arXiv research

The study shows stability of neckpinch singularities in mean curvature flows.

Study neckpinch singularities in Ricci flow with cylindrical symmetry.

The best known finite-time local Ricci flow singularity is the neckpinch, in which a proper subset of the manifold becomes geometrically close to a portion of a shrinking cylinder. In this paper, we prove precise asymptotics for rotationally symmetric Ricci flow neckpinches. We then compare these rigorous results with …

We study surfaces evolving by mean curvature flow (MCF). For an open set of initial data that are $C^3$-close to round, but without assuming rotational symmetry or positive mean curvature, we show that MCF solutions become singular in finite time by forming neckpinches, and we obtain detailed asymptotics of that singul…

Study weak super Ricci flow through neckpinch in metric measure spaces.

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…

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 study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is $C^3$-close at some time to a standard neck will develop a neckpinch singularity in finite time, will become asymptotically rotationally symmetric in a space-time neighborhood of its singul…

In earlier work, we derived formal matched asymptotic profiles for families of Ricci flow solutions developing Type-II degenerate neckpinches. In the present work, we prove that there do exist Ricci flow solutions that develop singularities modeled on each such profile. In particular, we show that for each positive int…

We study "warped Berger" solutions $\big(\mc S^1\times\mc S^3,G(t)\big)$ of Ricci flow: generalized warped products with the metric induced on each fiber $\{s\}\times\mathrm{SU}(2)$ a left-invariant Berger metric. We prove that this structure is preserved by the flow, that these solutions develop finite-time neckpinch …

Numerical simulations show stability of Type-II singularities in noncompact hypersurfaces.

In this paper, we construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches on S^(n+1), without performing an intervening surgery. In the restrictive context of rotational symmetry, this construction gives evidence in favor of Perelman's hope for a "can…

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…

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…

Gu and Zhu have shown that Type-II Ricci flow singularities develop from nongeneric rotationally symmetric Riemannian metrics on $S^m$, for all $m\geq 3$. In this paper, we describe and provide plausibility arguments for a detailed asymptotic profile and rate of curvature blow-up that we predict such solutions exhibit.

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a di…

Study shows different behaviors of noncompact hypersurfaces under mean curvature flow.

We construct examples of spherical space forms $(S^3/Γ,g)$ with positive scalar curvature and containing no stable embedded minimal surfaces, such that the following happens along the Ricci flow starting at $(S^3/Γ,g)$: a stable embedded minimal two-sphere appears and a non-trivial singularity occurs. We also give in d…

In this paper we consider the evolution of sets by a fractional mean curvature flow. Our main result states that for any dimension $n > 2$, there exists an embedded surface in $\mathbb R^n$ evolving by fractional mean curvature flow, which developes a singularity before it can shrink to a point. When $n > 3$ this resul…

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…

Graph Ricci flow reveals hidden hierarchies in stock market correlations.

It is well-known that the Ricci flow of a closed 3-manifold containing an essential minimal 2-sphere will fail to exist after a finite time. Conversely, the Ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. We discuss an intermediate case, that of …

Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further s…

Optimizes bounds for threefold singularity volumes.

Study describes singularities of height functions on specific singular surfaces.

The paper extends affine connection results to singular warped and twisted products.

New singularity concept in GR: volume singularities.

Study describes singularities of distance squared functions on singular surfaces.

In this paper, we define the set of singular grid diagrams $\mathcal{SG}$ which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set of all equivalence relations on $\mathcal{SG}$ which induce the bijection onto e…

Proves positive mass theorem for AF spin manifolds with conical singularities.

The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.

Rectifies flat singular points of area-minimizing currents with singularity degree > 1.

Paper generalizes a theorem for real analytic singularities.

The paper studies singularities of pedal curves of hyperbolic frontals.

The paper extends deformation theory to Calabi-Yau varieties with isolated log canonical singularities.

Maxfaces can have cuspidal edges near certain singularities.

New invariant distinguishes singular knots and links.

Study on singular twisted links and virtual braids, extending knot theory concepts.

Geodesics near singularities either hit or wind around, with winding number dependent on singularity type.

Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer g…

We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian …

Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.

Classifies neighborhoods around specific leaf structures.

We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic $T_7$ singularities. We define discrete symplectic invariants - the Lagrangian tangency orders. We use these invariants to distinguish symplectic singularities of classical $A-D-E$ singularities of planar…

The article investigates conditions for isomorphism of singular tangent bundles.

Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.

This thesis studies moduli spaces of singular connections on 3-manifolds and manifolds with cylindrical ends. A Chern-Simons functional is defined for singular connections on 3-manifolds which are singular along a knot. The critical points of that Chern-Simons functional are flat singular connections. The Hodge-de Rham…

Curvature criteria for A-simple singularities and their parallel curves identified.