Survey on algebraic minimal cones and nonassociative algebras.
problem Exploring algebraic minimal cones and nonassociative algebras.
method Recollections and recent developments in the field.
result Dedication to Vladimir Miklyukov's memory.
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A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,051 papers · 148 categories
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48 results for “algebraic minimal cones”
We construct two infinite families of algebraic minimal cones in . The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byp…
Observing a linear superposition principle, a family of new minimal hypersurfaces in Euclidean space is found, as well as that linear combinations of generalized helicoids induce new algebraic minimal cones of arbitrarily high degree.
Extends isoparametric foliations and area-minimizing cones in product manifolds.
problem Generalizing isoparametric foliations and area-minimizing cones in .
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.
The paper explores cone structures and their connections to parabolic geometries in complex manifolds.
problem Understanding cone structures and their properties in complex manifolds.
method Analyzes cone structures induced by parabolic geometries and VMRT structures, focusing on local invariants.
result Establishes a local differential-geometric version of a global algebraic-geometric recognition theorem.
The paper studies connections and Finsler geometry on JB-algebra structure groups.
problem Investigating geometric structures on JB-algebra structure groups.
method Endowing the structure group with a connection and Finsler metric, computing quantities, and proving minimality of paths.
result Established the Finsler metric and distance on the cone of a JB-algebra.
We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.
New algorithm for online optimization over symmetric cones, unifying previous methods.
problem Online convex optimization over symmetric cones.
method Symmetric-Cone Multiplicative Weights Update (SCMWU) algorithm.
result SCMWU is a no-regret algorithm.
New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.
problem Classifying Hsiang algebras and understanding their properties.
method Introducing quasicomposition and tripling constructions to study Hsiang algebras.
result The triple of a quasicomposition algebra is an exceptional Hsiang algebra.
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.
Proves algebraic cones for LCK manifolds with potential.
problem Characterizing algebraic cones for LCK manifolds.
method Analyzes LCK manifolds as complex submanifolds of Hopf manifolds and covers, proving algebraicity of the resulting cones.
result Affine algebraic structure on cones is independent of manifold choice.
Given a klt singularity , we show that a quasi-monomial valuation with a finitely generated associated graded ring is the minimizer of the normalized volume function , if and only if induces a degeneration to a K-semistable log Fano cone singularity. Moreover, such a mi…
Holomorphic tensors on algebraic cones are invariant under certain group actions.
problem Holomorphic tensors on products of algebraic cones
method Using algebraic structures and embeddings
result Holomorphic tensors are invariant under group actions
Study left-invariant pseudo-Riemannian metrics on Lie groups focusing on null cone Lie algebras.
problem Characterize left-invariant pseudo-Riemannian metrics on Lie groups in the null cone.
method Use bracket flow on Lie algebra to study metrics on Lie groups.
result Classify all cases of null cone Lie algebras in signatures (1,q) and (2,q).
Develops a new method to study algebraic tangent cones of sheaves using valuations.
problem Analyzing tangent cones of torsion-free sheaves on algebraic varieties.
method Introduces a slope stability theory and uses it to define a canonical tangent cone for quasi-regular valuations.
result Shows the existence of a canonical tangent cone for torsion-free sheaves, up to equivalence.
New method classifies special Vinberg cones of rank 4.
problem Classifying special Vinberg cones of rank 4.
method Using Clifford Nil-algebras and directed acyclic graphs.
result Explicit classification of rank 4 special Vinberg cones.
Kähler-Ricci flows' tangent cones are algebraic varieties.
problem Understanding the structure of Kähler-Ricci flows' tangent cones.
method Analyzing tangent cones as normal affine algebraic varieties and using Hörmander's estimate.
result The regular set of tangent cones coincides with the algebraic regular set.
We establish a classification of cubic minimal cones in case of the so-called radial eigencubics. Our principal result states that any radial eigencubic is either a member of the infinite family of eigencubics of Clifford type, or belongs to one of 18 exceptional families. We prove that at least 12 of the 18 families a…
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 …
Study of -vector cones in cluster algebras from weighted orbifolds.
problem Determine the closure of -vector cones in cluster algebras.
method Analyzing -vector cones in a cluster algebra defined from a weighted orbifold.
result Closure of the union of -vector cones is except for specific weighted orbifolds.
Study shows weak homotopy equivalences for complete minimal surfaces.
problem Understanding complete minimal surfaces and their properties.
method Analyzes algebraic null immersions and conformal minimal immersions.
result Inclusion and differential mappings are weak homotopy equivalences.
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.
Proves no minimal hypersurfaces in regions bounded by minimal cones.
problem Existence of minimal hypersurfaces in bounded regions.
method Analyzes minimal hypersurfaces in Euclidean space.
result No minimal hypersurfaces in regions bounded by unstable minimal cones.
The study finds conditions for area-minimizing cones over submanifolds.
problem Conditions for area-minimizing cones over submanifolds.
method General configuration results for area-minimizing cones.
result Cone over the minimal product of submanifolds and spheres are area-minimizing.
Associated with isoparametric foliations of unit spheres, there are two classes of minimal surfaces minimal isoparametric hypersurfaces and focal submanifolds. By virtue of their rich structures, we find new series of minimizing cones. They are cones over focal submanifolds and cones over suitable products among th…
Study left-invariant pseudo-Riemannian metrics on Lie groups using moving bracket approach.
problem Characterize left-invariant pseudo-Riemannian metrics on Lie groups with vanishing scalar curvature invariants.
method Using the moving bracket approach, analyze Lie algebras of dimensions ≤ 6 and semi-simple Lie algebras.
result All Lie algebras of dimension ≤ 6 except 3-dimensional solvable Lie algebra are in the null cone, leading to VSI metrics.
