arXiv research

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

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

Proves uniqueness of blowups for forced mean curvature flow.

We study singularity structure of Yang-Mills flow in dimensions $n \geq 4$. First we obtain a description of the singular set in terms of concentration for a localized entropy quantity, which leads to an estimate of its Hausdorff dimension. We develop a theory of tangent measures for the flow, which leads to a stratifi…

We will study the blowup behavior of a surface sequence immersed in $R^2$ with bounded Willmore functional and fixed genus.

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 …

Study Toda systems blowup masses linked to Weyl groups.

Global and local blowups of manifolds are proven equivalent.

We consider Riemannian 4-manifolds that Gromov-Hausdorff converge to a lower dimensional limit space, with the Ricci tensor going to zero. Among other things, we show that if the limit space is two dimensional then under some mild assumptions, the limiting four dimensional geometry away from the curvature blowup region…

Study on singularities in Lagrangian mean curvature flow with special Lagrangian cones.

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.

Stable blowup solutions found for supercritical Yang-Mills equations.

Stable blowup profile identified for wave maps in all dimensions.

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.

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…

In real algebraic geometry, Lojasiewicz's theorem asserts that any integral curve of the gradient flow of an analytic function that has an accumulation point has a unique limit. Lojasiewicz proved this result in the early 1960s as a consequence of his gradient inequality. Many problems in calculus of variations are que…

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

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

Uniqueness of nondegenerate blowups for planar networks shown.

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

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

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.

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…

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…

We prove the expansion formula for the classical Futaki invariants on the blowup of Kähler surfaces, which explains the balancing condition of Arezzo-Pacard. The relation with Stoppa's result is also discussed.

The blow-down map is studied in Lie algebroid cohomology.

We show the properties of the blowup limits of \KRf solutions on Fano surfaces if Riemannian curvature is unbounded. As an application, on every toric Fano surface, we prove that \KRf converges to a Kähler Ricci soliton metric if the initial metric has toric symmetry. Therefore we give a new Ricci flow proof of existen…

New method proves uniqueness in mean curvature flow.

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

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

Walker manifolds of signature (2,2) have been used to provide examples of Osserman and of conformal Osserman manifolds of signature (2,2). We study questions of geodesic completeness and Ricci blowup in this context.

We show that the blowup of an extremal Kahler manifold at a relatively stable point in the sense of GIT admits an extremal metric in Kahler classes that make the exceptional divisor sufficiently small, extending a result of Arezzo-Pacard-Singer. We also study the K-polystability of these blowups, sharpening a result of…

Parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces. In this paper, we show that the three notions of parabolic polystability, K-polystability and existence of constant scalar curvature Kähler metrics on the iterated blowup are equivalent, for certain polarizations…

It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking or translating) if…