Study on spheres with minimal equators.
problem Classifying metrics on spheres with minimal equators.
method Survey and discussion of related problems.
result Classification of metrics on spheres with minimal equators.
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48 results for “minimal spheres”
Proves existence of at least two minimal spheres in any 3D space.
problem Existence of minimal spheres in arbitrary 3D spaces.
method Iterative relative min-max constructions.
result Proves existence of at least two embedded minimal spheres.
Proves optimal regularity for sphere minimizers in 3-sphere.
problem Finding optimal regularity for sphere minimizers.
method Proves regularity for minimizers of prescribed mean curvature over isotopy classes.
result Proves optimal regularity for minimizers.
Minimal spheres found in ellipsoids with large axes.
problem Finding non-planar minimal spheres in ellipsoids.
method Quantitative proof of minimal spheres existence with constraints.
result Ellipsoids with large axes contain at least three non-planar minimal spheres.
Minimal submanifolds are stable in certain conformal spheres.
problem Stability of minimal submanifolds in conformal spheres.
method Analyzing -dimensional Riemannian spheres with specific curvature conditions.
result Closed stable minimal submanifolds are not found in -pinched conformal spheres.
Four minimal spheres found in sphere with special metric.
problem Existence of minimal spheres in spheres with specific metrics.
method Simon-Smith min-max theory for multiplicity one theorem.
result At least four embedded minimal 2-spheres proven.
Minimal surfaces in spheres found for any genus.
problem Finding minimal surfaces with arbitrary genus in 3-spheres.
method Topological structure analysis and embedding theorem.
result Every positive Ricci curvature 3-sphere contains a genus g surface.
Study on minimal hypersurfaces in a unit sphere, proving specific isometries.
problem Characterizing minimal hypersurfaces with constant scalar curvature.
method Analyzing -dimensional complete minimal hypersurfaces in a unit sphere with constant scalar curvature.
result Proves isometry to totally geodesic sphere or Clifford torus under certain conditions.
Minimal hypertori found in 4D sphere, solving Bernstein conjecture.
problem Finding minimal embedded hypertori in 4D sphere.
method Analyzing minimally embedded and immersed hypertori and hyperspheres.
result Infinitely many non-isometric minimally embedded hypertori and hyperspheres found.
5 minimal tori found in 3-spheres with positive Ricci curvature.
problem Existence of at least 5 embedded minimal tori in 3-spheres with positive Ricci curvature.
method Combination of min-max theory and mean curvature flow heuristics.
result Confirms B. White's conjecture for positive Ricci curvature.
Study finds many nonplanar minimal spheres in elongated ellipsoids.
problem Existence of nonplanar minimal spheres in elongated ellipsoids.
method Global bifurcation techniques to establish existence and quantify number.
result Arbitrarily many nonplanar minimal spheres exist in elongated ellipsoids.
In this note we complete the discussion of minimality of symplectic fiber sums. We find, that for fiber sums along spheres the minimality of the sum is determined by the cases discussed by M. Usher and one additional case: If the sum is the result of the rational blow-down of a symplectic -4-sphere in X, then it is non…
New minimal surfaces found in spheres and hyperbolic spaces.
problem Constructing minimal submanifolds in even-dimensional spheres and hyperbolic spaces.
method Using complex-valued harmonic morphisms.
result Explicit examples of minimal submanifolds in and .
The study of stable and index compact minimal submanifolds in Berger spheres.
problem Stability and index of compact minimal submanifolds in Berger spheres.
method Analyzing stability and index properties of compact minimal submanifolds in Berger spheres.
result Stable compact minimal submanifolds exist in Berger spheres for specific values of τ, and their classification is provided.
Minimal surfaces with negative curvature found in large spheres.
problem Existence of minimal surfaces with negative curvature in large dimensional spheres.
method Applied Song's strategy to closed Riemann surfaces with large automorphism groups, resulting in almost hyperbolic minimal surfaces.
result Existence of closed minimal surfaces with negative induced curvature in any sphere of large dimension.
