Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.
problem Estimating the Morse index of anisotropic minimal surfaces.
method Local analysis of Gauss map, conformal geometric techniques applied to the Gauss map.
result Upper and lower estimates for the Morse index of anisotropic minimal surfaces.
We study the Gauss map of minimal surfaces in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane H2. Conversely, any nowhere antiholomorphic harmonic map into $\mathbb{…
The paper studies the space of Gauss maps of complete minimal surfaces and their homotopy types.
problem Understanding the space of Gauss maps of complete minimal surfaces and their homotopy types.
method Proves the Gauss map assignment is a Serre fibration and determines the homotopy type of the space of meromorphic functions.
result The space of meromorphic functions on M that are the Gauss map of a complete full conformal minimal immersion has the same homotopy type as the space of all continuous maps from M to the 2-sphere. Harmonic and minimal great circle fibrations have special Gauss maps.
problem Characterizing Gauss maps of harmonic and minimal great circle fibrations.
method Analyzing the relationship between the Gauss map and the generating unit vector field.
result The Gauss map of a great circle fibration is harmonic (minimal) if and only if the generating unit vector field is harmonic (minimal).
Gauss map of complete minimal surfaces avoids certain hypersurfaces.
problem Characterizing the range of Gauss maps of minimal surfaces.
method Analyzing the degree of hypersurfaces omitted by the Gauss map.
result Gauss map can omit hypersurfaces of degree at most nn+2(n+1)n+2. Computes minimal dilatation for Thurston maps on surfaces.
problem Finding the minimal dilatation of Thurston maps on surfaces.
method Explicit computation using spectral radius in a congruence subgroup of PSL2(Z).
result Explicitly computes minimal dilatation for Thurston maps.
Minimal volume entropy vanishes for mapping tori over 3-manifolds.
problem Volume entropy of mapping tori over 3-manifolds.
method A variation of amenable category and minimal volume entropy of a homology class.
result Minimal volume entropy vanishes.
Paper proves uniqueness of minimal maps in curved spaces.
problem Proving uniqueness of minimal maps into Cartan-Hadamard manifolds.
method Proof based on convexity of functions in terms of squared singular values.
result Uniqueness theorem for minimal maps into Riemannian manifolds.
New stretch maps minimize distortion in geometric group theory.
problem Finding optimal maps in geometric group theory.
method Proving minimizers using modulus of curve families and MSP.
result Stretch maps are minimizers of mean quasiconformal distortion.
Let (S,h) be a closed hyperbolic surface and M be a quasi-Fuchsian 3-manifold. We consider incompressible maps from S to M that are critical points of an energy functional F which is homogeneous of degree 1. These "minimizing" maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic…
New inequality helps map stability in minimal surfaces.
problem Stability of minimal surfaces in Rn. method Developing new inequalities and perspectives on minimal surfaces.
result Reproves instability of classical minimal surfaces like Enneper.
The paper proves a Fenchel theorem for Gauss maps and shows circles and disks minimize certain energies.
problem Finding minimizers of nonlocal curvature energies.
method Combining Fenchel-type theorems with geometric analysis techniques.
result Circles and disks minimize specific energy functionals.
Study harmonic mappings and submanifolds using Bochner technique.
problem Classical theorems in harmonic mappings and submanifolds.
method Generalized Bochner technique.
result New insights into classical theorems.
New proof of timelike minimal surfaces using split-harmonic maps.
problem Interpolating a split-Fourier curve to a timelike minimal surface.
method Using split-harmonic maps to solve the singular Björling problem.
result Solved the interpolation problem for timelike minimal surfaces.
The Hopf fibration is rigid among minimal maps between spheres.
problem Characterizing minimal submersions between spheres.
method Analyzing the properties of the Hopf fibration and minimal maps.
result The Hopf fibration is the only minimal submersion from S3 to S2 under certain conditions. Study on unique generalized Gauss maps of minimal surfaces sharing hypersurfaces in projective varieties.
problem Uniqueness of generalized Gauss maps for minimal surfaces with shared hypersurfaces in projective varieties.
method Analysis of minimal surfaces in Rn+1 with inverse images of hypersurfaces in a projective subvariety. result Generalization and improvement of previous results on the uniqueness of generalized Gauss maps.
