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…
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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…
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.
Study BF invariants using simple type concepts.
problem Understanding Bauer--Furuta invariants for 4-manifolds.
method Extend simple type concept to BF blowup and BF homogeneous types, prove gluing formulae and adjunction inequality.
result Determine Bauer--Furuta invariant of a specific 4-manifold and give constraints on gluing decompositions.
Study of gauge theory blowups and Painlevé VI identity.
problem Understanding gauge theory blowups and their relation to Painlevé VI.
method Analyzing four-dimensional supersymmetric gauge theory with defects and proposing identities.
result Formula relating tau-function to conformal blocks of Liouville theory.
We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a -manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formu…
Curve shortening flow shrinks curves to points.
problem The behavior of curves under curve shortening flow.
method Nonlinear partial differential equations, maximum principle, monotonicity formulas, Harnack inequalities, blowup analysis.
result The curve shortening flow shrinks any closed embedded curve in the plane to a round point.
We consider the heat flow of corotational harmonic maps from to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self…
Universal functions derived for topological correlators in Yang-Mills theory.
problem Expressing correlation functions of topologically twisted Yang-Mills theory.
method Combining Seiberg-Witten geometry, u-plane integral, and blowup formula.
result Closed expressions for universal functions determined.
In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of int…
Study Toda systems blowup masses linked to Weyl groups.
problem Understanding blowup phenomena in Toda systems.
method Concrete examples of Toda systems solutions and blowup masses.
result Blowup masses correspond to Weyl groups.
In this expository paper, we explain a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a -manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applicatio…
We study the change of moduli spaces of Gieseker-semistable torsion free rank- sheaves on algebraic surfaces as we vary the polarizations. When the surfaces are rational with an effective anti-canonical divisor, the moduli spaces are linked by a series of flips (blowups and blowdowns). Using these results, we comput…
Paper analyzes blowup of regularized Jang solutions and constant expansion surfaces.
problem Blowup behavior of regularized solutions to Jang equation inside apparent horizons.
method Two geometric treatments: dilation and translation. Characterization of limits of rescaled and translated solutions.
result Limits of properly rescaled solutions are constant expansion surfaces.
Two types of differentials are shown equivalent for compactifying moduli spaces.
problem Compactifying moduli spaces of curves with prescribed orders of zeros and poles.
method Equivalence of multi-scale and logarithmic differentials, isomorphism of moduli stacks, explicit blowups.
result Multi-scale and logarithmic differentials are equivalent and isomorphic.
Global and local blowups of manifolds are proven equivalent.
problem Equivalence of global and local blowups in differential topology.
method Proof of equivalence between global and local constructions of blowups.
result Global and local constructions of blowups are shown to be equivalent.
Ancient ovals are key blowup limits in 3D Ricci flow near singularities.
problem Understanding blowup limits in 3D Ricci flow near singularities.
method Proving ancient ovals are blowup limits if and only if spherical singularities accumulate.
result Ancient ovals are necessary and sufficient for blowup limits in 3D Ricci flow.
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.
problem Singular foliations that cannot be resolved directly.
method Applying Nash modifications to the universal Lie ∞-algebroid of a singular foliation.
result Any singular foliation becomes a Debord foliation after one blowup.
Stable blowup solutions found for supercritical Yang-Mills equations.
problem Understanding blowup solutions for supercritical Yang-Mills equations.
method Investigated equivariant self-similar blowup solutions and their stability.
result Stability of blowup mechanism for odd dimensions greater than or equal to 5.
Stable blowup profile identified for wave maps in all dimensions.
problem Stability of blowup solutions for wave maps in supercritical energy.
method Novel stability analysis using similarity variables on the whole space.
result Global nonlinear stability of the corotational self-similar blowup profile.
Kähler blowups can have scalar curvature arbitrarily close to any given metric.
problem Understanding scalar curvature on Kähler blowups.
method Analyzing scalar curvature on blowups of Kähler manifolds.
result The scalar curvature of Kähler blowups can be made arbitrarily close to any given metric.
Strict type-II blowup in harmonic map flow is proven to have Hölder continuous body map.
problem Finite-time singularity of harmonic map flow.
method Analysis of outer energy scale and Hölder continuity proof.
result Strictly type-II blowup body map is Hölder continuous.
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…
Study the pullbacks and blowups of Lie algebroids and related structures.
problem Understanding the relationship between Lie algebroids, singular foliations, and Dirac structures under maps.
method Examine pullbacks and blowups of Lie algebroids and related structures under maps with constant rank or transversality assumptions.
result Establish the relation between the blowup of a Lie algebroid and its singular foliation.
The paper finds asymmetric Type-I blowup solutions for Yang-Mills flow.
problem Existence of asymmetric Type-I blowup solutions for Yang-Mills flow.
method Constructing an infinite-dimensional family of solutions for the Yang-Mills flow on for .
result Existence of asymmetric Type-I blowup solutions for the Yang-Mills flow.