This paper proves area-minimizing cones over products of Grassmannian manifolds.
problem Proving area-minimizing cones over products of Grassmannian manifolds.
method Using Hermitian orthogonal projectors and carefully computing the Jacobian.
result Cone over minimal products of Grassmannian manifolds are area-minimizing.
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…
The Unique Tangent Cone Property for Weakly Holomorphic Maps into Projective Algebraic Varietiesmath.DG
Paper proves unique tangent maps for complex maps into algebraic varieties.
problem Proving uniqueness of tangent maps for complex maps into algebraic varieties.
method Developed techniques to prove uniqueness of tangent maps for weakly holomorphic and locally approximable maps.
result Established unique tangent cone property for weakly holomorphic maps into projective algebraic varieties.
Tangent cones at infinitymath.GT
Study of tangent cones at infinity for algebraic sets.
problem Characterizing algebraic sets based on their tangent cones at infinity.
method Definition and analysis of tangent cones and , proving properties and relations.
result Affine linear subspace characterization based on 's dimension.
Constructs moduli spaces for Calabi-Yau cones and Sasaki-Einstein manifolds.
problem Proper moduli spaces for K-polystable Q-Fano cones and their links.
method Algebraic construction using local normalized volume and higher Θ-stable reduction.
result Alternative algebraic proof of proper moduli spaces for Q-Fano varieties.
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.
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.
Paper proves minimal surfaces near quadratic cones have specific smooth structure.
problem Characterize minimal surfaces near quadratic cones.
method Analyzes -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.
Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.
problem Proves existence and uniqueness of minimal capillary cones with bi-orthogonal symmetry.
method Solves a nonlinear free boundary equation parametrized by the contact angle and uses monotonicity properties.
result Demonstrates that minimizing capillary hypersurfaces can have singularities in codimension 7.
Cone structures over minimal products can't be calibrated smoothly.
problem Calibrating cones over minimal products with smooth calibrations.
method Extending a key result from [Zha26], showing obstruction.
result Cone structures over minimal products cannot be calibrated by smooth calibrations.
Unique minimal surfaces near quadratic cones are identified.
problem Identifying minimal surfaces near quadratic cones.
method Analyzing minimal hypersurfaces inside the unit ball with perturbed boundary conditions.
result Minimal surfaces are uniquely determined by their boundary conditions.
Bounded symmetric domains are biholomorphic to tube domains over Finsler symmetric cones.
problem Characterizing biholomorphic mappings between tube domains and bounded symmetric domains.
method Analyzing properties of Finsler symmetric cones and unital JB-algebras.
result Tube domains over Finsler symmetric cones are biholomorphic to bounded symmetric domains.
Minimal hypersurfaces with cylindrical tangent cones constructed and analyzed.
problem Constructing minimal hypersurfaces with specific geometric properties.
method Constructing minimal hypersurfaces with cylindrical tangent cones and proving unique continuation results.
result Existence and properties of minimal hypersurfaces with cylindrical tangent cones.
Study area minimizing currents in conformal cones, solving Dirichlet problems.
problem Area minimizing currents in conformal cones.
method Minimizing problem of area functionals, Dirichlet problem of minimal surface equations.
result Existence and uniqueness of area minimizing currents in a wide class of conformal manifolds.
Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.
problem Existence and classification of conical Calabi-Yau structures on horospherical cones.
method Variational approach to establish equivalence between volume minimization and existence of conical Calabi-Yau structures.
result Existence of many irregular horospherical cones with mild singularities.
Lightlike hypersurfaces in cone structures minimize time.
problem Finding time-minimizing paths in cone structures.
method Defining lightlike hypersurfaces and proving their foliation by cone geodesics.
result Lightlike hypersurfaces in globally hyperbolic spacetimes are time-minimizing.
It has been 40 years since Lawson and Osserman introduced the three minimal cones associated with Dirichlet problems in their 1977 Acta paper [LO77]. The first cone was shown area-minimizing by Harvey and Lawson in the celebrated paper [HL82]. In this paper, we confirm that the other two are also area-minimizing. In fa…
Unified rigidity theorem for Plateau surfaces in .
problem Rigidity of free-boundary minimal surfaces in .
method Analyzing conformal free-boundary minimal immersions of Plateau model cones.
result Every conformal free-boundary minimal immersion of the flat -cone into is congruent to the flat -cone.
Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.
problem Approximation and interpolation for regular immersions directed by algebraically elliptic cones.
method Uses homotopy-theoretic necessary and sufficient conditions for approximation and interpolation.
result Homotopy-theoretic conditions for approximation and interpolation are satisfied in many cases of interest.
The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of Lorentzian holonomy algebras was obtained recently. In the present paper weakly-irreduci…
In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved the existence of self-expanding hypersurfaces with prescribed tangent cones at infinity. If the cone is -regular and mean convex (but not area-minimizing…
We prove the existence of the analog of Lawson's minimal cones for a notion of nonlocal minimal surface introduced by Caffarelli, Roquejoffre and Savin, and establish their stability/instability in low dimensions. In particular we find that there are nonlocal stable minimal cones in dimension 7, in contrast with the ca…