Bound on equivariant index for min-max surfaces.
problem Bounding the index of equivariant min-max surfaces.
method Equivariant min-max procedure with group action.
result Equivariant index bound by number of parameters.
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48 results for “equivariant minimization”
New minimal discs and annuli found in ellipsoids.
problem Constructing minimal surfaces in ellipsoids.
method Equivariant variational methods.
result At least three distinct embedded free boundary minimal annuli in ellipsoids.
Minimal dimension equivariant embeddings of real and complex flag manifolds into Euclidean spacesmath.DG
Minimal dimensions found for flag manifolds embeddings.
problem Finding the smallest dimensions for flag manifolds embeddings.
method Equivariant embeddings of orthogonal and unitary groups acting on real and complex flag manifolds.
result Minimal dimensions achieved at isospectral models.
Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.
problem Existence and finiteness of -invariant minimal hypersurfaces.
method Equivariant min-max theory, compactness theorem, bumpy metrics theorem.
result Generalization of Morse index estimates to equivariant setting.
Proves minimization for Kähler manifolds with automorphisms.
problem Minimizing CM degree for Kähler manifolds with automorphisms.
method Equivariant CM minimization conjecture, asymptotic filtration Chow stability.
result Strict minimization by extremal filling.
New minimal hypersphere found in 4-sphere solving Bernstein problem.
problem Spherical Bernstein problem in
method Equivariant min-max theory for -invariant minimal hypersurfaces
result Construction of embedded non-equatorial minimal hypersphere
In this paper we continue our study of equivariant minimal Lagrangian surfaces in , characterizing the rotationally equivariant cases and providing explicit formulae for relevant geometric quantities of translationally equivariant minimal Lagrangian surfaces in terms of Weierstrass elliptic functions.
New minimal surfaces found in ball with boundary constraints.
problem Finding minimal surfaces with boundary conditions.
method Equivariant differential geometry approach.
result A family of free boundary minimal surfaces in the unit ball.
Minimal equivariant embedding found for flag manifolds.
problem Finding the smallest possible dimension for equivariant embeddings of flag manifolds.
method Proved the smallest possible dimension for -equivariant embeddings of .
result The smallest possible dimension is the optimal for -equivariant embeddings of .
The study proves a generic multiplicity one theorem for -invariant minimal hypersurfaces.
problem Proving a generic multiplicity one theorem for -invariant minimal hypersurfaces.
method Equivariant min-max theory and analysis of -homology classes.
result Shows a generic multiplicity one theorem for -invariant minimal hypersurfaces.
The minimal dimension of a sphere with an equivariant embedding of the bouquet of circles is math.GT
Minimal sphere dimension for equivariant embedding of circles.
problem Embedding a bouquet of circles into a sphere.
method Finding the minimal dimension of the sphere for equivariant embedding.
result The minimal dimension is .
In this note, we present a new look at translationally equivariant minimal Lagrangian surfaces in the complex projective plane via the loop group method.
Minimal rational curves on compactified symmetric spaces are orbit-closures of 1-parameter subgroups.
problem When are minimal rational curves on equivariant compactifications of symmetric spaces orbit-closures of 1-parameter subgroups?
method Combining algebraic geometry of minimal rational curves with differential geometry of symmetric spaces, showing Gauss-nondegeneracy of VMRT.
result The Gauss-nondegeneracy of VMRT implies that minimal rational curves on equivariant compactifications of symmetric spaces are orbit-closures of 1-parameter subgroups.
Construct minimal Lagrangian surfaces in complex projective plane via loop group method.
problem Construct minimal Lagrangian immersions from arbitrary Riemann surfaces into complex projective plane.
method Loop group method, perturbed equivariant minimal Lagrangian surfaces, Delaunay cylinders approximation.
result Construct a class of minimal Lagrangian cylinders approximating Delaunay cylinders.
Infinite -invariant minimal hypersurfaces found in Riemannian manifolds.
problem Finding minimal hypersurfaces in manifolds with group actions.
method New algorithm using multi-stage maximal cuttings.
result Each -homology class admits infinitely many distinct realizations by embedded minimal -hypersurfaces.
The equivariant CR minimal immersions from the round -sphere into the complex projective space have been classified by the third author explicitly (J London Math Soc 68: 223-240, 2003). In this paper, by employing the equivariant condition which implies that the induced metric is left-invariant,…
The study finds many Möbius bands and annuli on toroids.
problem Finding minimal surfaces on toroids.
method Equivariant variational methods.
result Linear growth of surface areas with symmetry order.
