arXiv research

Proves finite step termination of Kähler-Einstein metric singularity formation.

Study on Kähler-Ricci flow and conformal submersion singularity formation.

Study of singularity formation in dHYM flow on a blown-up CP^3.

The paper studies vector bundles over surfaces, focusing on singularity formation.

Unified view on big bang singularities from initial data.

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a di…

We study singularity formation of complete Ricci flow solutions, motivated by two applications: (a) improving the understanding of the behavior of the essential blowup sequences of Enders-Muller-Topping on noncompact manifolds, and (b) obtaining further evidence in favor of the conjectured stability of generalized cyli…

Lectures on mean curvature flow and its related equations.

We use numerical techniques to study the formation of singularities in Ricci flow. Comparing the Ricci flows corresponding to a one parameter family of initial geometries on S^3 with varying amounts of S^2 neck pinching, we find critical behavior at the threshold of singularity formation.

Study shows instability of naked singularities in scalar field models.

Model for associative submanifolds in K3 fibrations.

We study the formation of singularities for the mean curvature flow of monotone Lagrangians in $\C^n$. More precisely, we show that if singularities happen before a critical time then the tangent flow can be decomposed into a finite union of area-minimizing Lagrangian cones (Slag cones). When $n=2$, we can improve this…

Survey classifies singularity models in 3D Ricci flow.

We study mean curvature flow of smooth, axially symmetric surfaces in $\mathbb{R}^3$ with Neumann boundary data. We show that all singularities at the first singular time must be of type I.

Researchers find stable solutions for heat map flow in higher dimensions.

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…

Finite singular times for symmetric network curvature flow.

Localized big bang singularities found without background solutions.

Study on how soliton equations form singularities using L,A,B-triples.

Study investigates singularity formation in $\alpha$-Yang-Mills-Higgs fields on spheres.

Ricci flow modelled on specific singularities on closed manifolds.

We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in $\mathbb C^{n}$.

The paper proves conditions for curvature blow-up in quiescent big bang singularities.

We consider the motion by curvature of a network of curves in the plane and we discuss existence, uniqueness, singularity formation and asymptotic behavior of the flow.

We study singularity structure of Yang-Mills flow in dimensions $n \geq 4$. First we obtain a description of the singular set in terms of concentration for a localized entropy quantity, which leads to an estimate of its Hausdorff dimension. We develop a theory of tangent measures for the flow, which leads to a stratifi…

This paper studies rapidly forming singularities in the Yang-Mills flow. It is shown that a sequence of blow-ups near the singular point converges, modulo the gauge group, to a homothetically shrinking soliton with non-zero curvature. The proof uses Hamilton's monotonicity formula. Examples of homothetically shrinking …

We consider one of the generic regimes of formation of singularities. We obtain a detailed description of a possibly small, but fixed, neighborhood of the blowup point, up to (and including) the blowup time, and find that it is mean convex. This confirms a conjecture by Ilmanen. And we find that the singularity is isol…

We study surfaces evolving by mean curvature flow (MCF). For an open set of initial data that are $C^3$-close to round, but without assuming rotational symmetry or positive mean curvature, we show that MCF solutions become singular in finite time by forming neckpinches, and we obtain detailed asymptotics of that singul…

Study shows instability of naked singularities in perfect fluid models.

Study of null mean curvature flow on de Sitter lightcone, related to 2d-Ricci flow.

We study the provenance of singularity formation under mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that if the mean curvature is uniformly bounded on any finite time interval, then no singularities can develop during that time under both mean curvature flow an…

Stability of singularity formation in Yang-Mills fields in higher dimensions.

For gravitational collapse, we observe a correspondence between region close to past null infinity and region close to central singularity. In line with this philosophy, we construct a new ansatz, with which we first present a 40-page self-contained proof of trapped surface formation in a far-from-center region. A syst…

This paper develops a local analogue of the ADHM construction, which characterises ASD instantons defined over smooth bounded domains inside Euclidean $\mathbb{R}^4$ diffeomorphic to the 4-ball, in terms of infinite dimensional Hilbert spaces and bounded Hermitian linear operators satisfying an analogue of the ADHM equ…

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…

Study neckpinch singularities in Ricci flow with cylindrical symmetry.

In this work we initiate the mathematical study of naked singularities for the Einstein vacuum equations in $3+1$ dimensions by constructing solutions which correspond to the exterior region of a naked singularity. A key element is our introduction of a new type of self-similarity for the Einstein vacuum equations. Con…

Study on singularities in Lagrangian mean curvature flow with special Lagrangian cones.

We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows - surfaces of revolution - in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additio…

The paper studies singularities in a complex flow related to mean curvature.

We show that the twisted Kähler-Ricci flow on a complex manifold X converges to a flow of moving free boundaries, in a certain scaling limit. This leads to a new phenomenon of singularity formation and topology change which can be seen as a complex generalization of the extensively studied formation of shocks in Hamilt…

We study high codimension mean curvature flow of a submanifold $\mathcal{M}^n$ of dimension $n$ in Euclidean space $\mathbb{R}^{n+k}$ subject to the quadratic curvature condition $|A|^{2}\leq c_n |H|^{2}, c _n = \min\{ \frac{4}{3n} , \frac{1}{n-2}\}$. This condition extends the notion of two-convexity for hypersurface…

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…

Study on four-dimensional Ricci solitons and multiply warped Ricci flow solutions.

This work addresses the {\em singularity formation} of complete non-compact solutions to the conformally flat Yamabe flow whose conformal factors have {\em cylindrical behavior at infinity}. Their singularity profiles happen to be {\em Yamabe solitons}, which are {\em self-similar solutions} to the fast diffusion equat…

We initiate the mathematical study of spherical collapse of self-gravitating charged scalar fields. The main result gives a complete characterization of the future boundary of spacetime, providing a starting point for studying the cosmic censorship conjectures. In general, the boundary includes two null components, one…

The paper generalizes envelope constructions for chords in circles, revealing complex singularities.

Characterizes limits of Ricci flows and their singularities.