Invariant detects triple points in sphere immersions.
problem Detecting regular homotopy classes of triple-point-free sphere immersions.
method Defining an invariant using a directed tree and integer-valued function.
result Space of triple-point-free spheres has infinitely many regular homotopy classes.
The paper classifies biharmonic immersions and submersions in specific spheres.
problem Classifying biharmonic immersions and submersions in specific spheres.
method Analyzing biharmonic isometric immersions and Riemannian submersions from Berger 3-spheres.
result Complete classification of proper biharmonic Hopf tori in Berger 3-sphere.
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 paper classifies Sasakian immersions into odd spheres and finds new examples.
problem Classifying Sasakian immersions into odd-dimensional spheres.
method Analyzing compact Sasakian manifolds and using classification theorems.
result Infinite families of compact Sasakian η-Einstein manifolds cannot be immersed into odd spheres. Study on sphere immersions and their stability indices.
problem Analyzing the stability of sphere immersions.
method Calculation of Morse indices and stability indices for specific sphere immersions.
result Bounds on stability index of associative cone in R7. Study biharmonic conformal immersions into a 3D flat space, finding new examples and classifications.
problem Characterize and classify biharmonic conformal immersions into a conformally flat 3-space.
method Characterization of totally umbilical surfaces, method to produce biharmonic immersions, classification of maps, construction of examples.
result Construct many examples of biharmonic conformal immersions, including proper immersions and isometric ones.
2-spheres in 4-manifolds have complete concordance obstructions if they have immersed dual spheres.
problem Determining concordance of 2-spheres in 4-manifolds.
method Adapting Stong's methods to 4-manifolds, using self-intersection sets and concordance obstructions.
result Complete concordance obstructions for 2-spheres with immersed dual spheres in 4-manifolds.
The paper classifies isometric immersions between spheres of equal curvature.
problem Classifying isometric immersions between spheres of equal curvature.
method Defined coherent tangent bundles and classified wave fronts.
result Any isometric immersion of the n-sphere into the (n+1)-sphere of the same curvature is totally geodesic.
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.
The paper proves uniqueness of immersed spheres in three-manifolds.
problem Proving uniqueness of immersed spheres in three-manifolds.
method Solves the Hopf uniqueness problem for a class of immersed surfaces modeled by elliptic PDEs.
result Any compact immersed surface of genus zero in the class is a candidate sphere.
Study finds all conformal minimal immersions of 2-spheres in a complex Grassmann manifold with parallel second fundamental form.
problem Classifying conformal minimal immersions with parallel second fundamental form.
method Analyzing immersions in complex Grassmann manifold G(2,N;C). result Determined all conformal minimal immersions of 2-spheres with parallel second fundamental form.
Study on proper-biharmonic flat tori in spheres with CMC conditions.
problem Finding conditions for CMC proper-biharmonic immersions of tori in spheres.
method Analyzing rectangular and square tori, finding necessary and sufficient conditions, and explicit expressions.
result Explicit expressions of some CMC proper-biharmonic immersions of certain tori in spheres.
Characterizes 2-sphere unions from immersed circles, akin to Rosenstiehl's plane curve codes.
problem Characterizing unions of embedded disjoint circles in a 2-sphere.
method Uses interlacement of circles, Lippner's result, and directed interlacement graphs of paired trees.
result Higher-dimensional analogue of Rosenstiehl's characterization of Gauss codes.
Approximates compact and non-compact Sasakian manifolds in spheres.
problem Approximating Sasakian structures in spheres.
method CR immersions in standard spheres.
result Compact and non-compact Sasakian manifolds can be approximated.
Study classifies Morse functions with 4 critical points on immersed 2-spheres.
problem Classifying Morse functions with 4 critical points on immersed 2-spheres.
method Used dual graph of immersion and Reeb graphs to classify functions.
result Found all possible structures of the functions.
Study on minimal two-spheres in complex hyperquadric, proving non-congruence of constant curvature spheres.
problem Classifying minimal two-spheres of constant curvature in complex hyperquadric.
method Construction of non-homogeneous constant curved minimal two-spheres and classification theorem.
result Minimal two-spheres of constant curvature in Q4 are not congruent. 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.
