arXiv research

Study shows continuity and geometric regularity of Kähler-Ricci flow blow-up limits.

The paper examines the blow-up of Ricci curvatures in conformal metrics.

Corrected Monti's blow-up analysis for H-minimizing sets in Heisenberg group.

We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber and Enders-Müller-Topping that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result …

The paper analyzes high-dimensional sphere solutions to the Nirenberg problem with residual mass.

In this paper we prove a conjecture by Feldman-Ilmanen-Knopf in \cite{FIK} that the gradient shrinking soliton metric they constructed on the tautological line bundle over $\CP^1$ is the uniform limit of blow-ups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups…

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

Study uses blow-up method to analyze foliations in Riemannian geometry.

The paper shows translating solitons in $\mathbb{R}^4$ have $SO(2)$ symmetry.

We develop a theory of "minimal $θ$-graphs" and characterize the behavior of limit laminations of such surfaces, including an understanding of their limit leaves and their curvature blow-up sets. We use this to prove that it is possible to realize families of catenoids in euclidean space as limit leaves of sequences of…

Study on constant mean curvature 1-immersions into hyperbolic 3-manifolds.

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.

Ancient solutions arise in the study of parabolic blow-ups. If we can categorize ancient solutions, we can better understand blow-up limits. Based on an argument of Giga and Kohn, we give a Liouville-type theorem restricting ancient, type-I, non-collapsing two- dimensional mean curvature flows to either spheres or cyli…

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

Study on Yang-Mills fields blow-up in 4D, proving certain configurations impossible.

Curve shortening flow converges to a point with entropy bound.

We construct a sequence of compact embedded minimal disks in the unit ball in Euclidean 3-space whose boundaries are in the boundary of the ball and where the curvatures blow up at every point of a line segment of the vertical axis, extending from the origin. We further study the transversal structure of the minimal li…

We consider any pseudo holomorphic integral 2-cycle in an arbitrary almost complex manifold and perform a blow up analysis at an arbitrary point. Building upon a pseudo algebraic blow up (previously introduced by the author) we prove a geometric rate of decay for the mass ratio towards the limiting density, with an exp…

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

We extend a model of positive feedback and contagion in large mean-field systems, by introducing a common source of noise driven by Brownian motion. Although the driving dynamics are continuous, the positive feedback effect can lead to `blow-up' phenomena whereby solutions develop jump-discontinuities. Our main results…

Study regularity of Schrödinger eigenfunctions with Coulomb-type potentials.

We improve Gross-Wilson's local estimates to global ones. As an application, we study the blow-up limits of the degenerating Calabi-Yau metrics on singular fibers.

We construct a sequence of embedded minimal disks in a ball where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper.

Study of singularity formation in dHYM flow on a blown-up CP^3.

We define Type I singularities for the mean curvature flow associated to a density $ψ$ ($ψ$MCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the $ψ$-curvature due to the density. We describe a family of curves whose e…

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

We consider the Ricci flow on $\mathbb{CP}^n$ blown-up at one point starting with any $U(n)$-invariant Kähler metric. It is known that the Kähler-Ricci flow must develop Type I singularities. We show that if the total volume does not go to zero at the singular time, then any Type I parabolic blow-up limit of the Ricci …

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

The paper solves a Nirenberg problem on half spheres, finding multiple blow-ups.

Study on hyperbolic elastic flow, proving convergence and quantifying singularities.

Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathe'odory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure …

Proves a general connected sum formula for families Seiberg-Witten invariants.

We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient di…

Given a compact closed subset $M$ of a line segment in $\mathbb{R}^3$, we construct a sequence of minimal surfaces $Σ_k$ embedded in a neighborhood $C$ of the line segment that converge smoothly to a limit lamination of $C$ away from $M$. Moreover, the curvature of this sequence blows up precisely on $M$, and the limit…

The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.

Formula derived for holomorphic Poisson blow-ups.

Wave maps can have multiple bubbling solutions at blow-up points.

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

Formula derived for Bott-Chern classes in complex blow-ups.

Explains blow-ups for Lie groupoids and algebroids, comparing different methods.

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

I survey some of the developments in the theory of Ricci flow and its applications from the past decade. I focus mainly on the understanding of Ricci flows that are permitted to have unbounded curvature in the sense that the curvature can blow up as we wander off to spatial infinity and/or as we decrease time to some s…

Study on curvature blow-up rates in black hole interiors from gravitational collapse.

We show that the blow-up of a generalized Kahler 4-manifold in a nondegenerate complex point admits a generalized Kahler metric. As with the blow-up of complex surfaces, this metric may be chosen to coincide with the original outside a tubular neighbourhood of the exceptional divisor. To accomplish this, we develop a b…

Sharp curvature pinching for mean curvature flow in spheres proved.

We study the behaviour of Laplace-type operators H on a complex vector bundle E $\rightarrow$ M in the adiabatic limit of the base space. This space is a fibre bundle M $\rightarrow$ B with compact fibres and the limit corresponds to blowing up directions perpendicular to the fibres by a factor 1/$ε$. Under a gap condi…

We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.

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