Newly confirmed area-minimizing properties of Lawson-Osserman cones.
problem Verifying the area-minimizing property of Lawson-Osserman cones.
method Analyzing cones of type (n, p, 2) constructed in [XYZ].
result All Lawson-Osserman cones of type (n, p, 2) are area-minimizing.
Characterizes eigenfunctions of Lawson-Osserman cone and proves its integrability.
problem Eigenfunctions and integrability of Lawson-Osserman cone.
method Characterization of eigenfunctions and proof of integrability.
result Lawson-Osserman cone is integrable, generating all Jacobi fields via rotations and translations.
Study finds unique self-expanders for mean curvature flow.
problem Finding solutions to mean curvature flow.
method Derived equation based on generalized Lawson-Osserman cone and modified equilibria theory.
result Existence and uniqueness of self-expanders proved.
Study shows instability of specific cone solutions in high-dimensional spaces.
problem Unstable solutions of minimal graphs in high codimension.
method Min-max technique applied to Euclidean spaces.
result First examples of non-smooth unstable minimal graphs.
To study the Lawson-Osserman's counterexample to the Bernstein problem for minimal submanifolds of higher codimension, a new geometric concept, submanifolds in Euclidean space with constant Jordan angles(CJA), is introduced. By exploring the second fundamental form of submanifolds with CJA, we can characterize the Laws…
Develops a more general scheme for constructing non-parametric minimal cones.
problem Minimal surface equations and their solutions.
method Using harmonic Riemannian submersions and homothetic minimal immersions.
result Uncover a constellation of uncountably many non-parametric minimal cones.
Extends Smale's principle to produce minimal graphs with singularities.
problem Creating minimal graphs with isolated singularities in higher dimensions.
method Extends Smale's singular bridge principle to arbitrary codimension and applies it to specific minimal cones.
result Produces a minimal graph in 7D with any number of isolated singularities.
Recent progress on minimal surface system and cones in Euclidean spaces.
problem Exploring the Dirichlet problem for minimal surfaces and cones.
method Systematic developments and new families of minimizing cones.
result New families of minimizing cones of different types.
Proves planar Lipschitz critical points of area functional are smooth.
problem Lawson-Osserman conjecture about smoothness of critical points.
method Outer variations to prove smoothness of critical points.
result Proves conjecture for planar case.
New boundary functions yield infinitely many solutions for minimal surfaces.
problem Existence of multiple solutions for minimal surfaces.
method Lawson-Osserman constructions and Dirichlet problem analysis.
result Infinitely many analytic solutions and at least one nonsmooth solution.
We derive estimates of the Hessian of two smooth functions defined on Grassmannian manifold. Based on it, we can derive curvature estimates for minimal submanifolds in Euclidean space via Gauss map. In this way, the result for Bernstein type theorem done by Jost and the first author could be improved.
Introduces holed cone structures to generalize cone structures on 3-manifolds.
problem Generalizing cone structures to 3-manifolds with irreducible holonomy representations.
method Introduces holed cone structures and considers their deformation space.
result The deformation space of holed cone structures is a covering space of the character variety.
Paper shows examples of hyperbolic cone structures degenerating with decreasing cone angles.
problem Degeneration of hyperbolic cone structures with specific cone angles.
method Constructed examples of hyperbolic cone structures on a certain alternating link in the thickened torus.
result Example of degeneration of hyperbolic cone structures with decreasing cone angles less than 2π.
Unique cylindrical tangent cone for Simons' hypersurface found.
problem Uniqueness of cylindrical tangent cones for area-minimizing hypersurfaces.
method Developed a new Lojasiewicz inequality for non-isolated singularities.
result Cylindrical tangent cone for Simons' hypersurface is unique.
This paper studies geodesics and uniqueness of cscK cone metrics.
problem Uniqueness of constant scalar curvature Kahler cone metrics.
method Introduction of weighted function spaces, construction of cone geodesics, detailed asymptotic analysis of cscK cone metrics, linear theory for Lichnerowicz operator.
result The cscK cone metric is unique up to automorphisms.