Paper studies equatorial concentration of measure in sphere immersions and submersions.
problem Equatorial concentration of measure in sphere immersions and submersions.
method Analyzes concentration of measure phenomena in the sphere.
result Describes an equatorial concentration of measure for minimal immersions and submersions.
New examples of r-harmonic immersions into spheres constructed.
problem Constructing new examples of r-harmonic immersions into spheres.
method Polyharmonic maps generalization and specific constructions.
result Canonical inclusion is a proper r-harmonic submanifold if and only if the radius is 1/√r.
Study extends Ramanathan's result to isotropic surfaces in spheres of any dimension.
problem Minimal isometric immersions of isotropic surfaces in spheres.
method Analyzes compact and non-compact isotropic surfaces in spheres of arbitrary dimension.
result Extends Ramanathan's result to isotropic surfaces in spheres of any dimension.
The paper classifies minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.
problem Classifying minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.
method General construction of homogeneous minimal flat n-tori in spheres, detailed investigations of shortest vectors in lattices.
result There exists a 2-parameter family of non-congruent λ1-minimal flat 4-tori.
Proves a Hopf theorem for non-constant mean curvature spheres.
problem Proves uniqueness of spheres with non-constant mean curvature.
method Analyzes spheres in homogeneous three-manifolds, extending Hopf's theorem.
result Extends Hopf's theorem to non-constant mean curvature spheres.
A self-transverse immersion of the 2-sphere into 4-space with algebraic number of self intersection points equal to -n induces an immersion of the circle bundle over the 2-sphere of Euler class 2n into 4-space. Precomposing the circle bundle immersions with their universal covering maps, we get for n>0 immersions g_n o…
We construct two infinite sequences of immersions of the 3-sphere into 4-space, parameterized by the Dynkin diagrams of types A and D. The construction is based on immersions of 4-manifolds obtained as the plumbed immersions along the weighted Dynkin diagrams. We compute their Smale invariants and bordism classes of im…
The paper proves spheres in certain 3-manifolds are always rotational.
problem Characterizing spheres in homogeneous 3-manifolds with specific curvature relations.
method Analyzing surfaces with elliptic Weingarten equations and proving rotational symmetry.
result Spheres in the specified 3-manifolds are always rotational.
Minimal normal curvature immersions in the unit ball studied.
problem Minimal normal curvature immersions in the unit ball.
method Gromov's problem, differentiable sphere theorem, existence result.
result Determined the minimal possible value of the normal curvature of SnimesS1. Classifies minimal immersions from S2 into specific flag manifolds.
problem Classifying minimal immersions from S2 into specific flag manifolds. method Classification based on constant curvature and low-dimensional flag manifolds.
result Primitive minimal immersions of constant curvature from S2 into F2,1,1 and F2,2,1 are classified. Proves isometric embeddings in Euclidean spaces for RCD spaces.
problem Isometric immersions of RCD spaces in Euclidean spaces.
method Analyzes regular isometric immersions and eigenmaps of compact non-collapsed RCD spaces.
result Eigenmaps of compact non-collapsed RCD spaces are locally bi-Lipschitz embeddings to spheres.
Flat tori in 3-sphere have π extrinsic diameter under specific conditions.
problem Determining the extrinsic diameter of immersed flat tori in the 3-sphere.
method Analyzing asymptotic curves and Hopf fibration projections.
result The extrinsic diameter is π under certain topological conditions.
Study of Willmore energy on sphere sublevel sets and flow singularities.
problem Understanding the Willmore energy landscape and singularities of the Willmore flow.
method Gluing different instances of the Willmore flow and using an invariant for triple-point-free spheres.
result Classification of initial surfaces with energy at most 12π leading to unavoidable singularities.
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.
We study the problem to extend an immersed circle f in the 2-dimensional sphere to an immersion of the disc. We analyze existence and uniqueness for this problems in terms of the combinatorial structure of a word assigned to f. Our techniques are based on ideas of Blank who studied the extension problem in case of a pl…
A smooth four manifold is of finite type r if its Donaldson invariant satisfies D((x^2-4)^r)=0. We prove that every simply connected manifold is of finite type by using the structure of Donaldson invariants in the presence of immersed spheres. More precisely we prove that if a manifold X contains an immersed sphere w…
We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere minimizing the L^{2} integral of the second fundamental form. Assuming instead th…
Characterizes a specific type of convex curves on a 3-sphere.
problem Understanding convex curves on a 3-sphere.
method Decomposes curves on 3-sphere into 2-sphere curves, characterizes locally convex ones.
result Completely characterized a class of convex curves on the 3-sphere.
