We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the Weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n>10.
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Study on wave-breaking phenomena in solutions of the Camassa-Holm equation.
problem Wave-breaking phenomena in solutions of the Camassa-Holm equation.
method Investigation of pseudospherical surfaces and singularities of the metric.
result The metric blows up if and only if the solution breaks in finite time.
The paper finds multiple ways a special curvature can blow up in high dimensions.
problem Finding multiple metrics with constant Q-curvature in high dimensions.
method Constructing small perturbations of standard bubbles.
result Infinitely many smooth metrics with the same constant Q-curvature and arbitrarily large energy.
The paper examines the blow-up of Ricci curvatures in conformal metrics.
problem Characterizing the blow-up set of Ricci curvatures in conformal metrics.
method Analyzing the blow-up phenomena of Ricci curvatures on domains close to a limit set of lower dimension.
result Characterization of the blow-up set according to the Yamabe invariant of the manifold.
The Yamabe flow can blow up in infinite time with small perturbations.
problem Understanding the behavior of the Yamabe flow under small perturbations.
method Constructive proof using solutions of the Yamabe problem on the unit sphere as blow-up profiles.
result The Yamabe flow can blow up at multiple points on a Riemannian manifold in infinite time with small perturbations.
Let n be an integer such that 25 \leq n \leq 51. We construct a smooth metric g on S^n with the property that the set of constant scalar curvature metrics in the conformal class of g is not compact.
In this paper, we study the blow-up phenomena on the -harmonic map sequences with bounded uniformly -energy, denoted by $\{u_{α_k}: α_k>1 \quad \mbox{and} \quad α_k\searrow 1\}$, from a compact Riemann surface into a compact Riemannian manifold. If the Ricci curvature of the target manifold is of a positive l…
We extend a model of positive feedback and contagion in large mean-field systems, by introducing a common source of noise driven by Brownian motion. Although the driving dynamics are continuous, the positive feedback effect can lead to `blow-up' phenomena whereby solutions develop jump-discontinuities. Our main results…
The paper constructs metrics with blow-up solutions for a curvature equation.
problem Constructing metrics with blow-up solutions for a curvature equation.
method Analyzing the constant Q/R-curvature equation on .
result Constructs families of solutions exhibiting blow-up behavior.
Let (M,g) be a compact Riemannian manifold of dimension n \geq 3. The Compactness Conjecture asserts that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M,g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to this conjecture in dime…
New estimates quantify blow-up rates of spacelike singularities in gravitational collapse.
problem Quantifying the blow-up rates of spacelike singularities in gravitational collapse.
method Deriving new quantitative estimates using spherical symmetry and double-null coordinates.
result Polynomial blow-up rates for various quantities, with improved estimates for and .
Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equationmath.DG
We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of "lower energy" solutions to the above equation fails…
Wave maps can have multiple bubbling solutions at blow-up points.
problem Non-uniqueness of bubbling solutions in wave maps.
method Example construction of multiple bubbling solutions.
result First known example of non-uniqueness of bubbling for dispersive equations.
The paper solves a Nirenberg problem on half spheres, finding multiple blow-ups.
problem Finding conformal metrics with prescribed scalar curvature and zero boundary mean curvature on half spheres.
method Constructing finite energy solutions to a subcritical approximation of the problem on half spheres of dimension \( n \geq 5 \).
result The solutions exhibit multiple blow-up of cluster-type at the same boundary point.
Let (M,g) be a compact n-dimensional Riemannian manifold with boundary. This article is concerned with the set of scalar-flat metrics on M which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We construct examples of metrics on the unit ball, in dimensions n>=25, for wh…
Study on constant mean curvature 1-immersions into hyperbolic 3-manifolds.
problem Understanding CMC 1-immersions of surfaces into hyperbolic 3-manifolds.
method Parametrization of moduli space, analysis of blow-up phenomena, asymptotic analysis.
result Sharp condition for genus g=2 involving Kodaira map at six Weierstrass points.
Compactness fails for curvature equations in high dimensions.
problem Compactness of solutions to curvature equations fails in high dimensions.
method Chen and Wu constructed a smooth counterexample.
result Compactness fails in dimensions not less than 35.
