Minimal Lagrangian diffeomorphisms between spherical surfaces are rigid.
problem Rigidity of minimal Lagrangian diffeomorphisms between spherical surfaces.
method Proving that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry.
result Minimal Lagrangian diffeomorphisms between spherical surfaces are rigid (i.e., they are isometries).
Study of free boundary minimal Möbius bands in spherical caps.
problem Characterizing minimal surfaces with free boundary in spherical caps.
method Analyzing spectral properties and geometric constraints.
result Proves that any free boundary minimal Möbius band in spherical caps must be intrinsically rotationally symmetric.
Proves existence and regularity of spherical minimizers for lipid membrane energy.
problem Existence and regularity of minimizers for the Canham-Helfrich energy.
method Establishes lower semicontinuity and proves existence through weak convergence of immersions.
result Proves existence and regularity of minimizers for the Canham-Helfrich energy on spheres.
The study finds the optimal metrics for free boundary minimal surfaces in spherical caps.
problem Optimizing metrics for free boundary minimal surfaces in spherical caps.
method Introducing functionals based on eigenvalues of Steklov-type problems and proving maximizers are induced by immersions.
result Maximizing metrics are induced by free boundary minimal immersions in geodesic balls of a round sphere.
Study of immersions with Willmore energy leading to spherical and catenoid bubbles.
problem Classifying immersions with specific energy properties.
method Analyzing sequences of weak immersions with diverging conformal classes, applying Möbius transformations, and strong Wloc2,2-limits. result Obtaining spherical and catenoid bubbles as limits of immersions.
New minimal annuli found in unit ball, solving old problems.
problem Constructing free boundary minimal annuli in unit ball.
method Symmetric and foliated by spherical curvature lines.
result First non-embedded free boundary minimal annuli in unit ball.
Minimal products of spherical immersions are studied with geometric and dynamical properties.
problem Minimal products of spherical immersions and their geometric properties.
method Profile flow, Liouville integrability, phase map analysis, Routh completion, primitive order analysis.
result Minimal products have a scalar Sturm form plus two nonnegative squares in their Hessian.
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.
Study of homothetic solitons in inverse mean curvature flow.
problem Understanding the behavior of solitons in inverse mean curvature flow.
method Analyzing solutions that evolve by homotheties of a given submanifold.
result Classification of rotationally invariant Lagrangian homothetic solitons.
Study on membranes under confinement, proving existence and regularity of minimizers.
problem Existence and regularity of minimizers for constrained Helfrich energy.
method Elliptic system analysis, careful study of measure-valued Lagrange multiplier.
result Optimal regularity for solutions throughout branch points, rigid behavior for unit ball minimizers.
Study shows spherical embedding and immersion components are related to homotopy groups.
problem Understanding the connected components of spherical embeddings and immersions.
method Analyzing the spaces of spherical embeddings and immersions modulo immersions, and relating them to homotopy groups.
result The set of connected components of spherical embeddings and immersions modulo immersions is isomorphic to π_{n+1}(SG,SG_q).
Proves Li-Yau inequality for Helfrich functional, ensuring embeddedness in spherical cases.
problem Ensuring embeddedness of minimizers in the Canham-Helfrich model.
method Proves Li-Yau inequality for Helfrich functional, converting singular volume integral to explicit energy threshold.
result Existence of smoothly embedded minimizers in physically relevant cases.
A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n >= 3 a complete locally-convex (topological) immersion of a connected (n-1)-manifold into the n-sphere is a surjection onto the boundary of a c…
For an oriented isometric immersion f:M→Sn the spherical Gauss map is the Legendrian immersion of its unit normal bundle UM⊥ into the unit sphere subbundle of TSn, and the geodesic Gauss map γ projects this into the manifold of oriented geodesics in Sn (the Grassmannian of oriented 2-planes in $\ma…
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.
Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.
problem Investigating singularities of evolutes and focal surfaces of pseudo-spherical framed immersions.
method Introduced pseudo-spherical non-null framed curves, defined moving frames, and analyzed evolutes and focal surfaces.
result Evolutes of pseudo-spherical framed immersions are the sets of singular points of their focal surfaces.
