "strongly self-transverse". The theorem also holds for hypersurfaces of constant mean curvature or, more generally, of prescribed mean curvature. The paper also proves that for a generic ambient metric, every $2$-dimensional surface (integral current or flat chain mod $2$) without boundary that minimizes area in its homology class has support equal to a smoothly embedded minimal surface.