arXiv research

Unique cylindrical tangent cone for Simons' hypersurface found.

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

Paper constructs flows converging to cones and foliations.

Optimizes the first $p$-Laplacian eigenvalue on hypersurfaces.

Researchers compute Chern-Simons invariants for a specific type of knot orbifolds.

Uniqueness proven for cylindrical tangent cones in high dimensions.

In this paper, we prove that, up to similarity, there are only two minimal hypersurfaces in $\mathbb{R}^{n+2}$ that are asymptotic to a Simons cone, i.e. the minimal cone over the minimal hypersurface $\sqrt{\frac pn}\mathbb{S}^p\times \sqrt{\frac{n-p}n} \mathbb{S}^{n-p}$ of $\mathbb{S}^{n+1}$

Paper proves minimal surfaces near quadratic cones have specific smooth structure.

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

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

New special Lagrangian submanifolds with multiple conical singularities found.

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

Extends isoparametric foliations and area-minimizing cones in product manifolds.

The paper proves regularity for minimal surfaces near polyhedral cones.

In a seminal paper published in $1968$, J. Simons proved that, for $n\leq 5$, the Euclidean (minimal) cone $CM$, built on a closed, oriented, minimal and non totally geodesic hypersurface $M^n$ of $\mathbb S^{n+1}$ is unstable. In this paper, we extend Simons' analysis to {\em warped} (minimal) cones built over a close…

Study calculates volumes and Chern-Simons invariants for 2-bridge knot orbifolds.

We compute Chern-Simons invariants for a specific knot's hyperbolic orbifolds.

Study shows expanding Ricci solitons from specific metric cones.

Partial boundary regularity for area-minimizing currents at tangential boundary points.

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 …

Extends a Liouville theorem for stable minimal hypersurfaces.

Study confirms $C^1$ regularity for convex functionals with bounded degeneracy set.

In the paper [1] (arXiv:math/0408333) the authors discuss two possible definitions of the relative Cheeger-Simons characters, the second one fitting into a long exact sequence. Here we relate that picture to the one of the relative Deligne cohomology groups, defined via the mapping cone: we show that there are three me…

Rigidity theorem for scalar curvature on odd-dimensional singular manifolds.

New epiperimetric inequality for cones with improved regularity results.

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…

We calculate the Chern-Simons invariants of the twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of the twist knot cone-manifold structures. Following the general instruction of Hilden, Lozano, and Montesinos-Amilibia, we here present the concrete formulae and calc…

Study removes singularities from area-minimizing surfaces.

Stable saddle solutions found for specific dimensions of the Allen-Cahn equation.

Paper analyzes Velázquez's solution to a singularity in mean curvature flow.

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 area minimizing currents in Riemannian manifolds, proving unique structure and decay.

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

New solutions found for a free boundary problem using refined methods.

Smooth approximations near singularities of constant mean curvature surfaces are found.

The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.

The paper studies stability and minimizing properties of higher codimensional surfaces in Euclidean space.

A brief review on the progress made in the study of Chern-Simons gauge theory since its relation to knot theory was discovered ten years ago is presented. Emphasis is made on the analysis of the perturbative study of the theory and its connection to the theory of Vassiliev invariants. It is described how the study of t…

We introduce certain relative differential characters which we call Cheeger-Chern-Simons characters. These combine the well-known Cheeger-Simons characters with Chern-Simons forms. In the same way as the Cheeger-Simons characters generalize Chern-Simons invariants of oriented closed manifolds, the Cheeger-Chern-Simons …

Proves Simon's conjecture for 3-manifolds.

Computing Chern-Simons action for perturbed Dirac triples

New approach connects 3D Chern-Simons theory to spectral networks.

In this paper we study several aspects of the geometry of conformally stationary Lorentz manifolds, and particularly of GRW spaces, due to the presence of a closed conformal vector field. More precisely, we begin by extending to these spaces a result of J. Simons on the minimality of cones in Euclidean space, and apply…

Resurgence analysis of $SU(2)$ Chern-Simons on a specific homology sphere.

Alternative proof of Michael-Simon-Sobolev inequality using optimal transport.

The paper finds Chern-Simons forms for specific classes in simplicial de Rham complex.

Geometrically constructs dilogarithm from Chern-Simons theory.

Formulae for volume and Chern-Simons invariant of hyperbolic knot orbifolds.