arXiv research

The paper proves rigidity theorems for Type II singularities in Lagrangian flows.

Strict type-II blowup in harmonic map flow is proven to have Hölder continuous body map.

Kähler-Ricci flow shows type II singularity on Fano threefolds.

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

In this paper we investigate the singularities of Lagrangian mean curvature flows in $\mathbf{C}^m$ by means of smooth singularity models. Type I singularities can only occur at certain times determined by invariants in the cohomology of the initial data. In the type II case, these smooth singularity models are asympto…

In this paper we prove the existence of Type II singularities for the Ricci flow on $S^{n+1}$ for all $n\geq 2$.

We study almost-calibrated, $O(n)$-equivariant Lagrangian mean curvature flow in $\mathbb{C}^n$, and prove structural theorems about the Type I and Type II blowups of finite-time singularities. In particular, we prove that any Type I blowup of such a flow must be a special Lagrangian pair of transversely intersecting p…

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…

We study the convergence of complete non-compact conformally flat solutions to the Yamabe flow to Yamabe steady solitons. We also prove the existence of Type II singularities which develop at either a finite time $T$ or as $t \to +\infty$.

In this paper, we prove that any solution of Kähler-Ricci flow on a Fano compactification $M$ of semisimple complex Lie group, is of type II, if $M$ admits no Kähler-Einstein metrics. As an application, we found two Fano compactifications of $\mathrm{SO}_4(\mathbb{C})$ and one Fano compactification of $\mathrm{Sp}_4(\m…

Classifies certain 3D knots with specific properties.

In this paper we prove that a certain class of embedded unknotted curves in $\mathbb{R}^3$ evolving under curve shortening flow do not form singularities Type II before collapsing to a point. Our proof uses tools of the minimal surface theory to study a suitable isoperimetric ratio.

The paper examines translating solitons and their relation to Lagrangian mean curvature flows with zero Maslov class.

Local singularity analysis for Ricci flows with applications to bounded scalar curvature.

We show that a rescale limit at any degenerate singularity of Ricci flow in dimension 3 is a steady gradient soliton. In particular, we give a geometric description of type I and type II singularities.

Constructs finite-time singularities in Lagrangian mean curvature flow with precise dynamics.

We begin by presenting a symmetric version of the circle equivariant T-duality result in a joint work of the second author with Siye Wu, thereby generalising the results there. We then initiate the study of twisted equivariant Courant algebroids and equivariant generalised geometry and apply it to our context. As befor…

J.J.L. Vel$\acute{a}$zquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type $\mathrm{II}$ singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled…

This work concerns with the existence and detailed asymptotic analysis of Type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up a…

We provide a condition for spatial curves which rules out the development of a type I singularity. The condition is that after the last time for which an inflection point develops, if the torsion is ever everywhere non-negative, the curve cannot develop a type I singularity.

On a compact Kähler manifold with semi-ample canonical line bundle and Kodaira dimension one, we observe a relation between the infinite-time singularity type of the Kähler-Ricci flow and the characteristic indexes of singular fibers of the semi-ample fibration.

Curve Shortening Flow preserves circularity for convex projections.

We present an explicit formula for the topology and H-flux of the T-dual of a general type II compactification, significantly generalizing earlier results. Our results apply to T-dualities with respect to any circle action on spacetime. As before, T-duality exchanges type IIA and type IIB string theories. A new consequ…

We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi-Hanson space. In particular, we prove that starting from a class of asymptotically cylindrical $U(2)$-invariant initial metrics on $TS^2$, a Type II singularity modeled on the Eguchi-Hanson space develops in finite time. Furthe…

Study magnetic fields on special Lie groups, proving non-existence of certain types.

Study curve shortening flow in high dimensions with boundary constraints.

Study of curve shortening flow for twisted curves, defining curvature-torsion entropy.

In this paper, we study curvature behavior at the first singular time of solution to the Ricci flow on a smooth, compact n-dimensional Riemannian manifold $M$, $\frac{\partial}{\partial t}g_{ij} = -2R_{ij}$ for $t\in [0,T)$. If the flow has uniformly bounded scalar curvature and develops Type I singularities at $T$, us…

In this paper we mainly study the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a K ähler-Einstein surface. We show the relation between the maximum of the Kähler angle and the maximum of $|H|^2$ on the limit flow.

Study shows instability of specific cone solutions in high-dimensional spaces.

Ricci flows with bounded scalar curvature cannot develop Type I singular points.

Paper solves the minimal generating set problem for singular Reidemeister moves.

The article explores surfaces and soliton equations using spinors.

We study the Ricci flow on $\mathbb{R}^{4}$ starting at an SU(2)-cohomogeneity 1 metric $g_{0}$ whose restriction to any hypersphere is a Berger metric. We prove that if $g_{0}$ has no necks and is bounded by a cylinder, then the solution develops a global Type-II singularity and converges to the Bryant soliton when su…

In this paper, we introduce a monotonicity formula for the mean curvature flow. We also apply this monotonicity formula to study the asymptotic behavior of eternal solutions.

We consider the Kaehler-Ricci flow on complete finite-volume metrics that live on the complement of a divisor in a compact Kaehler manifold X. Assuming certain spatial asymptotics on the initial metric, we compute the singularity time in terms of cohomological data on X. We also give a sufficient condition for the sing…

Study on network flow singularities, focusing on Type-0 singularities.

For any $n$-dimensional smooth manifold $Σ$, we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in $Σ$ are cylindrical (of convex type) if the flow converges to a smooth hypersurface $M_{\infty}$ (maybe empty) at infinity. Previously this was shown (i) for $n\leq 7$,…

We prove that a closed immersed plane curve with total curvature $2πm$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2πm$ whose entropy is less than $m$ …

Study cohomogeneity-one Lagrangian mean curvature flow in complex spaces.

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

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 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…

We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial c…

It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian subspace. In particular this result applies to the Whitney spheres.

We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first…

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 consider the initial value problem $u_t = Δ\log u$, $u(x,0) = u_0(x)\ge 0$ in $\R^2$, corresponding to the Ricci flow, namely conformal evolution of the metric $u (dx_1^2 + dx_2^2)$ by Ricci curvature. It is well known that the maximal (complete) solution $u$ vanishes identically after time $T= \frac 1{4π} \int_{\R^…