arXiv research

Modified curve shortening flow constructs $λ$-Angenent curve.

New comparison theorems for rotationally symmetric self-shrinkers help in proving the uniqueness of the Angenent torus.

We show that the only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals, completing the classification initiated by Daskalopoulos, Hamilton and Sesum and X.-J. Wang

We prove that the only self-similar surfaces of Euclidean 3-space which are foliated by circles are the self-similar surfaces of revolution discovered by S. Angenent and that the only ruled, self-similar surfaces are the cylinders over planar self-similar curves.

We consider an embedded convex ancient solution $Γ_t$ to the curve shortening flow in $\mathbb{R}^2$. We prove that there are only two possibilities: the family $Γ_t$ is either the family of contracting circles, which is a type I ancient solution, or the family of evolving Angenent ovals, which correspond to a type II …

New minimal tori found in curved spaces.

We construct a class of compact ancient solutions to the mean curvature flow in Euclidean space with high codimension. In particular, we construct higher codimensional ancient curve shortening flows. Moreover, we characterize the asymptotic behavior of these solutions. Add on remark: the construction in this paper has …

To study the singularities that appear in mean curvature flow, one must understand self-shrinkers, surfaces that shrink by dilations under mean curvature flow. The simplest examples of self-shrinkers are spheres and cylinders. In 1989, Angenent constructed the first nontrivial example of a self-shrinker, a torus. A key…

Researchers develop a numerical method to compute the index of self-shrinkers, finding it to be 5 for the Angenent torus.

The curve shortening flow transforms figure-eight curves into bowties.

We extend two celebrated theorems on closed geodesics of Riemannian 2-spheres to the larger class of reversible Finsler 2-spheres: Lusternik-Schnirelmann's theorem asserting the existence of three simple closed geodesics, and Bangert-Franks-Hingston's theorem asserting the existence of infinitely many closed geodesics.…

We prove a comparison theorem for the isoperimetric profiles of simple closed curves evolving by the normalized curve shortening flow: If the isoperimetric profile of the region enclosed by the initial curve is greater than that of some `model' convex region with exactly four vertices and with reflection symmetry in bo…

Ancient curve flows classified into specific types.

Numerically estimates Colding-Minicozzi entropies of self-shrinkers.

Variational approximations for curve flows on Riemannian manifolds.

In this paper, we introduce a concept of B-minimal sub-manifolds and discuss the stability of such a sub-manifold in a Riemannian manifold $(M,g)$. Assume $B(x)$ is a smooth function on $M$. By definition, we call a sub-manifold $Σ$ {\em B-minimal} in $(M,g)$ if the product sub-manifold $Σ\times S^1$ is a {\em minimal}…

In this article, we consider the Angenent-Caputo-Knopf's Ricci Flow through neckpinch singularities. We will explain how one can see the A-C-K's Ricci flow through a neckpinch singularity as a flow of integral current spaces. We then prove the continuity of this weak flow with respect to the Sormani-Wenger Intrinsic Fl…

We construct closed, embedded, ancient mean curvature flows in each dimension $n\ge 2$ with the topology of $S^1 \times S^{n-1}$. These examples are not mean convex and not solitons. They are constructed by analyzing perturbations of the self-shrinking doughnuts constructed by Drugan and Nguyen (or, alternatively, Ange…

In previous work, Angenent, Isenberg, and Knopf created type-II Ricci flow neckpinch singularities. In this paper we construct solutions to Ricci flow whose initial data is the singular metric resulting from these singularities. We show in particular that the curvature decreases at the same rate at which it blew up. Th…

For each $n\geq 2$ we construct a new closed embedded mean curvature self-shrinking hypersurface in $\mathbb{R}^{2n}$. These self-shrinkers are diffeomorphic to $S^{n-1}\times S^{n-1}\times S^1$ and are $SO(n)\times SO(n)$ invariant. The method is inspired by constructions of Hsiang and these surfaces generalize self-s…

We study the compact noncollapsed ancient convex solutions to Mean Curvature Flow in $\mathbb{R}^{n+1}$ with $O(1)\times O(n)$ symmetry. We show they all have unique asymptotics as $t\to -\infty$ and we give precise asymptotic description of these solutions. In particular, solutions constructed by White, and Haslhofer …

In this work, we are going to find sufficient conditions on the initial triaxial Bianchi IX metric on some 4-dimensional manifolds foliated by homogeneous S3 for a Type I singularity to occur when it is flowed under the Ricci flow. This work generalises the study on rotationally symmetric manifolds done by Angenent and…

In this paper we apply techniques from optimal transport to study the neckpinch examples of Angenent-Knopf which arise through the Ricci flow on $\mathbb{S}^{n+1}$. In particular, we recover their proof of 'single-point pinching' along the flow. Using the methods of optimal transportation, we are able to remove the ass…

Study neckpinch singularities in Ricci flow with cylindrical symmetry.

Under mean curvature flow, a closed, embedded hypersurface $M(t)$ becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time $T$ and the limit set "$M(T)$", with respect to initial data. We employ an Angenent-like neck-pinching argument to…

We construct many closed, embedded mean curvature self-shrinking surfaces $Σ_g^2\subseteq\mathbb{R}^3$ of high genus $g=2k$, $k\in \mathbb{N}$. Each of these shrinking solitons has isometry group equal to the dihedral group on $2g$ elements, and comes from the "gluing", i.e. desingularizing of the singular union, of th…

Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are dimensionally agnostic. These SRF equations were tested numerically and…

Proves mean curvature flow from conical singularities to shrinkers.

The paper studies nodal sets of solutions to parabolic equations, proving finiteness and monotonicity properties.

The paper confirms conjectures about ancient ovals and provides counterexamples.

New self-shrinkers found in higher dimensions.

Constructs self-shrinkers with unique asymptotic behavior.

Enhanced estimates for ancient ovals and translators in 3D and 4D.

Gradient estimate for linearized translator equation in R^4.

We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus $g$, and are non-compact with one end. Each has $4g+4$ symmetries and comes from desingularizing the inters…

We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. SRF was recently introduced as an unstructured mesh formulation of Hamilton's Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this …

The study explores Bertrand and Mannheim curves in 4D Euclidean space for framed curves.

The study examines Bertrand Legendre curves in the unit tangent bundle over Euclidean plane.

Method for generating new curves from plane curves on cylinders.

In this study, we introduce a new approach to curve pairs by using integral curves. We consider the direction curve and donor curve to study curve couples such as involute-evolute curves, Mannheim partner curves and Bertrand partner curves. We obtain new methods to construct partner curves of a unit speed curve and giv…

The paper characterizes curves in pseudo-Galilean 4-space.

In this paper, we introduce a new approach to non-lightlike curve pairs by using integral curves in Minkowski 3-space. We consider direction curve and donor curve to study non-lightlike curve couples such as involute-evolute curves, Mannheim partner curves and Bertrand partner curves. We obtain new methods to construct…

The paper explores Bertrand and framed curves in 3D space.

The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.

Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…

Flow deforms locally convex curves to curves of constant k-order width.

Study on CR curves in 3-sphere, focusing on critical curves integration and existence.

Study rectifying curves in 3D multiplicative Euclidean space.