Flow preserves isoperimetric ratio for immersed surfaces.
problem Preserving isoperimetric ratio in Willmore flow.
method Non-local L2-gradient flow for Willmore energy. result Long-time existence and convergence for spherical initial data.
New equations for pseudo-spherical surfaces found, with unique isometric immersions.
problem Finding isometric immersions for pseudo-spherical surfaces described by k-th order evolution equations.
method Investigating the relationship between pseudo-spherical surfaces and k-th order evolution equations, proving the existence of unique isometric immersions.
result There is only one type of k-th order evolution equations that admit local isometric immersions, with universal coefficients of the second fundamental form.
Researchers found multiple spherical Ricci metrics on tori with rotational symmetry.
problem Constructing and analyzing spherical Ricci metrics with rotational symmetry.
method Explicitly constructed a two-parameter family of metrics with rotational symmetry and showed their existence on tori.
result Infinitely many non-isometric spherical Ricci metrics can be realized on the same torus.
The paper lifts spherical Morse functions to immersions and embeddings.
problem Lifting spherical Morse functions to other maps.
method New methods to lift to special generic maps with non-positive codimensions.
result Constructs most lifts to special generic maps.
We consider the class of differential equations that describe pseudo-spherical surfaces of the form u_t=F(u,u_x,u_xx) and u_xt=F(u,u_x) given in Chern-Tenenblat \cite{ChernTenenblat} and Rabelo-Tenenblat \cite{RabeloTenenblat90}. We answer the following question: Given a pseudo-spherical surface determine…
The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat [3], is characterized by the property that to each solution of a differential equation, within the class, there corresponds a 2-dimensional Riemannian metric of curvature equal to −1. The class of differe…
For an (m+1)-dimensional space-time (Xm+1,g), define a mapped null hypersurface to be a smooth map ν:Nm→Xm+1 (that is not necessarily an immersion) such that there exists a smooth field of null lines along ν that are both tangent and g-orthogonal to ν. We study relations between mapped null hyp…
Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.
problem Characterize free boundary minimal submanifolds in geodesic balls of hyperbolic and spherical spaces.
method Define and analyze functionals related to critical metrics and spectral indices.
result Critical metrics of defined functionals arise from free boundary minimal immersions in geodesic balls of hyperbolic and spherical spaces.
Free boundary minimal submanifolds with boundaries on concentric spheres
problem Finding minimal submanifolds with boundaries on concentric spheres in Euclidean space
method Using a Steklov problem with an indefinite weight
result Exact Morse index of an m-dimensional flat annulus in an n-dimensional spherical shell Study on unique vortex equations and their geometric implications.
problem Uniqueness of vortex equations involving entire functions.
method Analyzing entire functions on the complex plane and showing geometric applications.
result Uniqueness of harmonic maps and affine spherical immersions with polynomial differential constraints, but failure for non-polynomial entire functions.
The paper proves rigidity theorems for hypersurfaces in spherical space forms.
problem Proving rigidity of hypersurfaces in spherical space forms under certain curvature conditions.
method Topological and homotopical methods, including diffeomorphisms and weak homotopy equivalences.
result The universal cover of the hypersurface is diffeomorphic to the n-sphere and fundamental group bounds.
The paper proves a stronger Frankel theorem for minimal hypersurfaces of Sk.
problem Proving the intersection of minimal hypersurfaces in Sk. method Analyzing the intersection of two compact minimal hypersurfaces in Sk. result Two compact minimal hypersurfaces in Sk intersect in a hemisphere. New invariant for CR maps from spheres discovered.
problem Identifying CR maps from spheres.
method Introducing a CR analogue of the Ahlfors derivative.
result The invariant distinguishes many sphere maps and vanishes for linear embeddings.
Minimal surfaces in spherical space forms have limited genus.
problem Understanding minimal surfaces in spherical space forms.
method Analyzing orientable index one minimal surfaces with large fundamental groups.
result Surfaces have genus at most two, confirming a conjecture.
