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).
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.
Existence and uniqueness of spherical helicoidal surfaces in 3-sphere via spherical curves.
problem Existence and uniqueness of spherical helicoidal surfaces in 3-sphere.
method Continuous function of distance to axis, spherical angular momentum of spherical curves.
result Existence and uniqueness theorem for spherical helicoidal surfaces in 3-sphere.
Study investigates minimal surfaces in spherical caps, extending previous findings.
problem Characterizing minimal surfaces with free boundaries and capillary conditions in spherical caps.
method Extending previous half-space intersection properties to warped products and capillary minimal surfaces in high codimension.
result Established a dual operation relating free boundary and capillary minimal surfaces.
Study finds surfaces in spherical caps that maximize modified energy.
problem Geometry of surfaces with free boundaries and capillary conditions.
method Monotonicity formulae and energy maximization analysis.
result Capillary minimal surfaces maximize a modified energy in their conformal orbit.
Study shows negatively curved manifolds' spherical volume equals minimal surface area.
problem Understanding the spherical volume of negatively curved manifolds.
method Combining metric currents theory and limits of hyperbolic groups' representations.
result Spherical volume of negatively curved manifolds equals minimal surface area.
Smooth minimizers found for Willmore energy surfaces.
problem Finding minimizers for Willmore energy surfaces.
method Existence and smoothness established through axially symmetric surfaces with prescribed isoperimetric ratio.
result Existence and smoothness of minimizers proven.
Minimal surfaces in 4D space are stable if their Gauss map spherical area is less than 2π.
problem Stability of minimal surfaces in 4D space.
method Geometric criteria based on the Gauss map of minimal surfaces in terms of the spherical area.
result Minimal surfaces in 4D space are stable if their Gauss map spherical area is less than 2π.
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 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.
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.
We prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two. This confirms a conjecture of R. Schoen for an infinite class of 3-manifolds.
This paper connects Laguerre minimal surfaces to Weierstrass representations.
problem Understanding the relationship between Laguerre minimal surfaces and Weierstrass representations.
method Defining spherical mean curvature and providing Weierstrass-type representations for two classes of surfaces.
result Laguerre minimal surfaces are related to H2-surfaces, providing a new Weierstrass-type representation. For a family of spherical minimal catenoids C_a in the hyperbolic 3-space, there exist two constants 0<a_c<a_l such that the following are true: (1) C_a is an unstable minimal surface with index one if a<a_c, (2) C_a is a stable minimal surface if a>=a_c, and (3) C_a is a least area minimal surface in the sense of Meek…
We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hami…
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.
We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.
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.
This thesis is devoted to the Differential Geometry of curves and surfaces along with applications in Quantum Mechanics. In the 1st part we introduce the well known Frenet frame. Later, we show that the curvature function is a lower bound for the scalar angular velocity of any other orthonormal moving frame, from which…
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.
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.
The isoperimetric ratio of an embedded surface in R3 is defined as the ratio of the area of the surface to power three to the squared enclosed volume. The aim of the present work is to study the minimization of the Willmore energy under fixed isoperimetric ratio when the underlying abstract surface has fixed genus $…
The abstract proves spherical surface decompositions with conical singularities.
problem Decomposing surfaces with spherical metrics and conical singularities.
method Geometric triangulations and irreducible components of standard shapes.
result Spherical polygons, including half-spherical concave polygons, can be arbitrarily complicated.
Small deformations of marginally (outer) trapped surfaces are considered by using their stability operator. In the case of spherical symmetry, one can use these deformations on any marginally trapped round sphere to prove several interesting results. The concept of 'core' of a black hole is introduced: it is a minimal …
A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obt…
Study proves uniqueness of certain minimal surfaces in spherical and hyperbolic spaces.
problem Proving uniqueness of free boundary minimal annuli in geodesic balls.
method Using Steklov problem frequency and antipodal map invariance.
result Minimal annuli are congruent to a critical rotational annulus.
The paper explores rigid geometric structures near surfaces with equality in area-charge inequalities.
problem Geometric constraints near surfaces with equality in area-charge inequalities.
method Investigation of equality in area-charge inequalities for spherical minimal surfaces and MOTS within the Einstein-Maxwell equations framework.
result Equality in area-charge inequalities imposes rigid geometric structures, including normal electric and magnetic fields and isometric Riemannian products.