Proves conditions for minimal surfaces in complex hyperbolic space.
problem Conditions for finite energy equivariant minimal surfaces in complex hyperbolic space.
method Analyzes peripheral holonomy and uses Higgs bundles.
result Explicit parametrization and construction of minimal surfaces.
Equivalence proven between algebraic stability and geometric stability.
problem Equivalence of algebraic and geometric stability criteria.
method Algebraic proof of equivalence, existence and uniqueness of minimal centers.
result Existence and uniqueness of minimal optimal destabilizing centers.
New Zoll families of minimal spheres found in spheres and projective spaces.
problem Finding new Zoll families of minimal spheres in various spaces.
method Equivariant constructions using Nash-Moser-Hamilton implicit function theorem.
result First examples of metrics on real projective spaces with Zoll families of minimal projective hyperplanes.
The paper proves the regularity of cohomogeneity two problems and constructs minimal hypersurfaces on spheres.
problem Cohomogeneity two equivariant isotopy minimization problems and minimal hypersurfaces with large Betti numbers.
method Developed cohomogeneity two equivariant min-max theory for minimal hypersurfaces.
result Constructs minimal hypersurfaces on spheres with large Betti numbers and specific symmetries.
Equivariant primitive harmonic maps into symmetric spaces, with applications to Willmore surfacesmath.DG
The study constructs equivariant harmonic maps into symmetric spaces with applications to Willmore surfaces.
problem Constructing harmonic maps into symmetric spaces.
method Equivariant primitive harmonic maps construction.
result Examples of -equivariant Willmore Moebius strips in .
We show that the equivariant chain complex associated to a minimal CW-structure X on the complement M(A) of a hyperplane arrangement A, is independent of X. When A is a sufficiently general linear section of an aspheric arrangement, we explain a new way for computing the twisted homology of M(A).
Characterizes area-minimizing maps for surfaces of genus ≥ 2.
problem Equivariant area-minimizing maps on surface covers.
method Classifies minimal surfaces in Hilbert spheres with constant negative Gaussian curvature.
result Characterizes all equivariantly area-minimizing maps from the universal cover of a surface to a Hilbert sphere.
The paper generalizes free boundary min-max theory to equivariant settings.
problem Existence of minimal hypersurfaces with free boundary in manifolds with boundary.
method Free boundary min-max theory generalized to equivariant settings.
result Existence of nontrivial smooth almost properly embedded -invariant minimal hypersurfaces with free boundary.
Equivariant min-max theorymath.DG
We develop an equivariant min-max theory as proposed by Pitts-Rubinstein in 1988 and then show that it can produce many of the known minimal surfaces in up to genus and symmetry group. We also produce several new infinite families of minimal surfaces in proposed by Pitts-Rubinstein. These …
Unstable minimal surfaces in n-space link to hyperbolic products.
problem Characterizing unstable minimal surfaces in and their product counterparts.
method Lifting to -trees, deforming to hyperbolic products, and proving instability equivalence.
result Unstable minimal surfaces in imply unstable surfaces in product hyperbolic spaces.
Permutation-equivariant neural networks improve auction mechanisms by reducing regret and sample complexity.
problem Designing optimal auction mechanisms that balance revenue and bidders' regret.
method Introduced permutation-equivariant neural networks to auction mechanisms.
result Permutation-equivariant neural networks decrease expected ex-post regret and improve model generalizability.
Genus one critical catenoidmath.DG
Constructs a unique surface in a ball with specific properties.
problem Creating a minimal surface with specific topological and geometric constraints.
method Variational methods, equivariant min-max theory, nontrivial sweepout.
result First genus one critical catenoid in a unit ball.
We construct and analyze minimal disc stackings with bounds on their Morse index.
problem Constructing and analyzing minimal free boundary disc stackings.
method Constructing minimal free boundary disc stackings in a three-dimensional Euclidean unit ball, proving bounds on their Morse index.
result Uniform, linear bounds on the Morse index of all such surfaces.
New minimal surfaces found in 3D sphere with low genus.
problem Finding minimal surfaces in 3D sphere with low genus.
method Equivariant min-max procedure applied to a novel sweepout.
result Discovered new minimal surfaces with low genus.
