arXiv research

Study on curvature blow-up rates in black hole interiors from gravitational collapse.

New estimates quantify blow-up rates of spacelike singularities in gravitational collapse.

This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.

The paper constructs solutions with infinite-time singularities in Lagrangian mean curvature flow.

Level set flow's singularities are type I under 2-convexity, leading to specific curvature blow-up rates.

The study finds polynomial upper bounds for singularities in Einstein-scalar field system.

The blow-up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow. In particular, this generalizes a theorem of M.E. Gage and R.S. Hamilton about mean curvature flow in $\mathbb{R}^{2}$.

Study on half spheres solves Nirenberg problem with complex blow-up analysis.

Local singularity analysis for Ricci flows with applications to bounded scalar curvature.

Study identifies numerical signs of blow-up in hydrodynamic equations.

We continue the study, initiated by the first two authors in \cite{IW19}, of Type-II curvature blow-up in mean curvature flow of complete noncompact embedded hypersurfaces. In particular, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics near the "va…

We consider any pseudo holomorphic integral 2-cycle in an arbitrary almost complex manifold and perform a blow up analysis at an arbitrary point. Building upon a pseudo algebraic blow up (previously introduced by the author) we prove a geometric rate of decay for the mass ratio towards the limiting density, with an exp…

The paper analyzes high-dimensional sphere solutions to the Nirenberg problem with residual mass.

In each dimension $N\geq 3$ and for each real number $λ\geq 1$, we construct a family of complete rotationally symmetric solutions to Ricci flow on $\mathbb{R}^{N}$ which encounter a global singularity at a finite time $T$. The singularity forms arbitrarily slowly with the curvature blowing up arbitrarily fast at the r…

We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show …

This work concerns with the existence and detailed asymptotic analysis of Type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up a…

In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based …

Study on instability of extreme Reissner-Nordström spacetime perturbations.

For any manifold $N^p$ admitting an Einstein metric with positive Einstein constant, we study the behavior of the Ricci flow on high-dimensional products $M = N^p \times S^{q+1}$ with doubly-warped product metrics. In particular, we provide a rigorous construction of local, type II, conical singularity formation on suc…

We describe the moduli space of stable rank 2 parabolic bundles over an elliptic curve with 3 marked points.

Study precise asymptotics of noncompact Type-IIb solutions to mean curvature flow.

We study the one-parameter family of twisted Kahler Taub-NUT metrics (discovered by Donaldson), along with two exceptional Taub-NUT-like instantons, and understand them to the extend that should be sufficient for blow-up and gluing arguments. In particular we parametrize their geodesics from the origin, determine curva…

We prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex man…

We study the contraction of a convex immersed plane curve with speed (1/α)k^{α}, where αin(0,1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. We also discuss a special symmetric case of type two blow-up and show that it converge…

Constructs finite-time singularities in Lagrangian mean curvature flow with precise dynamics.

We study the local curvature estimates of long-time solutions to the normalized Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundles. Using these estimates, we prove that on such a manifold, the set of singular fibers of the semi-ample fibration on which the Riemann curvature blows up at…

Study pinches curvature under Laplacian G_2 flow, proving Weyl tensor norm blows up.

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe…

Gu and Zhu have shown that Type-II Ricci flow singularities develop from nongeneric rotationally symmetric Riemannian metrics on $S^m$, for all $m\geq 3$. In this paper, we describe and provide plausibility arguments for a detailed asymptotic profile and rate of curvature blow-up that we predict such solutions exhibit.

Shallow neural networks can represent polynomials efficiently.

The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.

Random covers of hyperbolic surfaces have a spectral gap with polynomial rate.

Formula derived for holomorphic Poisson blow-ups.

Polynomial networks converge to Gaussian processes at a rate of O(n^(-1/2)).

Formula derived for Bott-Chern classes in complex blow-ups.

Study on Monge-Ampère equations with polynomial growth rates.

Explains blow-ups for Lie groupoids and algebroids, comparing different methods.

The paper derives a formula for Chow weights of toric blow-ups.

Compact mean curvature flow solutions with bounded curvature in high dimensions are constructed.

The paper examines the blow-up of Ricci curvatures in conformal metrics.

New Calabi-Yau metrics converge polynomially to Calabi model space.

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…

In a singular Type I Ricci flow, we consider a stratification of the set where there is curvature blow-up, according to the number of the Euclidean factors split by the tangent flows. We then show that the strata are characterized roughly in terms of the decay rate of their volume, which in our context plays the role o…

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 studies a 1D diffusion equation with nonlinear Robin boundary conditions and finds conditions for global and finite time blow-up or blow-down.

Paper addresses inefficiency in converting EFGs to NFGs for learning.

The study examines exceptional sets for radial limits of superharmonic functions on curved manifolds.

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 …