Given a closed orientable Euclidean cone 3-manifold C with cone angles less than or equal to pi, and which is not almost product, we describe the space of constant curvature cone structures on C with cone angles less than pi. We establish a regeneration result for such Euclidean cone manifolds into spherical or hyperbo…
Study strict stability of cones with isolated singularities.
problem Stability of cones with isolated singularities.
method Analyzes special Lagrangian and coassociative cones, provides examples for the complex case.
result Proves strict stability for special Lagrangian and coassociative cones.
The paper lifts Lagrangian immersions to cones in complex space.
problem Creating Lagrangian cones from immersions in complex projective space.
method Developing a method to lift immersions to cones, producing examples and analyzing projections.
result Examples of Lagrangian cones and special cones are produced, with projections showing few transverse double points.
Weiss and, independently, Mazzeo and Montcouquiol recently proved that a 3--dimensional hyperbolic cone-manifold (possibly with vertices) with all cone angles less than 2π is infinitesimally rigid. On the other hand, Casson provided 1998 an example of an infinitesimally flexible cone-manifold with some of the cone an…
The study finds billiard trajectories with infinitely many reflections in certain cones.
problem Existence of billiard trajectories with infinitely many reflections.
method Analysis of C3 convex cones and elliptic cones in R3. result Existence of C2 convex cones with billiard trajectories having infinitely many reflections. New cones found in sphere foliations, minimizing in most dimensions.
problem Finding new minimizing cones in sphere foliations.
method Analyzing isoparametric foliations and their associated minimal surfaces.
result Most cones over focal submanifolds and products of minimal isoparametric hypersurfaces are minimizing.
The SCMU algorithm computes cone factorizations for symmetric cones, improving upon existing methods.
problem Computing cone factorizations for symmetric cones in optimization.
method Introduces and analyzes the symmetric-cone multiplicative update (SCMU) algorithm.
result The SCMU algorithm non-decreases the squared loss objective.
We describe some properties of noncompact Euclidean cone manifolds with cone angles less than c less than 2pi and singular locus a submanifold. More precisely, we describe its structure outside a compact set. As a corollary we classify those with cone angles less than 3pi/2 and those with all cone angles equal to 3pi/2…
We prove that every closed oriented 3-manifold admits a hyperbolic cone-manifold structure with cone-angle arbitrarily close to 2pi.
Smooth convergence to a sphere from cone-shaped surfaces.
problem Evolution of surfaces inside a cone using inverse mean curvature flow.
method Inverse mean curvature flow applied to star-shaped hypersurfaces meeting a convex cone perpendicularly.
result Smooth convergence to a sphere over time.
New Calabi-Yau metrics with conical singularities are created near complex lines.
problem Creating Calabi-Yau metrics with conical singularities near complex lines.
method Using branched covering arguments to construct metrics with conical singularities.
result Calabi-Yau metrics with unstable conical singularities are successfully created.
Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.
problem Finding metrics for toric Kähler cones with conical singularities.
method Parametrized family of Calabi-Yau cone metrics with conical singularities.
result Any toric Calabi-Yau cone metric with conical singularities belongs to this optimal family.
The Lee-Gauduchon cone is a convex cone of cohomology classes for complex manifolds.
problem Understanding the Lee-Gauduchon cone for complex manifolds.
method Analyzing the Lee-Gauduchon cone as a convex cone of cohomology classes.
result The Lee-Gauduchon cone is a bimeromorphic invariant.
The paper solves area minimizing problems in special geometric cones.
problem Area minimizing problems in conformal cones.
method Defining NCM condition, proving existence of minimal graphs, solving in specific cones.
result Existence of minimal graphs in mean convex conformal cones.
The paper computes fundamental groups of warped cones and finds expanders.
problem Understanding the fundamental groups of warped cones.
method Computing discrete fundamental groups of warped cones.
result Warped cones can be coarsely non-equivalent to box spaces.
Lower bounds on cone density for nontrivial complements in low dimensions.
problem Finding density limits for minimal cones with nontrivial complements.
method Proving lower bounds on cone density for cones of dimensions less than seven with nontrivial complements.
result Established lower bounds on cone density for minimal cones with nontrivial complements in dimensions less than seven.
Criterion for surfaces in Heisenberg group to be graphs using flat cones.
problem Characterizing surfaces in Heisenberg group as graphs.
method Using planar cones to define intrinsic rectifiability.
result Criterion for topological surfaces to be intrinsic Lipschitz graphs.