We illustrate the flow or wave character of the metrics and curvatures of evolving manifolds, introducing the Riemann flow and the Riemann wave via the bialternate product Riemannian metric. This kind of evolutions are new and very natural to understand certain flow or wave phenomena in the nature as well as the geomet…
Study on existence and uniqueness of 1-immersions of surfaces into hyperbolic 3-manifolds.
problem Existence and uniqueness of mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds.
method Analyzing the asymptotic behavior of minimizers of the Donaldson functional as t approaches 0 from the positive side.
result First existence and uniqueness result about (CMC) 1-immersions of surfaces of genus 2 into hyperbolic 3-manifolds.
(CMC) 1-immersions in hyperbolic 3-manifolds often develop singularities.
problem Existence and uniqueness of (CMC) 1-immersions of surfaces into hyperbolic 3-manifolds.
method Analysis of blow-up phenomena and orthogonality conditions.
result Existence and uniqueness of (CMC) 1-immersions for surfaces of any genus.
In this paper we prove that all initially-smooth solutions of the Euler-Weil-Petersson equation, which describes geodesics on the universal Teichmüller space under the Weil-Petersson metric, will remain smooth for all time. This extends the work of Escher-Kolev for strong Riemannian metrics to the borderline case of $H…
For a compact manifold with boundary we introduce the -fold scattering stretched product which is a compact manifold with corners for each coinciding with the previously known cases for It is constructed by iterated blow up of boundary faces and boundary faces of multi-diagonals i…
Study on half spheres solves Nirenberg problem with complex blow-up analysis.
problem Prescribing scalar curvature on half spheres.
method Refined blow-up analysis of finite energy approximated solutions.
result Complex blow-up points and vortex problems reveal new connections.
The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.
problem Existence and uniqueness of almost complex blow-ups on almost complex manifolds.
method Definition and construction of almost complex blow-ups, proving their existence and uniqueness.
result Existence and uniqueness of almost complex blow-ups on 4D almost complex manifolds.
Formula derived for holomorphic Poisson blow-ups.
problem Invariance of Koszul-Brylinski homology under Poisson blow-ups.
method Blow-up formula derivation for holomorphic Koszul-Brylinski homologies.
result Invariance of -degeneracy of Dolbeault-Koszul-Brylinski spectral sequence.
Study identifies numerical signs of blow-up in hydrodynamic equations.
problem Determining if numerical results of blow-up are genuine or artifacts.
method Geometrically consistent spatiotemporal discretization of complexified Euler equations.
result Identification of a signature based on supremum norm growth rates of vorticity.
Formula derived for Bott-Chern classes in complex blow-ups.
problem Calculating Bott-Chern classes in blow-ups of complex manifolds.
method Proved blow-up formula for Bott-Chern classes, established Riemann-Roch without denominators.
result Formula for Bott-Chern classes in blow-ups.
Explains blow-ups for Lie groupoids and algebroids, comparing different methods.
problem Blow-ups of Lie groupoids and algebroids.
method Detailed explanation of various blow-up constructions.
result Different blow-up constructions for Lie groupoids and algebroids are shown to fit into a general geometric framework.
The paper derives a formula for Chow weights of toric blow-ups.
problem Chow weights of toric blow-ups.
method Combinatorial formula derived from toric manifold and Delzant polytope.
result Explicit formula for Chow weights of blow-ups.
Study on curvature blow-up rates in black hole interiors from gravitational collapse.
problem Understanding curvature blow-up rates in black hole interiors during gravitational collapse.
method Investigation of spherically symmetric Einstein-scalar field spacetimes, focusing on blow-up rates of curvature and mass.
result Kretschmann scalar blows up faster than in Schwarzschild setting, indicating a new blow-up phenomenon.
We show that the blow-up of a generalized Kahler 4-manifold in a nondegenerate complex point admits a generalized Kahler metric. As with the blow-up of complex surfaces, this metric may be chosen to coincide with the original outside a tubular neighbourhood of the exceptional divisor. To accomplish this, we develop a b…
This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.
problem Determining the RLCT of sum-of-products polynomials through blow-up.
method Investigates a specific blow-up algorithm for sop polynomials to resolve their singularities.
result It is possible to resolve the singularities of sop polynomials using a specific blow-up algorithm.
