arXiv research

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…

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…

Estimates mean curvature flow with geometric bounds.

We develop some estimates under the Ricci flow and use these estimates to study the blowup rates of curvatures at singularities. As applications, we obtain some gap theorems: $\displaystyle \sup_X |Ric|$ and $\displaystyle \sqrt{\sup_X |Rm|} \cdot \sqrt{\sup_X |R|}$ must blowup at least at the rate of type-I. Our estim…

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 …

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

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…

Study Toda systems blowup masses linked to Weyl groups.

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

Global and local blowups of manifolds are proven equivalent.

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

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.

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.

Rotationally invariant Ricci flows are constructed and shown to converge to spacetimes.

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.

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.

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…

Study on bending energy of surfaces with curvature concentration, deriving new lower bounds.

Study on the $L^p$ behavior of helix curves' energy density.

Proves uniqueness of blowups for forced mean curvature flow.

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…