The study of limit cones for multi-Fuchsian representations in (PSL2R)d.
problem Characterizing the structure of limit cones for multi-Fuchsian representations.
method Analysis of normalized multi-lengths and convex cones in R≥0d. result Different regimes of limit cones exist, with some having finite sides and others dense extremal rays.
We prove 3-dimensional hyperbolic cone-manifolds are geometrically inflexible: a cone-deformation of a hyperbolic cone-manifold determines a bi-Lipschitz diffeomorphism between initial and terminal manifolds in the deformation in the complement of a standard tubular neighborhood of the cone-locus whose pointwise bi-Lip…
We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on selfdual cones, Weyl groups of certain Kac-Moody algebras …
By a classical theorem of Gallot (1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is dec…
Given a geometrically finite hyperbolic cone-manifold, with the cone singularity sufficiently short, we construct a one parameter family of cone-manifolds decreasing the cone angle to zero. We also control the geometry of this one parameter family via the Schwarzian derivative of the projective boundary and the length …
Study connects contact structures to cone geodesics and contactomorphisms.
problem Understanding contact structures on cone geodesics.
method Review and generalize cone geodesics to contact manifolds, establish correspondence with contactomorphisms.
result Established correspondence between contactomorphisms and cone structures.
Researchers prove uniqueness of certain cylindrical tangent cones for special Lagrangians.
problem Proving uniqueness of cylindrical tangent cones for special Lagrangians.
method Analyzing exact special Lagrangian submanifolds with multiplicity one and cylindrical tangent cones.
result The cylindrical tangent cones are unique under specific conditions.
Analytic sets with unique infinite tangent cone are algebraic.
problem Characterizing analytic sets with unique infinite tangent cones.
method Analytic and algebraic set properties, degree of complex algebraic sets.
result Degree of Lipschitz normally embedded sets equals their infinite tangent cone degree.
The paper proves conditions for C1 regularity of definable sets using tangent cones and paratangent cones.
problem Conditions for C1 regularity of definable sets in o-minimal structures. method Analysis of tangent and paratangent cones to establish C1 regularity. result Equivalence of three conditions for C1 regularity of definable sets. We study special Lagrangian cones in $\C^n$ with isolated singularities. Our main result constructs an infinite family of special Lagrangian cones in $\C^3$ each of which has a toroidal link. We obtain a detailed geometric description of these tori. We prove a regularity result for special Lagrangian cones in $\C^3$ wi…
Researchers describe a new Thom form for mapping cones.
problem Developing a new Thom form for mapping cones.
method Using the mapping cone covariant derivative and Berezin integral, they explicitly write down the Thom form.
result The Thom form is closed with respect to the mapping cone differentiation, integrates to 1 along the fiber, and satisfies the transgression formula.
Unified rigidity theorem for Plateau surfaces in Bn.
problem Rigidity of free-boundary minimal surfaces in Bn. method Analyzing conformal free-boundary minimal immersions of Plateau model cones.
result Every conformal free-boundary minimal immersion of the flat T-cone into Bn is congruent to the flat T-cone. Uniqueness proven for cylindrical tangent cones in high dimensions.
problem Proving uniqueness of cylindrical tangent cones in high dimensions.
method Analyzing area-minimizing hypersurfaces in R^9.
result Uniqueness of cylindrical tangent cones Cp,qimesR in R9. The paper determines the automorphism group of p-cones and shows they are not self-dual for peq2.
problem The non-self-duality of p-cones for peq2. method Analyzes the automorphism group of p-cones and investigates the duality theory of p-cones under different inner products. result No inner product can make a p-cone self-dual for peq2. Find limiting sets for digital cones and suspensions.
problem Digital topology cone and suspension constructions.
method Identify (m, n)-limiting sets, especially (0, 0)-freezing sets.
result Discover (0, 0)-limiting sets for digital cones and suspensions.
Hyperkahler manifolds with round Kahler cones have unique bimeromorphic models.
problem Existence of round Kahler cones in hyperkahler manifolds.
method Analyzing the Kahler cone and its relation to the Bogomolov-Beauville-Fujiki form.
result Maximal holonomy hyperkahler manifolds with b2>4 have deformations with round Kahler cones.