The paper constructs metrics on spheres with families of minimal hypersurfaces.
problem Finding Riemannian metrics on spheres with specific families of minimal hypersurfaces.
method Used Nash-Moser Inverse Function Theorem in the tame maps setting.
result Generalized Guillemin's theorem for Zoll families of minimal hypersurfaces.
The paper proves the existence of at least 4 embedded minimal tori in a three-sphere with positive Ricci curvature.
problem Proving the existence of embedded minimal tori in three-spheres with positive Ricci curvature.
method The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory.
result There exist at least 4 distinct embedded minimal tori in the three-sphere with positive Ricci curvature.
A minimal hypersurface in a sphere is uniquely determined.
problem Characterizing closed minimal hypersurfaces in spheres.
method Proving strong rigidity of closed minimal hypersurfaces.
result Closed minimal hypersurfaces in spheres are uniquely determined.
Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable comp…
Study finds bound on energy of minimal spheres on complex manifolds.
problem Finding bounds on energy of minimal spheres on complex manifolds.
method Proving existence of harmonic spheres with Morse index bound one.
result Sum of energies of minimal spheres realizes a geometric invariant width.
Minimal surfaces reflect across spheres, proving annulus uniqueness.
problem Uniqueness of free boundary minimal annuli in balls.
method Reflection principle applied to minimal surfaces meeting spheres at 90 degrees.
result Every embedded free boundary minimal annulus in a ball is the critical catenoid.
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.
New formulas for minimal surfaces with specific end conditions.
problem Existence and explicit formulas for minimal surfaces with embedded planar ends.
method Provided new explicit formulas for genus 0 minimal surfaces in R^3 with 2k+1 embedded planar ends.
result Existence and explicit formulas for minimal surfaces with 2k+1 embedded planar ends for all k ≥ 4.
New gravitational instantons and non-holomorphic minimal spheres discovered.
problem Existence of non-holomorphic minimal spheres in ALF spaces.
method Gluing construction of ALF gravitational instantons using specific metrics.
result Existence of non-holomorphic minimal spheres in a general class of ALF spaces.
Study finds the volume of unit vector fields on a punctured sphere and shows their images match minimally immersed Klein bottles.
problem Finding the volume of unit vector fields on a punctured sphere.
method Analyzes the volume of unit vector fields on an antipodally punctured unit 2-sphere and shows their images coincide with minimally immersed Klein bottles.
result The images of minimizing vector fields on the punctured sphere match those of minimally immersed Klein bottles.
The paper examines deformations of pseudoholomorphic curves in a nearly Kähler sphere.
problem Investigating rigidity and deformability of pseudoholomorphic curves in .
method Analyzing moduli space of minimal surfaces isometric to pseudoholomorphic curves.
result Describes the moduli space of noncongruent minimal surfaces isometric to pseudoholomorphic curves.
In this paper, we prove that the -sphere endowed with an arbitrary Riemannian metric either contains at least two embedded minimal -spheres or admits an optimal foliation by -spheres. This generalizes recent results by Haslhofer-Ketover (Duke Math. J. 2019), where the existence of optimal foliations and minima…
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.
New findings on minimal isometric immersions of flat n-tori into spheres.
problem Conditions for minimal isometric immersions of flat n-tori into spheres.
method Analyzes rationality conditions and derives upper bounds for algebraic irrationality degree.
result Upper bound for algebraic irrationality degree of minimal isometric immersions is sharp and equals 4 for n=3.
The study finds conditions for minimal spheres into ellipsoids using eigenfunctions.
problem Conditions for branched minimal immersions of spheres into ellipsoids to be embedded.
method Using eigenfunctions with respect to a critical metric, the study provides sufficient conditions for embeddings and constructions of non-planar minimal spheres.
result Conditions for embeddings and constructions of non-planar minimal spheres using eigenfunctions.