Note on minimal maps' uniqueness via singular values.
problem Uniqueness of minimal maps into \(\mathbb{R}^n\).
method Using singular values and convexity of area functional, proving local linearity of singular value vectors.
result Improved uniqueness theorem for minimal graphs.
Study proves existence of non-trivial harmonic map flows to hemispheres.
problem Existence of non-trivial harmonic map flows to hemispheres.
method Construction of infinitely many weak solutions to harmonic map flow starting from non-minimizing but stationary maps.
result Proves existence of non-trivial self-expanding harmonic map flows to hemispheres.
This paper proves compactness of conformal Chern-minimal surfaces in Hermitian surfaces.
problem Compactness of conformal Chern-minimal surfaces in Hermitian surfaces.
method Proves compactness through bubble tree limit analysis.
result Compactness of conformal Chern-minimal surfaces is established with bounded area.
Study energy-minimizing maps in projective spaces, proving sharp bounds.
problem Finding optimal mappings in projective spaces.
method Proving lower bounds and characterizing energy-minimizing maps.
result Sharp lower bounds and characterization of energy-minimizing maps.
We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal resu…
Minimal vector fields on oscillator groups studied, with specific conditions for minimality.
problem Characterizing minimal left-invariant unit vector fields on oscillator groups.
method Analyzing structure constants and harmonic maps into the unit tangent bundle.
result Minimal vector fields defined by specific conditions on oscillator groups.
Study on harmonic maps on weighted Riemannian foliations.
problem Characterize harmonic maps on weighted foliations.
method Analyze transversally f-harmonic and (F,F′)f-harmonic maps. result Equivalence of transversally f-harmonic and (F,F′)f-harmonic maps in minimal foliations. The paper shows that the Gauss map of minimal surfaces is open and meagre in the space of holomorphic maps.
problem Characterizing the set of minimal surfaces with a specific Gauss map.
method Analyzing the spaces of conformal minimal immersions and holomorphic maps, and using topological properties.
result The Gauss map assignment is an open map, and the set of minimal surfaces satisfying the Osserman curvature estimate is meagre.
In this paper, we study the Lorentzian minimal surfaces in the Minkowski space-time with finite type Gauss map. First, we obtain the classification of this type of surfaces with pointwise 1-type Gauss map. Then, we proved that there are no Lorentzian minimal surface in the Minkowski space-time with null 2-type Gauss ma…
In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent Lp bounds for ∇kf that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in W4,p for all…
We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal surfaces. In the genus 0 case, our flow is just the harmonic map flow, and it tries…
Minimal maps from surfaces to torus found for various genus values.
problem Finding minimal degree maps from genus g surfaces to the torus. method Constructing simplicial degree d maps from a triangulation of a genus g surface to the 7-vertex triangulation of the torus. result Minimal maps exist for g≥1 and ∣d∣≥2g−1 for g≥3. Harmonic maps intersect all minimal surfaces with bounded curvature.
problem Intersection of harmonic maps with minimal surfaces.
method Nonconstant conformal harmonic maps intersecting bounded curvature minimal surfaces.
result Harmonic maps intersect every nonflat properly embedded minimal surface of bounded curvature.
Minimal action of mapping class group on character variety.
problem Character variety of Deroin-Tholozan representations.
method Geometric perspective using symplectic structure.
result Infinite mapping class group orbits are dense.
New proof of harmonic map uniqueness with analytic targets.
problem Uniqueness of energy-minimizing harmonic maps with analytic targets.
method Symmetric (log)-epiperimetric inequality for harmonic maps with analytic targets.
result Tangents at infinity of energy-minimizing harmonic maps are unique.
