arXiv research

Paper analyzes blowup of regularized Jang solutions and constant expansion surfaces.

Stable blowup solutions found for supercritical Yang-Mills equations.

The paper finds asymmetric Type-I blowup solutions for Yang-Mills flow.

Study Toda systems blowup masses linked to Weyl groups.

Stable blowup profile identified for wave maps in all dimensions.

The aim of this paper is to collect some facts about the blowup of Jang's equation. First, we discuss how to construct solutions that blow up at an outermost MOTS. Second, we exclude the possibility that there are extra blowup surfaces in data sets with non-positive mean curvature. Then we investigate the rate of conve…

The paper constructs and analyzes self-similar blowup solutions for a wave map equation.

In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove …

Study proves energy critical heat equation solutions are Type I blowups for n ≥ 7.

We consider the energy-supercritical harmonic map heat flow from $\mathbb{R}^d$ into $\mathbb{S}^d$, under an additional assumption of 1-corotational symmetry. We are interested by the 7 dimensional case which is the borderline between the Type I blowup regime. We construct for this problem a stable finite time blowup …

Estimates for VPMCF show ancient MCF solutions and finite-time behavior.

We study singularity formation in spherically symmetric solutions of the charge-one and charge-two sector of the (2+1)-dimensional S^2 sigma-model and the (4+1)-dimensional Yang-Mills model, near the adiabatic limit. These equations are non-integrable, and so studies are performed numerically on rotationally symmetric …

Stability of singularity formation in Yang-Mills fields in higher dimensions.

Blowups of Kähler manifolds with extremal metrics inherit such metrics under stability conditions.

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

Study on unique solutions to one-phase free boundary problems.

We study the heat flow for Yang-Mills connections on $\mathbb R^d \times SO(d)$. It is well-known that in dimensions $5 \leq d \leq 9$ this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an explicit example given by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stab…

We study the formation of generic singularities of mean curvature flow by combining the different approaches, specifically the methods in studying blowup of nonlinear heat equations, the techniques used by the author and the collaborators for mean curvature flow, and these invented by Colding and Minicozzi. We study th…

Global and local blowups of manifolds are proven equivalent.

Solves complex Monge-Ampère equation for Kähler-Ricci solitons.

Ancient ovals are key blowup limits in 3D Ricci flow near singularities.

We study singularity formation of complete Ricci flow solutions, motivated by two applications: (a) improving the understanding of the behavior of the essential blowup sequences of Enders-Muller-Topping on noncompact manifolds, and (b) obtaining further evidence in favor of the conjectured stability of generalized cyli…

Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $\partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so that under these new metrics, $K$ is the scalar curvature and $c$ is …

We give conditions under which the blowup of an extremal Kähler manifold along a submanifold of codimension greater than two admits an extremal metric. This generalizes work of Arezzo-Pacard-Singer, who considered blowups in points.

The paper resolves singular foliations through a series of blowups.

Kähler blowups can have scalar curvature arbitrarily close to any given metric.

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

Study the pullbacks and blowups of Lie algebroids and related structures.

Uniqueness of nondegenerate blowups for planar networks shown.

Wave maps with noise can lead to self-similar blowup from arbitrary initial data.

Functor connects symplectic and contact structures via cutting and blowups.

Study deformed Hermitian-Yang-Mills equation on complex projective space blowup.

Stability of weighted extremal manifolds proven through blowups.

We construct minimal laminations with prescribed singularities on a line segment using perturbation techniques and PDE methods. In addition to the singular set, the rate of curvature blowup is also prescribable in our construction, and we show that all curvature blowup rates between quadratic and quartic arise. Our res…

Characterizes blowups of Dirac structures on manifolds.

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…

We present natural and general ways of building Lie groupoids, by using the classical procedures of blowups and of deformations to the normal cone. Our constructions are seen to recover many known ones involved in index theory. The deformation and blowup groupoids obtained give rise to several extensions of $C^*$-algeb…

We describe a method for recursively calculating Gromov-Witten invariants of all blowups of the projective plane. This recursive formula is different from the recursive formulas due to Göttsche and Pandharipande in the zero genus case, and Caporaso and Harris in the case of no blowups. We use tropical curves and a recu…

Blowups of Kähler manifolds can inherit extremal metrics.

Study constant mean curvature tori in R^3 using spectral data and Whitham deformations.

We prove that on a closed surface, for any $c>0$, our min-max theory for prescribing mean curvature produces a solution given by a curve of constant geodesic curvature $c$ which is almost embedded, except for finitely many points, at which the solution is a stationary junction with integer density. Moreover, each smoot…

Yamabe invariants of certain non-Kähler surfaces are zero.

Proves invariance of weighted extremal Kähler metrics under smooth blowups.

We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant…

Study curvature of blown-up manifold, finds negative holomorphic sectional curvature.

Consider a vector bundle over a Kähler manifold which admits a Hermitian Yang-Mills connection. We show that the pullback bundle on the blowup of the Kähler manifold at a collection of points also admits a Hermitian Yang-Mills connection, for Kähler classes on the blowup which make the exceptional divisors small. Our p…

Global existence and smoothing effects for reaction-diffusion equations with blowup in infinite time.

In the paper we formulate and derive the family blowup formula of family Seiberg-Witten invariants. The formula has been used in the enumerative application of counting singular curves on algebraic surfaces. We first give a topological derivation of the formula by using family index theorem. Then we define the algebrai…