The study proves local rigidity of minimal 2-spheres in electrovacuum spacetimes.
problem Proving local rigidity of minimal 2-spheres in electrovacuum spacetimes under certain conditions.
method Analyzing electrovacuum spacetimes and using constraints on charged Hawking mass and area minimization.
result Local rigidity of minimal 2-spheres in electrovacuum spacetimes, with isometric neighborhoods to specific spacetimes.
In this paper, we study the rigidity theorem of closed minimally immersed Legendrian submanifolds in the unit sphere. Utilizing the maximum principle, we obtain a new characterization of the Calabi torus in the unit sphere which is the minimal Calabi product Legendrian immersion of a point and the totally geodesic Lege…
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.
Spiral Minimal Productsmath.DG
Paper constructs new minimal submanifolds in spheres by spinning given ones.
problem Creating new minimal submanifolds in spheres from given ones.
method Spin given minimal submanifolds by a curve in a balanced way.
result Generates spiral minimal products forming a two-dimensional family.
Study minimal networks on spheres and balls near standard metrics.
problem Existence of minimal networks in spheres and balls with metrics close to standard.
method Finite-dimensional reduction method, inspired by configuration of networks and triods.
result Existence of minimal networks in spheres and balls for metrics close to standard.
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 paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both and the hyperquadric of . The moduli space of all those noncongruent ones is introduced, which can be described by certain complex symmetric matrices…
Paper proves pinching theorem for minimal surfaces in spheres.
problem Pinching rigidity of minimal surfaces in spheres.
method Simon conjecture and Simons-type integral inequalities.
result New proof of pinching theorem for minimal surfaces in spheres.
Minimal Surfaces in the Three-Dimensional Sphere and Minimal Hypersurfaces of Type Number Twomath.DG
We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson equation. We obtain a classification theorem for bi-umbilical hypersurfaces of t…
We prove that any manifold diffeomorphic to and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the exi…
Study of minimal immersions from a sphere to a complex hyperquadric.
problem Characterizing minimal immersions from a sphere to a complex hyperquadric.
method Analysis through harmonic maps and linear full reducibility.
result Determination of reducible conformal minimal immersions with constant curvature.
New method for minimal submanifolds in spheres using eigenfunctions.
problem Constructing minimal submanifolds in spheres.
method Eigenfunctions of the Laplace-Beltrami operator to describe minimal submanifolds.
result New proof of -eigenfunctions on spheres.
Minimal submanifolds in spheres can be produced via Clifford type minimal products, and their Morse indices and nullities are calculated.
problem Understanding the properties of minimal submanifolds in spheres via Clifford products.
method Analyzing the first eigenfunctions and Morse indices of minimal products of minimal submanifolds.
result The Morse index and nullity of the minimal product are calculated and shown for specific cases.
Using the Lawson's existence theorem of minimal surfaces and the symmetries of the Hopf fibration, we will construct symmetric embedded closed minimal surfaces in the three dimensional sphere. These surfaces contain the Clifford torus, the Lawson's minimal surfaces, and seven new minimal surfaces with genera 9, 25, 49,…
Minimal surfaces in 3-sphere created by reflections from polygons, with new examples based on pentagons.
problem Constructing minimal surfaces in 3-sphere using reflections.
method Minimal -gon solves free boundary problem; curvature lines combinatorics investigated.
result New examples of minimal reflection surfaces based on pentagons.
We show that the disk complex of a genus Heegaard surface for the 3-sphere is homotopy equivalent to a wedge of -dimensional spheres. This implies that genus Heegaard surfaces for the 3-sphere are topologically minimal with index .
Minimal surfaces in S3(2) linked to vector fields on punctured sphere.
problem Connecting minimal surfaces in S3(2) to vector fields on a punctured sphere.
method Established a correspondence between minimal surfaces and area-minimizing vector fields.
result Stability relation for Lawson cylinders in S3(2).
Improved lower bound for the first eigenvalue of embedded minimal hypersurfaces in the unit sphere
problem First eigenvalue of embedded minimal hypersurfaces
method Establishing an improved lower bound
result Better than Duncan-Sire-Spruck's bound