In [15] Robert Osserman proved that the image of the Gauss map of a complete, non flat minimal surface in R^3 with finite total curvature miss at most 3 points. In this paper we prove that the Gauss map of such a minimal immersions omit at most 2 points. This is a sharp result since the Gauss map of the catenoid omits …
Maps on surfaces can be embedded into spheres with minimal dimensions.
problem Embedding periodic maps of surfaces into spheres with the smallest possible dimensions.
method Determining the minimal dimensions m for embeddings of periodic maps of order n on surfaces of genus g into spheres Sm. result For each integer k>1, there exist infinitely many periodic maps such that the smallest possible m is equal to k. Minimal simplicial maps constructed for spheres and manifolds.
problem Constructing minimal simplicial maps of specific degrees.
method Triangulations and degree constructions for manifolds and spheres.
result Minimal triangulations for degree d self-maps of Sn−1imesS1. Study of Gauss maps for minimal surfaces in a specific 3D model.
problem Characterizing minimal surfaces in a non-standard 3D space.
method Defining and analyzing Gauss maps for surfaces in S2imesR, proving properties of these maps. result Minimal surfaces with the same non-constant Gauss map are related by specific isometries.
Dirac-harmonic maps are uncoupled under certain conditions.
problem Understanding the uncoupling of Dirac-harmonic maps.
method Critical points of a super-symmetric energy functional, with focus on harmonic maps.
result Dirac-harmonic maps are uncoupled under minimality assumption.
Minimal generating sets found for surface mapping groups.
problem Finding the smallest sets of elements needed to generate mapping class groups of surfaces.
method Analyzing surfaces with different genera and punctures to find minimal generating sets.
result Minimal generating sets found for Mg,p and Mg,p± with g≥3 and p≥0. Paper estimates Gaussian curvature of minimal graphs in a specific manifold.
problem Estimating Gaussian curvature of minimal graphs in MimesR. method Using Weierstrass representation via ℘−harmonic mappings and Schwarz lemma type results. result Proves Schwarz lemma type and Heinz type results for harmonic mappings.
In this note, we show that the solution to the Dirichlet problem for the minimal surface system in any codimension is unique in the space of distance-decreasing maps. This follows as a corollary of the following stability theorem: if a minimal submanifold Σ is the graph of a (strictly) distance-decreasing map, then $…
Rectifies singular set of harmonic maps into complex.
problem Regularity of harmonic maps into complex manifolds.
method Proves (m−2)-rectifiability of singular set. result Singular set is (m−2)-rectifiable. Characterizes harmonic morphisms preserving minimal submanifolds and finds novel area-minimising hypercones.
problem Understanding harmonic morphisms and their relationship to minimal submanifolds.
method Characterization of harmonic morphisms as weakly horizontally conformal maps preserving minimal submanifold equations, derivation of reduction properties for other co-dimensions, application to find novel area-minimising hypercones.
result Novel family of degree 4 area-minimising hypercones in R^m, m≥32.
In this article, we study the modified defect relations of the Gauss map of complete minimal surfaces in R3 and R4 on annular ends. We obtain results which are similar to the ones obtained by Fujimoto~[J. Differential Geometry \textbf{29} (1989), 245-262] for (the whole) complete minimal surfaces…
In this note, we prove a Schwarz-Pick type lemma for minimal maps between negatively curved Riemannian surfaces. More precisely, we prove that if f:M→N is a minimal map with bounded Jacobian between two complete negatively curved Riemann surfaces M and N whose sectional curvatures σM and σN satisfy $infσ_M …
The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the complex two-space. To achieve this purpose, we show the precise maximal number of exceptional values of the Gauss map for a complete minimal Lagran…
In this paper, we study the Gauss map of a free boundary minimal surface. The main theorem asserts that if components of the Gauss map are eigenfunctions of the Jacobi-Steklov operator, then the surface must be rotationally symmetric.
In this article, we study the ramification of the Gauss map of complete minimal surfaces in R^3 and R^4 on annular ends. We obtain results which are similar to the ones obtained by Fujimoto and Ru for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a c…
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.
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.