arXiv research

The paper studies Kähler-Ricci flow on Fano manifolds and finds examples of type II singularities.

Study on Yamabe flow convergence and singularities.

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

Study shows how solutions to Yamabe flow can develop Type II singularities.

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.

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

Study of Lagrangian mean curvature flow with equivariant symmetry.

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…

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

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 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.

Lagrangian spheres develop singularities under flow, matching Whitney spheres.

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

Ricci flow shows Eguchi-Hanson singularities in 4D.

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

Classifies certain 3D knots with specific properties.

T-duality in singular spacetime models with H-flux.

Study Ricci flow on R^4 starting at a specific metric, finding singularities and minimal spheres.

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.

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

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…

Curve Shortening Flow preserves circularity for convex projections.

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

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

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

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.

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…

Study Ricci flow on $\mathbb{R}^{n+1}$, focusing on asymptotic behavior and singularities.

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.

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.

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 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…

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

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^…

Researchers explore mirror symmetries for twisted $G_2$ manifolds.

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.

Study on curve shortening flow with Neumann boundary conditions, showing singularities and limits.

Ricci flow on warped products leads to curvature blow-up.

In each dimension $N\geq 3$ and for each real number $λ\geq 1$, we construct a family of complete rotationally symmetric solutions to Ricci flow on $\mathbb{R}^{N}$ which encounter a global singularity at a finite time $T$. The singularity forms arbitrarily slowly with the curvature blowing up arbitrarily fast at the r…

The study bounds entropy of plane curves and applies to curve shortening flow.

Classifies K-stable Fano varieties and finds new examples.

Paper proves uniqueness of Type II Yamabe metrics on manifolds.

Study curve shortening flow in high dimensions with boundary constraints.

We use the twistorial construction of D-instantons in Calabi-Yau compactifications of type II string theory to compute an explicit expression for the metric on the hypermultiplet moduli space affected by these non-perturbative corrections. In this way we obtain an exact quaternion-Kahler metric which is a non-trivial d…

Compact mean curvature flow solutions with bounded curvature in high dimensions are constructed.