arXiv research

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

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.

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…

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…

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

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…

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.

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.

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

Study Toda systems blowup masses linked to Weyl groups.

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

Global and local blowups of manifolds are proven equivalent.

Rotationally invariant Ricci flows are constructed and shown to converge to spacetimes.

Ancient ovals are key blowup limits in 3D Ricci flow near singularities.

Proves uniqueness of cylindrical tangent flows in mean curvature flow.

Study on bending energy of surfaces with curvature concentration, deriving new lower bounds.

We give conditions under which the blowup of an extremal Kähler manifold along a submanifold of codimension greater than two admits an extremal metric. This generalizes work of Arezzo-Pacard-Singer, who considered blowups in points.

The paper resolves singular foliations through a series of blowups.

Stable blowup solutions found for supercritical Yang-Mills equations.

Stable blowup profile identified for wave maps in all dimensions.

In real algebraic geometry, Lojasiewicz's theorem asserts that any integral curve of the gradient flow of an analytic function that has an accumulation point has a unique limit. Lojasiewicz proved this result in the early 1960s as a consequence of his gradient inequality. Many problems in calculus of variations are que…

We prove that on a closed surface, for any $c>0$, our min-max theory for prescribing mean curvature produces a solution given by a curve of constant geodesic curvature $c$ which is almost embedded, except for finitely many points, at which the solution is a stationary junction with integer density. Moreover, each smoot…

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

Study improves curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.

Study the pullbacks and blowups of Lie algebroids and related structures.

The paper finds asymmetric Type-I blowup solutions for Yang-Mills flow.

Wave maps with noise can lead to self-similar blowup from arbitrary initial data.

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

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

Functor connects symplectic and contact structures via cutting and blowups.