arXiv research

Study steady gradient Ricci solitons with cylindrical tangent flows at infinity.

Low-entropy surfaces can be flowed into spheres and cylinders.

In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in $\mathbb{R}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x,t)$, then there exists a positive $\varepsilon=\varepsilon(X)>0$ such that the flow is mean convex in a …

In this work we consider periodic spherically symmetric metrics of constant positive scalar curvature on the n-dimensional cylinder called pseudo-cylindric metrics. These metrics belong to the conformal class $[g_0]$ of the Riemannian product $S^1\times S^{n-1}$ : a circle of length $T$ crossed with the (n-1)-dimension…

Study resolves flow through cylindrical singularities, proving nonfattening.

We study the limiting case of the Krichever construction of orthogonal curvilinear coordinate systems when the spectral curve becomes singular. We show that the case when the curve is reducible and all its irreducible components are rational curves the construction procedure reduces to solving systems of linear equatio…

We prove the existence of Ricci flow starting from a class of metrics with unbounded curvature, which are doubly-warped products over an interval with a spherical factor pinched off at an end. These provide a forward evolution from some known and conjectured finite-time local singularities of Ricci flow, generalizing p…

This paper studies mean curvature flows near cylindrical singularities.

We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $2$-dimensiona…

Paper proves strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.

It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylin…

Proves uniqueness of cylindrical tangent flows in mean curvature flow.

Unique steady and expanding solitons with spherical links identified.

The paper studies cylindrical singularities in mean curvature flow and proves their local regularity.

We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding ve…

Cylindrical contact homology computed for links of simple singularities.

The paper proves cylindrical nature of singular minimal ruled surfaces.

This thesis studies moduli spaces of singular connections on 3-manifolds and manifolds with cylindrical ends. A Chern-Simons functional is defined for singular connections on 3-manifolds which are singular along a knot. The critical points of that Chern-Simons functional are flat singular connections. The Hodge-de Rham…

Researchers found cylindrical steady gradient solitons in 3D.

New special Lagrangian submanifolds with cylindrical tangent cones are constructed.

We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smoot…

Study neckpinch singularities in Ricci flow with cylindrical symmetry.

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…

Study on stability of cylindrical singularities in MCF of finite codimensions.

New bifurcation found in perturbations of non-generic closed self-shrinkers.

Resolves conjecture on cylindrical mean curvature flows in all dimensions.

The paper proves strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.

Study on parabolic points and cylindrical surfaces in Euclidean 3-space.

The abstract proves spherical surface decompositions with conical singularities.

Unique cylindrical tangent cone for Simons' hypersurface found.

New Ricci flows found with Einstein orbifolds at infinity.

Paper studies generic dynamics of MCFs with spherical singularities.

We prove a modified version of Turbiner's conjecture in three dimensions and we give a counter-example to the original conjecture. The Lie algebraic Schrödinger operators corresponding to flat metrics of a certain restricted type are shown to separate partially in either Cartesian, cylindrical or spherical coordinates.

Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In …

Minimal hypersurfaces with cylindrical tangent cones constructed and analyzed.

Classifies rank-one submanifolds in Euclidean space.

Study proves surfaces with constant curvature are simple shapes.

For the n-dimensional spherical pedal curve $ped_{γ,P}$ with respect to an n-dimensional spherical unit speed curve $γ$ and a given point $P \in S^n$, we define the spherical orthotomic curve of $γ$ relative to the point $P$, and classify singularities of spherical orthotomic curves.

Study proves uniqueness and rigidity of cylindrical self-shrinkers using Łojasiewicz inequalities.

We study the Yamabe invariants of cylindrical manifolds and compact orbifolds with a finite number of singularities, by means of conformal geometry and the Atiyah-Patodi-Singer $L^2$-index theory. For an $n$-orbifold $M$ with singularities $Σ_Γ = \{(\check{p}_1, Γ_1), ..., (\check{p}_s, Γ_s)\}$ (where each group $Γ_j<O…

New 1-parameter family of ovals identified in 4d Ricci flow classification.

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…

The study confirms a conjecture about critical points of smooth functions.

Study of spacelike singularities in spherical spacetimes with scalar matter.

Study null surfaces of pseudo-spherical curves in anti-de Sitter space.

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

Constructs a mean curvature flow with surgery for compact mean convex hypersurfaces.

Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.