Minimal dimensions for Riemann surface embeddings computed for specific groups.
problem Finding the minimal dimensions for embedding Riemann surfaces into Euclidean spaces.
method Representations of groups, equivariant triangulations, orbifold theory.
result Minimal dimension for Hurwitz action on Klein quartic is 8.
On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operatorsmath.DG
The paper studies geometric properties of group equivariant operators and their Riemannian structure.
problem Understanding the geometric structure of group equivariant operators.
method Endowing the space of group equivariant non-expansive operators with a Riemannian manifold structure and using gradient descent methods.
result Gradient descent methods can be applied to minimize cost functions on the space of group equivariant non-expansive operators.
New spectral estimates for minimal surfaces with boundary conditions.
problem Quantifying the Morse index of free boundary minimal surfaces.
method Adapted Montiel-Ros partitioning methods to compact manifolds with boundary, accounting for mixed and group actions.
result Explicit two-sided linear bounds on the Morse index for minimal surfaces.
Study minimal rational curves on complex manifolds with isotropic VMRT.
problem Understanding minimal rational curves tangent to distributions on complex manifolds.
method Partial equivariant compactification of metabelian groups.
result Any isotropic VMRT can be realized as VMRT of minimal rational curves tangent to a distribution.
Embedded minimal -bundles in math.DG
New minimal hypersurfaces in 4D sphere found.
problem Constructing embedded minimal hypersurfaces in .
method Equivariant min-max theory and suspended Hopf action.
result Infinitely many topological -bundles and Seifert fibered manifolds found.
The paper finds a special hypersurface in a manifold with positive Ricci curvature.
problem Finding a special hypersurface in a manifold with positive Ricci curvature.
method Equivariant min-max method applied to -manifolds.
result The hypersurface is a multiplicity one minimal -hypersurface.
Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
problem Finding generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
method Weyl asymptotic law for -equivariant volume spectrum, generic density result.
result Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
New minimal surface theory disproves a conjecture in symmetric spaces.
problem Proving the existence of unstable minimal maps in symmetric spaces.
method Using Hitchin representations and equivariant maps, with a new index bound.
result Disproves the Labourie conjecture for with .
Measures equivariance in vision models using Lie derivative.
problem Understanding the role of equivariance in recent vision models.
method Introducing Lie derivative to measure equivariance with strong mathematical foundations and minimal hyperparameters.
result Many violations of equivariance can be linked to spatial aliasing in network layers, and larger models tend to display more equivariance.
Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.
problem Characterizing and constructing minimal Lagrangian surfaces in complex hyperbolic quadric.
method Loop of flat connections, isometric deformations, DPW-type representation.
result Explicit examples of minimal Lagrangian surfaces, including catenoid-type examples.
Extends equivariant contact structure results to mod p L-spaces.
problem Equivariant contact structures on minimal L-spaces.
method Uses Serre spectral sequence of Borel Floer cohomology.
result Introduces two new numerical invariants.
New framework for conformal equivariant cycles in KK-theory.
problem Tackles conformal equivariance in unbounded KK-theory.
method Extends unbounded Kasparov theory with novel perturbation theory.
result Defines new unbounded representatives of Kasparov classes.
We present a deformation for constant mean curvature tori in the 3-sphere. We show that the moduli space of equivariant constant mean curvature tori in the 3-sphere is connected, and we classify the minimal, the embedded, and the Alexandrov embedded tori therein. We conclude with an instability result.
Minimal surfaces with symmetries exist under certain group actions.
problem Existence of conformal minimal immersions with group symmetries.
method Identifying conditions for -equivariant conformal minimal immersions.
result Existence of such immersions under specific group actions.
In this article we introduce a definition for the moduli space of equivariant minimal immersions of the Poincaré disc into a non-compact symmetric space, where the equivariance is with respect to representations of the fundamental group of a compact Riemann surface of genus at least two. We then study this moduli space…
The paper studies minimal surfaces in complex hyperbolic spaces, showing moduli space connected components.
problem Understanding the structure of moduli spaces of minimal surfaces in complex hyperbolic spaces.
method Relating the moduli space to nilpotent cones in Higgs bundles, analyzing limit points of actions.
result Connected components of the moduli space of minimal immersions in are indexed by the Toledo invariant and the Euler number of the normal bundle.
A projection maps geodesic currents to Teichmüller space.
problem Mapping geodesic currents to Teichmüller space.
method Equivariant, length-minimizing projection from filling currents to Teichmüller space.
result The projection is well-behaved and maps geodesic currents to Teichmüller space.