We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideal, which allows one to define a blow-up in the category of smooth manifolds. We then investigate which generalized complex submanifolds are suitable for blowing up. Two classes naturally appear; generalized Poisso…
The paper analyzes high-dimensional sphere solutions to the Nirenberg problem with residual mass.
problem The Nirenberg problem on high-dimensional spheres with residual mass.
method Analysis of subcritical approximations and blowing up solutions.
result Comprehensive description of blowing up solutions, including blow-up points and rates.
The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up …
In this paper, we study the blow-up of a locally conformal symplectic manifold.We show that there exists a locally conformal symplectic structure on the blow-up of a locally conformal symplectic manifold along a compact induced symplectic submanifold.
Study on spinor field equation on spheres, focusing on blow-up analysis.
problem Spinorial Yamabe problem on spheres.
method Variational methods, blow-up analysis.
result Blow-up profile for the spinorial Yamabe type equation on .
Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.
problem Investigate the homotopy of blow ups in algebraic and symplectic geometry.
method Develops fibrewise surgery theory and a purely homotopy theoretic approach.
result Obtained homotopy decompositions of the based loop space on blow ups.
Study on blow-up behavior of sign-changing solutions for Yamabe equation.
problem Blow-up behavior of sign-changing solutions for Yamabe equation.
method Construction of a smooth metric on space forms to prove blow-up at lowest energy level.
result Blow-up occurs at the lowest energy level for sign-changing solutions in dimensions 11 to 24.
Proves inextendibility of weak null singularities from curvature blow-up.
problem Inextendibility of weak null singularities in the context of curvature blow-up.
method Introduces a new strategy to infer -inextendibility from curvature blow-up.
result Expected to contribute to the resolution of strong cosmic censorship conjecture.
Desingularizes singular foliations with a locally compact groupoid.
problem Handling singularities in foliations.
method Blow-up construction of smooth manifolds and groupoids.
result Locally compact locally Hausdorff groupoid desingularizes singular foliations.
Corrected Monti's blow-up analysis for H-minimizing sets in Heisenberg group.
problem Blow-up analysis of H-minimizing sets in Heisenberg group with corrected partial differential equation.
method Revised Monti's results on blow-ups of H-perimeter minimizing sets in and corrected the partial differential equation for the limit function.
result Corrected the partial differential equation for the limit function of blow-ups in Heisenberg group.
Researchers found solutions to a complex equation on spheres, overcoming a key difficulty.
problem Finding solutions to a specific equation on spheres with a background metric.
method Constructed a smooth metric invariant under antipodal map, used a noncompact family of solutions, and addressed the loss of ellipticity.
result Provided solutions to the -Yamabe equation for and beyond, overcoming a main difficulty.
The article proves conditions for blow-ups of lcK spaces to remain lcK.
problem Conditions for blow-ups of locally irreducible lcK spaces to remain lcK.
method Proves conditions for blow-ups of lcK spaces to remain lcK.
result Blow-ups of locally irreducible lcK spaces are lcK if and only if the original space is induced gcK.
Finite-time blow-up in Yang-Mills flow for small energy initial connections.
problem Finite-time blow-up of Yang-Mills flow solutions.
method Analyzing the Yang-Mills flow on Riemannian and Kähler manifolds.
result Finite-time blow-up occurs for small energy initial connections.
The aim of this article is to study expansions of solutions to an extremal metric type equation on the blow-up of constant scalar curvature Kähler surfaces. This is related to finding constant scalar curvature Kähler (cscK) metrics on K-stable blow-ups of extremal Kähler surfaces
Study shows continuity and geometric regularity of Kähler-Ricci flow blow-up limits.
problem Geometric regularity of blow-up limits of the Kähler-Ricci flow.
method Established geometric regularity for Type I blow-up limits based on sequences of Ricci vertices.
result The limiting flow is continuous in time in Gromov-Hausdorff and Gromov- distance.
We derive a blow-up formula for the de Rham cohomology of a local system of complex vector spaces on a compact complex manifold. As an application, we obtain the blow-up invariance of -degeneracy of the Hodge-de Rham spectral sequence associated to a local system of complex vector spaces.