arXiv research

We study almost-calibrated, $O(n)$-equivariant Lagrangian mean curvature flow in $\mathbb{C}^n$, 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…

Study shows different behaviors of noncompact hypersurfaces under mean curvature flow.

Strict type-II blowup in harmonic map flow is proven to have Hölder continuous body map.

Numerical simulations show stability of Type-II singularities in noncompact hypersurfaces.

Study cohomogeneity-one Lagrangian mean curvature flow in complex spaces.

Study proves energy critical heat equation solutions are Type I blowups for n ≥ 7.

We consider the energy-supercritical harmonic map heat flow from $\mathbb{R}^d$ into $\mathbb{S}^d$, under an additional assumption of 1-corotational symmetry. We are interested by the 7 dimensional case which is the borderline between the Type I blowup regime. We construct for this problem a stable finite time blowup …

Kähler blowups can have scalar curvature arbitrarily close to any given metric.

The paper studies Ricci flow with specific curvature and volume constraints, proving convergence and curvature bounds.

Study curvature of blown-up manifold, finds negative holomorphic sectional curvature.

Uniqueness of nondegenerate blowups for planar networks shown.

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…

New method proves uniqueness in mean curvature flow.

Proves uniqueness of blowups for forced mean curvature flow.

The paper proves rigidity theorems for Type II singularities in Lagrangian flows.

Estimates mean curvature flow with geometric bounds.

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…

We consider the energy supercritical harmonic heat flow from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear heat equation $$\partial_t u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \…

Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly resemble cones. For mean curvature flow, by the combined work of Huisken, Ilmanen, and White, singularities weakly resembl…

Complete shrinking soliton found on a specific complex surface.

Estimates for VPMCF show ancient MCF solutions and finite-time behavior.

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…

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…

We show uniqueness of cylindrical blowups for mean curvature flow in all dimension and all codimension. Cylindrical singularities are known to be the most important; they are the most prevalent in any codimension. Mean curvature flow in higher codimension is a nonlinear parabolic system where many of the methods used f…

Ancient solutions of Lagrangian mean curvature flow in C^n naturally arise as Type II blow-ups. In this extended note we give structural and classification results for such ancient solutions in terms of their blow-down and, motivated by the Thomas-Yau Conjecture, focus on the almost calibrated case. In particular, we c…

The paper examines translating solitons and their relation to Lagrangian mean curvature flows with zero Maslov class.

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

We show that the blowup of an extremal Kahler manifold at a relatively stable point in the sense of GIT admits an extremal metric in Kahler classes that make the exceptional divisor sufficiently small, extending a result of Arezzo-Pacard-Singer. We also study the K-polystability of these blowups, sharpening a result of…

Blowups of Kähler manifolds with extremal metrics inherit such metrics under stability conditions.

We develop some estimates under the Ricci flow and use these estimates to study the blowup rates of curvatures at singularities. As applications, we obtain some gap theorems: $\displaystyle \sup_X |Ric|$ and $\displaystyle \sqrt{\sup_X |Rm|} \cdot \sqrt{\sup_X |R|}$ must blowup at least at the rate of type-I. Our estim…

Proves existence of shrinkers via mean curvature flow.

Ancient Ricci flows are identified without curvature sign condition.

In this paper we investigate the singularities of Lagrangian mean curvature flows in $\mathbf{C}^m$ by means of smooth singularity models. Type I singularities can only occur at certain times determined by invariants in the cohomology of the initial data. In the type II case, these smooth singularity models are asympto…

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 …

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…

Study constant mean curvature tori in R^3 using spectral data and Whitham deformations.

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

Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $\partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so that under these new metrics, $K$ is the scalar curvature and $c$ is …

We use elementary methods to construct a minimal lamination of the interior of a positive cone in R3.

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…

Study of curvature blow-up in noncompact hypersurfaces using mean curvature flow.

Study on line bundle flow on Kähler surfaces converging to a singular solution.

Study shows instability of specific cone solutions in high-dimensional spaces.

Study of curve shortening flow for twisted curves, defining curvature-torsion entropy.

In previous work, Angenent, Isenberg, and Knopf created type-II Ricci flow neckpinch singularities. In this paper we construct solutions to Ricci flow whose initial data is the singular metric resulting from these singularities. We show in particular that the curvature decreases at the same rate at which it blew up. Th…

Study Toda systems blowup masses linked to Weyl groups.

In this paper, we introduce a monotonicity formula for the mean curvature flow. We also apply this monotonicity formula to study the asymptotic behavior of eternal solutions.

Paper analyzes blowup of regularized Jang solutions and constant expansion surfaces.