We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called {\it horizontal} area functional associated to the canonical {\it horizontal} area form. We derive the intrinsic equation in the general case…
In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized spherical curves in Euclidean (n+1)−space En+1. Further, we introduce some kind of generalized spherical surfaces in Euclidean spaces E3 and respect…
We prove some related results concerning blow-up solutions for the Jang equation. First: it has been shown that, given an outermost marginally outer trapped surface (MOTS) Σ, there exists a solution to Jang's equation which blows up at Σ. Here we show that in addition, large classes of spherically symmetric initial dat…
Defines spherical type surfaces via support function and classifies them.
problem Characterizing surfaces via support function.
method Weierstrass type representation for SS-surfaces with prescribed Gauss map.
result Every compact and connected SS-surface is the sphere.
The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.
problem Understanding surfaces with spherical curvature lines and their generation mechanisms.
method The approach involves Lie sphere transformations, Legendre curves, and polynomial conserved quantities of connections.
result Lie applicable surfaces with exactly one family of spherical curvature lines are generated by the lift of constrained elastic curves.
Nearly spherical, positively curved surfaces are mapped from a sphere.
problem Mapping nearly spherical, positively curved surfaces from a sphere.
method Combines Ricci flow, Kim-Milman construction, and Bakry-Émery criterion.
result Every nearly spherical, positively curved surface is the contractive image of a round sphere.
We prove that any class VII surface with b2=1 has curves. This implies the "Global Spherical Shell conjecture" in the case b2=1: Any minimal class VII surface with b2=1 admits a global spherical shell, hence it is isomorphic to one of the surfaces in the known list. The main idea of the proof is to show th…
Stability conditions on K3 surfaces are linked to the masses of spherical objects.
problem Determining stability conditions on K3 surfaces.
method Using the masses of spherical objects and lax stability conditions associated to spherical bundles.
result Stability conditions on K3 surfaces are determined by the masses of spherical objects up to a natural C-action. Develops spherical density-equalizing maps for closed surfaces.
problem Lack of methods for genus-0 closed surfaces.
method Conformal parameterization onto unit sphere, density equalization, quasi-conformal theory, harmonic energy, landmark constraints.
result Landmark-aligned spherical density-equalizing maps balancing different distortion measures.
Small deformations of marginally outer trapped surfaces (MOTS) are studied by using the stability operator introduced by Andersson-Mars-Simon. Novel formulae for the principal eigenvalue are presented. A characterization of the many marginally outer trapped tubes (MOTT) passing through a given MOTS is given, and the po…
This study classifies quadric surfaces in 3-sphere as Weingarten surfaces.
problem Extension of quadric surfaces of revolution to 3-sphere.
method Rigorous classification and characterization using spherical angular momentum.
result Spherical ellipsoids, hyperboloids, and paraboloids are Weingarten surfaces with a specific cubic relation between principal curvatures.
Low-entropy surfaces can be flowed into spheres and cylinders.
problem Proving mean curvature flow for low-entropy hypersurfaces.
method Low-entropy density drop argument and recent work on hypersurfaces.
result Closed hypersurfaces with entropy ≤ 2 can be flowed into spherical and cylindrical shapes.
In this paper, we study the stability of catenoids and helicoids in the hyperbolic 3-space H3. (1) For a family of spherical minimal catenoids {Ca}a>0 in H3, there exist two constants 0<ac<al such that ∙ Ca is an unstable minimal surface with index o…
Let (M,g) be a closed oriented Riemannian 3-manifold and suppose that there is a strongly irreducible Heegaard splitting H. We prove that H is either isotopic to a minimal surface of index at most one or isotopic to the stable oriented double cover of a non-orientable minimal surface with a vertical handle atta…
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.
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.
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.
In this article we address a number of features of the moduli space of spherical metrics on connected, compact, orientable surfaces with conical singularities of assigned angles, such as its non-emptiness and connectedness. We also consider some features of the forgetful map from the above moduli space of spherical sur…
The paper extends minimal network theory to the sphere, proving local minimality.
problem Finding networks of minimal length on the sphere.
method Adapted spherical geometry, calibration method, and local metric perturbation estimates.
result Spherical minimal networks composed of great-circle arcs are locally length-minimizing within small geodesic balls.
Study dihedral spherical surfaces and their foliations.
problem Characterize dihedral spherical surfaces and their foliations.
method Define and analyze dihedral surfaces and their foliations, introduce geometric decompositions and deformations.
result Determine the dimension of the moduli space for dihedral surfaces.
Study on spherical conical metrics and their reducibility on compact Riemann surfaces.
problem Existence and geometric structure of reducible spherical conical metrics.
method Analysis of monodromy groups and geometric cutting of surfaces.
result Existence of reducible spherical conical metrics with saddle points on the same geodesic.