We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a de…
The study restricts stable minimal immersions in product spaces to specific configurations.
problem Prohibiting stable minimal immersions in certain product spaces.
method Analyzing stable minimal immersions in products of complex, quaternionic, and octonionic projective spaces.
result The only stable compact minimal immersions in the product of a quaternionic projective space with any other Riemannian manifold are the products of quaternionic projective subspaces with compact stable minimal immersions of the second manifold.
Paper introduces spherical knot mosaics for knot and link invariants.
problem Representing knots on a sphere with tiles.
method Tiling a 2-sphere with 11 knot mosaic tiles to define new invariants.
result New knot invariants derived from spherical mosaic tiling.
New minimal surfaces found with spherical curvature lines.
problem Finding minimal surfaces with specific curvature lines.
method Constructing surfaces parametrized by rhombic lattices.
result Found new examples of minimal annuli with free boundaries.
In this paper, we exploit a subtle indeterminacy in the definition of the spherical Kervaire-Milnor invariant which was discovered by R. Stong to construct non-spin 4-manifolds with even intersection form and prescribed signature.
Proves nonnegatively curved hypersurfaces on a sphere are convex disks.
problem Characterizing nonnegatively curved hypersurfaces with free boundary on a sphere.
method Analyzes hypersurfaces in Euclidean space with constant mth mean curvature. result Compact hypersurfaces are embedded convex disks.
An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. A prominent result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which characterizes minimal total absolute curvature immersions, and tight…
Paper proves convergence for Willmore immersions with minimal bubbles.
problem Proving convergence of Willmore immersions with minimal bubbles.
method Replaces total curvature control with local Willmore energy control.
result Proves convergence result for sequences of Willmore immersions.
Loops in surfaces and chord diagrams are studied with graph factorizations and grammars.
problem Understanding loops in surfaces and their properties.
method Factorization of filoops into spheric and toric sums, and grammars generating chordiagraphs.
result Minimal genus of filoops and stability properties under factorizations.
Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.
problem Understanding eigenvalues and gaps in spherical triangles.
method Explicit computation of Dirichlet eigenvalues and eigenfunctions for spherical lunes and triangles.
result Fundamental gap of spherical triangles increases as the angle of the lune decreases.
Sharp lower bound found for area of vector fields on spherical annuli.
problem Finding the minimum area of unit vector fields on spherical annuli.
method Established a sharp lower bound through mathematical analysis.
result Sharp lower bound for the area of unit vector fields on spherical annuli.
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. The paper disproves compactness for high-energy Willmore immersions and finds minimal bubbles on Willmore surfaces.
problem Compactness for high-energy Willmore immersions of Willmore energy above 16π. method Explicit construction of minimal bubbles and analysis of limit sequences of Willmore immersions.
result Compactness for immersed Willmore tori of energy below 12π is proven. Characterizes spherical and plane curves using RM frames.
problem Understanding the geometric properties of curves in different spaces.
method Employing rotation minimizing frames to study curvature and torsion.
result Characterizes curves as those whose position vector lies on a moving plane.
Researchers resolve the compactness problem for helicoidal minimal surfaces in 3-sphere.
problem Determine when helicoidal minimal surfaces are compact.
method Proved compactness conditions using explicit integrals and quotient surfaces.
result Compact members of the family are characterized by rational pitch and/or rational ratio of parameters.
Study minimal Kähler submanifolds in product of space forms.
problem Existence and properties of minimal isometric immersions of Kähler manifolds into product spaces.
method Analyse obstruction conditions and prove existence results for minimal immersions into Sm−1imesR and Hm−1imesR. result Only minimal isometric immersions of Riemannian surfaces into Sm−1imesR and Hm−1imesR exist. Every meromorphic function maps to a minimal surface in 3D.
problem Understanding the mapping properties of meromorphic functions to minimal surfaces.
method Proving that every meromorphic function on a Riemann surface is the Gauss map of a conformal minimal immersion into \(\mathbb{R}^3\).
result Meromorphic functions on Riemann surfaces are realized as the Gauss maps of conformal minimal immersions.
Spherical Plateau problem studies minimal surfaces in quotients of spheres.
problem Minimal surfaces in quotients of spheres.
method Metric currents, barycenter map method.
result Intrinsic uniqueness of solutions for negatively curved manifolds.