In this paper we considerably extend the class of known α-minimizing hypercones using sub-calibration methods. Indeed, the improvement of previous results follows from a careful analysis of special cubic and quartic polynomials.
We show the area-minimality property of all homogeneous area-minimizing hypercones in Euclidean spaces (classified by Lawlor) following Lawson's original idea in his 72' Trans. A.M.S. paper "The equivariant Plateau problem and interior regularity". Moreover, each of them enjoys (coflat) calibrations singular only at th…
Characterizes harmonic morphisms preserving minimal submanifolds and finds novel area-minimising hypercones.
problem Understanding harmonic morphisms and their relationship to minimal submanifolds.
method Characterization of harmonic morphisms as weakly horizontally conformal maps preserving minimal submanifold equations, derivation of reduction properties for other co-dimensions, application to find novel area-minimising hypercones.
result Novel family of degree 4 area-minimising hypercones in R^m, m≥32.
The study proves that certain stable minimal hypersurfaces must be cylindrical.
problem Characterizing stable minimal hypersurfaces in Euclidean space.
method Analyzing the density at infinity and using stable area minimizing hypercone properties.
result Stable minimal hypersurfaces with specific conditions are cylindrical.
Rectifying curves on hypercones are geodesics, characterized in higher dimensions.
problem Characterizing rectifying curves in higher-dimensional spaces.
method Extending results from Chen (2017) to higher dimensions, using hypercones and hyperplanes.
result Rectifying curves on hypercones are geodesics, and these curves can be mapped to spherical curves in higher dimensions.
Study compares nodal sets of solutions to the Allen-Cahn equation.
problem Comparing nodal sets of solutions to the Allen-Cahn equation with conical asymptotics.
method Maximum principle for linearized operator on unbounded domains.
result Positive phase uniquely determines the solution and enforces global ordering.
Extends isoparametric foliations and area-minimizing cones in product manifolds.
problem Generalizing isoparametric foliations and area-minimizing cones in SnimesSn. method Analyzes isoparametric foliations and area-minimizing cones, extending known results.
result Extends known area-minimizing cones to codimension-two cases, yielding infinitely many families of area-minimizing subcones.
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].
Smooths out complex shapes into simpler forms.
problem Transforming complex shapes into simpler, smooth forms.
method Perturbing minimizing hypercones and viscosity mean convex cones into smooth, properly embedded hypersurfaces.
result Properly embedded smooth minimizing hypersurfaces and self-expanders are achieved.
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 …
Paper constructs flows converging to cones and foliations.
problem Understanding mean curvature flow convergence to cones and foliations.
method Constructs a family of mean curvature flows converging to cones and foliations under specific conditions.
result Flow converges to area minimizing, strictly stable hypercone and Hardt-Simon foliation of the cone.
Construct locally minimizing (1,2)-clusters with prescribed asymptotic geometry.
problem Minimizing clusters with prescribed asymptotic geometry.
method Develop a refined construction using the Hardt-Simon foliation.
result Produce a countably infinite family of distinct locally minimizing clusters asymptotic to a singular area-minimizing hypercone.
Extends a Liouville theorem for stable minimal hypersurfaces.
problem Stable minimal hypersurfaces in cylindrical cones.
method Analyzes the density and foliation of minimal hypersurfaces.
result Shows that stable minimal hypersurfaces are cylindrical.
The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.
problem Characterizing canal hypersurfaces in Lorentz-Minkowski 4-space.
method Analyzing geometric invariants and conditions for canal hypersurfaces.
result Characterizations of canal hypersurfaces, including flatness and minimality conditions.
The study characterizes canal hypersurfaces in Euclidean spaces and their curvature properties.
problem Characterizing canal hypersurfaces in Euclidean spaces.
method Analyzing canal hypersurfaces in Euclidean n-space, focusing on E4, computing curvature properties, and proving specific cases.
result Flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones, and minimal canal hypersurfaces are only generalized catenoids.
Paper proves minimal surfaces near quadratic cones have specific smooth structure.
problem Characterize minimal surfaces near quadratic cones.
method Analyzes n-varifolds in the unit ball close to a minimizing quadratic cone. result Singularities modeled on these cones determine the local structure of nearby minimal surfaces.
The paper studies stability and minimizing properties of higher codimensional surfaces in Euclidean space.
problem Stability and minimizing properties of higher codimensional surfaces in Euclidean space.
method Analyzes surfaces associated with the weighted area-functional and proves stability and minimization properties under specific conditions.
result Minimal cones with globally flat normal bundles are f-stable, and highly singular determinantal varieties and Pfaffian varieties are f-minimizing.