Study uniform K-stability and its connection to log Fano pairs' plt blowups.
problem Testing uniform K-stability of log Fano pairs.
method Evaluate volume function invariants for all plt blowups.
result Establishes uniform K-stability criteria for log Fano pairs.
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.
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.
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.
The paper proves stability of a blowup solution for Yang-Mills heat flow.
problem Stability of blowup solutions for Yang-Mills heat flow.
method Small perturbation analysis and explicit self-similar blowup solution.
result Stability of the explicit self-similar blowup solution under perturbations.
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.
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.
Stability of singularity formation in Yang-Mills fields in higher dimensions.
problem Stability of self-similar blowup profiles for Yang-Mills equations in (1+d)-dimensions. method Analysis of explicitly known equivariant self-similar blowup solution and small equivariant perturbations.
result Global-in-space asymptotic stability of the self-similar blowup solution for Yang-Mills equations in (1+d)-dimensions for d≥5. 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.
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…
Let (X,L) be a polarised manifold. We show that K-stability and asymptotic Chow stability of the blowup of X along a 0-dimensional cycle are closely related to Chow stability of the cycle itself, for polarizations making the exceptional divisors small. This can be used to give (almost) a converse to a result of Arezzo …
Stable shrinkers found for heat flow of harmonic maps.
problem Stability analysis of self-similar blowup in parabolic evolution equations.
method Systematic, robust, and constructive approach avoiding delicate techniques.
result Nonlinear asymptotic stability of a self-similar shrinker proved.
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.
Wave maps into negatively curved targets can blow up stably.
problem Existence and stability of blowup for wave maps.
method Construction of a self-similar wave map for a negatively curved target.
result Stable blowup mechanism for wave maps in high dimensions.
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.
Study shows different behaviors of noncompact hypersurfaces under mean curvature flow.
problem Analyzing stability of noncompact hypersurfaces with curvature blowup.
method Numerical overlap method to construct global solutions.
result Existence of near and far classes of initial data leading to distinct behaviors.
Researchers prove existence of a stable self-similar blowup solution.
problem Proving the spectral gap conjecture for harmonic map heat flow.
method Existence of a monotone self-similar solution using interval arithmetic for rigorous computer-assisted estimates.
result Mathematically rigorous proof of the stability of a self-similar blowup solution.
Stability of specific solitons proven in higher dimensions.
problem Stability of homothetically shrinking Yang-Mills solitons in higher dimensions.
method Heat flow for Yang-Mills connections, small equivariant perturbations, general framework for spectral problems.
result Nonlinear asymptotic stability of the Weinkove solution in higher dimensions.
This is a continuation of the work of Arezzo-Pacard-Singer and the author on blowups of extremal Kähler manifolds. We prove the conjecture stated in [32], and we relate this result to the K-stability of blown up manifolds. As an application we prove that if a Kähler manifold M of dimension greater than 2 admits a cscK …
We study the J-flow from the point of view of an algebro-geometric stability condition. In terms of this we give a lower bound for the natural associated energy functional, and we show that the blowup behavior found by Fang-Lai is reflected by the optimal destabilizer. Finally we prove a general existence result on com…
Stable type II blowup solutions found for a specific heat flow equation.
problem Stability of type II blowup solutions for a harmonic heat flow.
method Reduction to a finite-dimensional problem, modulation techniques, and contradiction argument.
result Stable type II blowup solutions constructed for the energy supercritical harmonic heat flow.
Researchers find stable solutions for heat map flow in higher dimensions.
problem Stability of shrinkers for harmonic map heat flow in higher dimensions.
method Construction of specific target manifolds allowing for stable shrinkers.
result Existence of corotational self-similar shrinkers representing stable blowup mechanisms.
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.
Rotationally invariant Ricci flows are constructed and shown to converge to spacetimes.
problem Constructing and understanding Ricci flows through surgery on rotationally invariant manifolds.
method Rotationally invariant Ricci flow through surgery, convergence to spacetimes, blowup rate analysis.
result Rotationally invariant Ricci flows converge to spacetimes with controlled curvature blowup.
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.
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.
Study singularity formation in Ricci flow solutions.
problem Understanding singularity behavior in noncompact manifolds.
method Analyzing complete Ricci flow solutions.
result Evidence for stability of generalized cylinders as singularity models.
Extremal metrics found on specific manifold operations.
problem Conditions for extremal metrics on blowups.
method Analyzes blowups of extremal Kähler manifolds.
result Extremal metrics exist on blowups of higher codimension.
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.
The paper solves symplectic embedding problems in higher dimensions, proving new embedding conditions.
problem Symplectic embedding problems in higher dimensions.
method Symplectic blowup construction, h-principle for symplectic surfaces, stabilization of pseudoholomorphic curves.
result New embedding conditions for symplectic balls and surfaces in higher dimensions.
Study of Lagrangian mean curvature flow with equivariant symmetry.
problem Understanding singularities in Lagrangian mean curvature flow.
method Structural theorems about blowups of finite-time singularities.
result Classification of singularities in equivariant case.
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.
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 RnimesSO(n) for 5≤n≤9. result Existence of asymmetric Type-I blowup solutions for the Yang-Mills flow.
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.
New methods for constructing Lie groupoids and related K-theory computations.
problem Building Lie groupoids and computing K-theory.
method Blowups and deformations to the normal cone.
result Recovery of known constructions and new extensions of C∗-algebras. Shows uniqueness of cylindrical blowups in mean curvature flow.
problem Uniqueness of cylindrical blowups in mean curvature flow in higher codimension.
method Developed new methods to prove uniqueness of cylindrical blowups.
result Implication of regularity of the singular set for the system.
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.
Hermitian Yang-Mills connections on blown-up manifolds are shown to exist.
problem Existence of Hermitian Yang-Mills connections on blown-up Kähler manifolds.
method Gluing techniques to construct connections explicitly.
result Existence of Hermitian Yang-Mills connections on blowups for small exceptional divisors.
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.
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 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…
Constructs a stable finite-time blowup solution for a specific harmonic map heat flow problem.
problem Energy-supercritical harmonic map heat flow with 1-corotational symmetry in 7 dimensions.
method Constructs a stable finite time blowup solution under corotational symmetry.
result Constructs a stable finite time blowup solution with concentration of the universal profile.
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.