Revisits Weyl's problem on isometric immersions of spheres into 3D manifolds.
problem Isometric immersions of S² into 3D Riemannian manifolds with non-negative Gauss curvature.
method Utilizes the framework of J-holomorphic curves developed by Labourie.
result Exhibits a sufficient condition for the existence of global C¹¹ isometric immersions.
Constructs surfaces with constant mean curvature from Bessel equation.
problem Creating surfaces with constant mean curvature.
method From the Bessel equation, constructs immersions of the twice-punctured Riemann sphere into R^3.
result Family of constant mean curvature surfaces constructed.
We classify all regular three-dimensional convex cones which possess an automorphism group of dimension at least two, and provide analytic expressions for the complete hyperbolic affine spheres which are asymptotic to the boundaries of these cones. The affine spheres are represented by explicit hypersurface immersions …
The article studies integrability conditions for surfaces in Lie groups.
problem Existence of isometric immersions with prescribed Gauss map and curvature.
method Analyzes Lie groups like Euclidean sphere and Riemann sphere, establishing conditions for positive extrinsic curvature.
result A surface isometrically immersed in S3 has positive constant extrinsic curvature if its Gauss map is harmonic. Constructs immersions into pseudo-Riemannian spaces from equiaffine immersions.
problem Creating immersions into pseudo-Riemannian spaces from equiaffine immersions.
method Explicit construction using para-Sasaki metric and principal R-bundle structure. result Maximal spacelike submanifolds in Hn+1,n with specific boundary conditions. The Whitney sphere has a unique energy gap for a specific equation.
problem Energy gap phenomenon for the Whitney sphere.
method Solving the equation abla∗T=0 on Lagrangian surfaces. result Proves a gap theorem for the Whitney sphere.
New characterization of Calabi torus in unit sphere found.
problem Rigidity of closed minimally immersed Legendrian submanifolds in unit sphere.
method Maximum principle and Simons' type integral inequality.
result New characterization of Calabi torus in unit sphere.
Nonpositive towers property in 3-manifolds spines.
problem Properties of 3-manifold spines.
method Analysis of 2-dimensional spines in aspherical 3-manifolds and 3-ball.
result Nonpositive towers property in 2-dimensional spines of aspherical 3-manifolds.
Analytic saddle spheres in S^3 are equators.
problem Characterizing saddle-shaped minimal surfaces in 3-sphere.
method Purely geometric approach, no PDE imposed.
result Analytic saddle spheres in S^3 are equators.
The paper explores null hypersurfaces in Lorentzian manifolds using geometric immersions.
problem Understanding the geometry of null hypersurfaces in Lorentzian manifolds.
method The approach involves isometric immersions of leafs of the screen distribution into semi-Euclidean spheres or hyperbolic spaces.
result Null hypersurfaces are shown to be umbilic and screen totally umbilic under certain geometric conditions.
The paper proves conditions for closed minimally immersed hypersurfaces in a 5-sphere.
problem Conditions for closed minimally immersed hypersurfaces in a 5-sphere.
method Analyzes hypersurfaces with constant scalar curvature and A3. result Closed minimally immersed hypersurfaces in a 5-sphere are isoparametric and can only have specific scalar curvature values.
We consider critical points of the functionals Π and Ψ defined as the global L2-norm of the second fundamental form and mean curvature vector of isometric immersions of compact Riemannian manifolds into a background Riemannian manifold, respectively, as functionals over the space of deformations of the immersion…
Existence of minimal annuli in 3-sphere with boundary on geodesic spheres.
problem Existence of free boundary minimal annuli in 3-sphere.
method One-parameter family of complete minimal immersions of R × S^1 into S^3, analysis of Otsuki tori.
result Existence of embedded free boundary minimal annuli contained in geodesic balls.