arXiv research

Extends Smale's principle to produce minimal graphs with singularities.

Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.

Study on minimizing singular capillary cones with stability and instability results.

We show that every area-minimizing hypercone and every oriented Lawlor cone in [Law91] can be realized as a tangent cone at a point of some homologically area-minimizing singular compact submanifold. In particular this generalizes the result of N. Smale [Sma99].

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…

Study proves uniqueness of tangent cones for area-minimizing currents in higher codimensions.

Study on singularities in area-minimizing currents, proving unique tangent cones and rectifiability.

The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.

Study area-minimizing hypersurfaces in singular manifolds with nonnegative scalar curvature.

Uniqueness proven for stable hypersurface tangent cones.

Paper finds formulas for minimizing perimeter in special spaces, proving key dimensions and existence.

Study on flat singular points of area-minimizing currents, defining a singularity degree.

We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (e.g. those of Reifenberg, Taylor and Whit…

The paper studies deformations of singular minimal hypersurfaces in dimensions 7 and above.

Construct locally minimizing $(1,2)$-clusters with prescribed asymptotic geometry.

Hardt-Simon proved that every area-minimizing hypercone $\mathbf{C}$ having only an isolated singularity fits into a foliation of $\mathbb{R}^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $\mathbf{C}$. In this paper we prove that if a stationary $n$-varifold $M$ in the unit ball $B_1 \subset \mathbb{R}^…

This paper solves minimal surface equations near Hardt-Simon foliations.

Minimal hypersurfaces with cylindrical tangent cones constructed and analyzed.

The study examines singularities in flows with curvature bounds and identifies unique tangent flows.

Study confirms boundedness of certain singularities in log Fano geometry.

Minimal Lagrangian diffeomorphisms between spherical surfaces are rigid.

New proof of a unique 3-part partition in 8D space.

We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one n-dim…

Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.

Study properties of solutions with singularities in the negative cone.

We prove the existence of a minimal diffeomorphism isotopic to the identity between two hyperbolic cone surfaces $(Σ,g_1)$ and $(Σ,g_2)$ when the cone angles of $g_1$ and $g_2$ are different and smaller than $π$. When the cone angles of $g_1$ are strictly smaller than the ones of $g_2$, this minimal diffeomorphism is u…

We prove that a minimizer of the Yamabe functional does not exist for a sphere $\mathbb{S}^n$ of dimension $n \geq 3$, endowed with a standard edge-cone spherical metric of cone angle greater than or equal to $4π$, along a great circle of codimension two. When the cone angle along the singularity is smaller than $2π$, …

Unified rigidity theorem for Plateau surfaces in $\mathbb{B}^n$.

Rectifies flat singular points for area-minimizing currents.

We study singularities of Lagrangian mean curvature flow in $\C^n$ when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonia…

We extend the results of Hardt and Simon on area-minimizing cones to prove that isolated singularities of stationary one-sided area-minimizing hypersurfaces can be locally perturbed away on the side that they are minimizing.

New cones in 4D space found with minimal mass.

We adapt the method of Simon [JDG '93] to prove a $C^{1,α}$-regularity theorem for minimal varifolds which resemble a cone $\bf{C}_0^2$ over an equiangular geodesic net. For varifold classes admitting a "no-hole" condition on the singular set, we additionally establish $C^{1,α}$-regularity near the cone $\bf{C}_0^2 \ti…

We show that directed minimal cones in (n+1)-dimensional Euclidean space which have at most one singularity are - besides the trivial cases: empty set, whole space - half spaces. Using blow-up techniques, this result can be used to get C^{1,lambda}-regularity for the measure-theoretic boundary of almost minimal Cacciop…

Given a klt singularity $x\in (X, D)$, we show that a quasi-monomial valuation $v$ with a finitely generated associated graded ring is the minimizer of the normalized volume function $\widehat{\rm vol}_{(X,D),x}$, if and only if $v$ induces a degeneration to a K-semistable log Fano cone singularity. Moreover, such a mi…

The Kähler-Ricci flow near conical singularities is described with a $C/t$ curvature bound.

Unique cylindrical tangent cone for Simons' hypersurface found.

Constructs graphs with singularities in a special space.

New Ricci flow solutions found with rotational symmetry and cone-like singularities.

Paper removes singularities from compact area minimizers in positive scalar curvature manifolds.

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

We prove that tangent cones at singular boundary points of a two-dimensional current almost area minimizing are unique. Following the ideas exposed by White in [8], the result is achieved by combining a suitable epiperimetric inequality and an almost-monotonicity formula for the mass at boundary points.

Paper proves rigidity and index of Y-cones in unit ball.

Minimal surfaces in spheres constructed from symmetry reductions of ODEs.

We prove that the density of a topologically nontrivial, area-minimizing hypercone with an isolated singularity must be greater than the square root of 2. The Simons' cones show that this is the best possible constant. If one of the components of the complement of the cone has nontrivial kth homotopy group, we prove a …

We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) convex Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than $π$. We interpret this result in terms of Teichmüller theory, and prove the existence of …

We analyze the asymptotic behavior of a $2$-dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Rie…

In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or e…