arXiv research

Finite-time blow-up in Yang-Mills flow for small energy initial connections.

Study finite time singularities in Ricci flow with bounded scalar curvature.

In this note we establish that finite-time singularities of the mean curvature flow of compact Riemannian submanifolds are characterised by the blow up of the mean curvature.

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

Study of separatrix configurations in holomorphic flows with real time.

Classifies self-similar solutions for heat equations with positive speed.

We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = Δu + |\nabla u|^2 u \quad \text{in } Ω\times(0,T) \\ u &= \varphi \quad \text{on } \partial Ω\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } Ω, \end{align*} where $Ω$ is a bounded, smooth do…

Researchers found a new type of singularity in surface evolution equations.

We consider the porous medium equation with power-type reaction terms $u^p$ on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If $p>m$, small data give rise to global-in-time solutions while solutions associated to large data blow up in finite t…

In this note we study finite-time singularities in the Chern-Ricci flow. We show that finite-time singularities are characterized by the blow-up of the scalar curvature of the Chern connection.

In this short paper, we show that Kähler-Ricci flows over closed manifolds would have scalar curvature blown-up for finite time singularity. Certain control of the blowing-up is achieved with some mild assumption.

Curve shortening flow converges to a point with entropy bound.

The blow-up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow. In particular, this generalizes a theorem of M.E. Gage and R.S. Hamilton about mean curvature flow in $\mathbb{R}^{2}$.

Study identifies numerical signs of blow-up in hydrodynamic equations.

We study the wave analog of the Liouville equation and the constant mean curvature equations in 2 space dimensions, which are energy critical. We exhibit a blow-up criteria for the former using tools from conformal geometry, and we exhibit finite time blow-up for the latter under suitable assumptions on the initial dat…

Curves become nearly circular over time without initial assumptions.

We prove that the scalar curvature of a homogeneous Ricci flow solution blows up at a forward or backward finite-time singularity.

The paper studies a 1D diffusion equation with nonlinear Robin boundary conditions and finds conditions for global and finite time blow-up or blow-down.

In this paper, we partially settle down the long standing open problem of the finite time blow-up property about the nonlinear Schr$\ddot{o}$dinger equations on some Riemannian manifolds like the standard 2-sphere $S^2$ and the hyperbolic 2-space $H^{2}(-1)$. Using the similar idea, we establish such blow-up results on…

Local existence and uniqueness of Bakry-Émery Ricci flow solutions on finite graphs.

We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3.

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…

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 paper analyzes finite-time singularities in Spin(7)-structure flows using Shi-type estimates.

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = Δu + |\nabla u|^2 u \quad \text{in } Ω\times(0,T) \\ u &= u_b \quad \text{on } \partial Ω\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } Ω, \end{align*} with $u(x,t): \bar Ω\times [0,T) \to …

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

We study the local curvature estimates of long-time solutions to the normalized Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundles. Using these estimates, we prove that on such a manifold, the set of singular fibers of the semi-ample fibration on which the Riemann curvature blows up at…

We show that the norm of the Riemann curvature tensor of any smooth solution to the Ricci flow can be explicitly estimated in terms of its initial values on a given ball, a local uniform bound on the Ricci tensor, and the elapsed time. This provides a new, direct proof of a result of Sesum, which asserts that the curva…

The paper derives a formula for Chow weights of toric blow-ups.

It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time.

Anisotropic curvature flow of networks shows unique solutions and behavior under finite time.

We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve h…

We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with larg…

Study on half spheres solves Nirenberg problem with complex blow-up analysis.

New study confirms some mean curvature flow solutions have bounded mean curvature.

In this note we investigate the behaviour at finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M^m_t\hookrightarrow (N^{m+n}, h). We show that they are characterized by the blow-up of a trace A = H \cdot II of the square of the second fundamental form.

The Yamabe flow can blow up in infinite time with small perturbations.

This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.

We investigate the scalar curvature behavior along the normalized conical Kähler-Ricci flow $ω_t$, which is the conic version of the normalized Kähler-Ricci flow, with finite maximal existence time $T<\infty$. We prove that the scalar curvature of $ω_t$ is bounded from above by $C/(T-t)^2$ under the existence of a con…

The paper studies how submanifolds in Gaussian space behave under mean curvature flow, showing they typically blow up.

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of the nor…

Study on wave-breaking phenomena in solutions of the Camassa-Holm equation.

The paper studies how to transform a sequence of cmc planes into a minimal surface.

We prove that for a mean curvature flow of a compact symplectic surface in a compact Kaehler-Einstein surface, the tangent cone at the first blow-up time consists of a finite union of more than two 2-planes in $R^4$ which are complex in a complex structure on $R^4$.

Proves existence of Yamabe metrics on conical 4-manifolds using min-max method.

Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.

In this paper we study the blow up sequence of mean curvature flow of surfaces in $\mathbb R^3$ with additional forces. We prove that the blow up limit of a mean curvature flow of smoothly embedded surfaces with additional forces with finite entropy is a smoothly embedded self-shrinker.

This paper is concerned with the existence of constant scalar curvature Kaehler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kaehler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already car…