Uniqueness of nondegenerate blowups for planar networks shown.
problem Uniqueness of nondegenerate blowups for the motion by curvature of planar networks.
method Proof based on Lojasiewicz-Simon gradient inequality applied to stability properties of critical points of the length functional.
result Uniqueness of nondegenerate compact blowups for the motion by curvature of planar networks.
Wave maps with noise can lead to self-similar blowup from arbitrary initial data.
problem Analyzing self-similar blowup in wave maps with additive noise.
method Stochastic perturbation of wave maps in supercritical dimensions.
result Self-similar blowup with positive probability for arbitrary corotational initial data.
Study proves energy critical heat equation solutions are Type I blowups for n ≥ 7.
problem Analyzing blowup behavior of energy critical nonlinear heat equations.
method Reverse inner-outer gluing mechanism and bubbling behavior analysis.
result Proves all blowups are of Type I for n ≥ 7.
Functor connects symplectic and contact structures via cutting and blowups.
problem Establishing a functorial relationship between symplectic and contact structures.
method Developed a cutting procedure and its inverse for manifolds with boundary and equivariant transverse maps, then applied it to non-symplectic and non-contact structures.
result Obtained an inverse functor for equivariant radial-squared blowups.
Study deformed Hermitian-Yang-Mills equation on complex projective space blowup.
problem Solving the deformed Hermitian-Yang-Mills equation on complex projective space blowup.
method Expressed the equation as an ODE and solved it using combinatorial methods under an algebraic stability condition.
result Evidence supporting a conjecture on general compact Kahler manifolds.
Stability of weighted extremal manifolds proven through blowups.
problem Stability of weighted extremal manifolds.
method Blowup technique to analyze weighted extremal Kähler manifolds.
result Proves weighted extremal manifolds are relatively weighted K-polystable.
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…
In this note we study the localization of Futaki-Morita integrals at isolated degenerate zeros by giving a streamlined exposition in the spirit of Bott and implement the localization procedure for a holomorphic vector field on with a maximally degenerate zero, giving an essentially unique formula for the Futaki-…
Blowups of Dirac structuresmath.SG
Characterizes blowups of Dirac structures on manifolds.
problem Characterizing when a Dirac structure lifts to a blowup.
method Analyzes the properties of submanifolds and Lie algebras.
result Lifts of Dirac structures are possible under specific conditions.
We study almost-calibrated, -equivariant Lagrangian mean curvature flow in , 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 -algeb…
Blowups of Kähler manifolds can inherit extremal metrics.
problem Extending extremal metrics to blowups of Kähler manifolds.
method Analyzing the action of a torus on blowups and weighted extremal metrics.
result Blowups of Kähler manifolds can inherit weighted extremal metrics.
Yamabe invariants of certain non-Kähler surfaces are zero.
problem Determining the sign of Yamabe invariants for non-Kähler surfaces.
method Analyzing Inoue surfaces and Kodaira surfaces, their blowups, and applying Seiberg-Witten theory.
result Yamabe invariants of Inoue surfaces and their blowups are all zero.
Proves invariance of weighted extremal Kähler metrics under smooth blowups.
problem Invariance of weighted extremal Kähler metrics under smooth blowups.
method Uniform coercivity estimate for the (relative, weighted) Mabuchi energy on blowups.
result 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.
problem Proving positive curvature of blowups of manifolds with positive holomorphic sectional curvature.
method Calculates curvature tensor of a specific metric on the blowup's exceptional divisor.
result Holomorphic sectional curvature is negative in some directions for small enough t.
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…
The paper proves smoothness of transition layers in the Allen-Cahn equation.
problem Proving uniform regularity for transition layers.
method Utilizes Allen-Cahn monotonicity formula, Lipschitz approximation, and blowups.
result Shows uniform regularity for transition layers converging to smooth mean curvature flows.
The blow-down map is studied in Lie algebroid cohomology.
problem Computing Lie algebroid cohomology of blowups.
method Developed a Gysin sequence for Lie algebroids and used it to compute cohomology.
result Generalized Mazzeo-Melrose theorem to Lie algebroids.
New method proves uniqueness in mean curvature flow.
problem Proving uniqueness in mean curvature flow.
method Arguments from [CM2] to get stronger uniqueness.
result Stronger effective version of uniqueness of blowups.
Blowups of Kähler manifolds with extremal metrics inherit such metrics under stability conditions.
problem Extremal Kähler metrics on blowups of Kähler manifolds.
method Analyzing K-stability and geometric invariant theory.
result 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.
problem Existence and stability of self-similar blowup solutions for a wave map equation.
method Construction of self-similar solutions, detailed nonlinear stability analysis, spectral analysis of linearized operators.
result Sharp semigroup bounds and nonlinear stability of all discretely self-similar profiles in all dimensions.