Study horospheres in hyperbolic 3-manifolds with degenerate ends.
problem Understanding geodesics in degenerate hyperbolic 3-manifolds.
method Analyze almost minimizing geodesics to explore horospheres.
result Identify geodesics passing through thin parts of the manifold.
For Finsler metrics (no reversibility assumed) on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove in particular, that for almost all asymptotic directions the minimal rays with these directions laminate the universal cove…
A major breakthrough in the theory of topological algorithms occurred in 1992 when Hyam Rubinstein introduced the idea of an almost normal surface. We explain how almost normal surfaces emerged naturally from the study of geodesics and minimal surfaces. Patterns of stable and unstable geodesics can be used to character…
The minimal area of Finsler disks with minimizing geodesics is at least 6/π r^2.
problem Finding the minimal area of Finsler disks with minimizing geodesics.
method Discretizing the Finsler metric using random geodesics and applying integral geometry formulas.
result The Holmes--Thompson area of Finsler disks with minimizing geodesics is at least 6/π r^2, with examples showing the inequality is sharp.
The study examines gradient almost Yamabe solitons in warped product manifolds and their geometric properties.
problem Investigating the geometry of gradient almost Yamabe solitons in warped product manifolds.
method Presenting geometric rigidity results, investigating existence conditions, and classifying specific solitons.
result Classification of rotational gradient almost Yamabe solitons in RimesfRn. The paper examines the geometry of specific submanifolds in flag manifolds.
problem Understanding the geometry of invariant almost semi Kähler submanifolds.
method Analyzing homogeneous spaces as almost Hermitian submanifolds of flag manifolds.
result Minimal and totally geodesic properties of certain submanifolds.
Paper proves minimality of stable submanifolds in flat neighbourhoods.
problem Proving the uniqueness of minimizers of elliptic functionals in flat neighbourhoods.
method Introducing a penalized minimization problem to exploit almost minimizers' regularity.
result Quantitative estimates of minimizers in flat neighbourhoods.
We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolut…
Almost Zoll affine surface found on cylinder.
problem Finding surfaces with special geodesic properties.
method Exhibited an affine structure on a cylinder.
result Affine structure on cylinder is almost Zoll.
In this paper we extend a recent result of Collin-Rosenberg ({\it a solution to the minimal surface equation in the Euclidean disc has radial limits almost everywhere}) to a large class of differential operators in Divergence form. Moreover, we construct an example (in the spirit of \cite{CR2}) of a minimal graph in $\…
Researchers study invariants of specific mappings in affine spaces.
problem Investigating invariants of second type almost geodesic mappings in non-symmetric affine spaces.
method Used computational methods to derive invariants, including generalized Thomas and Weyl projective tensors.
result Generalized invariants of second type almost geodesic mappings were successfully derived.
Study on surface group representations in PU(2,1) leading to convex-cocompact examples.
problem Nonmaximal representations of surface groups in PU(2,1).
method Analysis of convex-cocompact representations with unique equivariant minimal surfaces.
result Existence of convex-cocompact representations with specific properties.
New invariants found for a specific type of mapping in generalized Riemannian space.
problem Understanding transformations of Christoffel symbols in generalized Riemannian space.
method Analysis of third type almost geodesic mappings.
result Obtained new invariants that are projective parameters and tensors.
In this paper we study geodesics of left-invariant sub-Riemannian metrics on SO(3) and almost-Riemannian metrics on S2. These structures are connected with each other, and it is possible to use information about one of them to obtain results about another one. We give an explicit parameterization of sub-Riemannian g…
The paper proves a gap theorem for almost non-negatively curved manifolds.
problem Proving a gap theorem for almost non-negatively curved manifolds.
method Two novel technical tools: controlling the spreading of minimal geodesics and Ricci flow smoothing.
result Closed manifolds with bounded geometry are diffeomorphic to torus bundles.
The paper proves a transformation theorem under a monotone property of almost Euclidean factors of geodesic balls.
problem The non-increasing property of numbers of almost Euclidean factors of geodesic balls.
method Proves a transformation theorem under a non-decreasing property compared to the non-increasing property.
result Shows that for a manifold with nonnegative Ricci curvature, if its universal cover is polar at infinity and the number of almost Euclidean factors is monotone, then its fundamental group is finitely generated and virtually abelian.
Study on geodesics on high genus expander surfaces, proving filling and non-simple properties.
problem Properties of geodesics on expander surfaces of high genus.
method Adapting Margulis' counting strategy to low length scales.
result Almost every geodesic of certain lengths is filling or non-simple.
New findings on hypersurfaces with specific curvature properties in space forms.
problem Characterizing hypersurfaces with almost constant curvature in space forms.
method Analyzing starshaped hypersurfaces with various curvature conditions.
result Closed starshaped hypersurfaces with almost constant mean curvature or higher order mean curvature are close to geodesic spheres.
Geodesic lines on nearly Kähler S^3×S^3 found.
problem Finding geodesic lines on nearly Kähler manifolds.
method Explicit parameterization of geodesic lines on S^3×S^3.
result Explicit parameterization of geodesic lines on S^3×S^3.
Study of complex surfaces in a specific pseudo-Riemannian space.
problem Classifying complex surfaces in a particular pseudo-Riemannian space.
method Investigation of totally geodesic and parallel second fundamental form complex surfaces.
result Complete classification of totally geodesic and parallel second fundamental form complex surfaces.
For modelling of various physical processes, geodesic lines and almost geodesic curves serve as a useful tool. Trasformations or mappings between spaces (endowed with a metric or connection) which preserve such curves play an important role in physics, particularly in mechanics, and in geometry as well. Our aim is to c…
Study and classify totally geodesic submanifolds in nearly Kaehler flag manifold.
problem Classifying totally geodesic submanifolds in nearly Kaehler flag manifold.
method Developed structural approach to nearly Kaehler flag manifold, expressed curvature tensor in terms of nearly Kaehler structure and canonical complex structures.
result Classified almost complex totally geodesic submanifolds of nearly Kaehler flag manifold and its semi-Riemannian counterpart.
Introduces new types of submersions from quaternionic Hermitian manifolds.
problem Exploring new types of submersions in quaternionic geometry.
method Definition and investigation of h-conformal and almost h-conformal slant submersions.
result Properties and conditions for these submersions are studied.
In this paper almost complex surfaces of the nearly Kähler S3×S3 are studied in a systematic way. We show that on such a surface it is possible to define a global holomorphic differential, which is induced by an almost product structure on the nearly Kähler S3×S3. We also find a correspondence betwe…
We find the minimal value of the length in de Sitter space of closed space-like curves with non-vanishing non-space-like geodesic curvature vector. These curves are in correspondence with closed almost-regular canal surfaces, and their length is a natural magnitude in conformal geometry. As an application, we get a low…
Study new submersion types on quaternionic manifolds.
problem Explore new submersion types on quaternionic manifolds.
method Introduce and study h-conformal semi-invariant submersions and almost h-conformal semi-invariant submersions.
result Properties of these submersions, including foliations and geodesic conditions.
Study classifies totally geodesic surfaces in nearly Kähler space.
problem Classifying totally geodesic surfaces in nearly Kähler space.
method Detailed description of nearly Kähler space and classification of surfaces.
result Classification of totally geodesic almost complex surfaces.
Survey of recent results in weak almost contact structures.
problem New geometric structures replacing complex structure in contact manifolds.
method Survey of recent findings in weak almost contact manifolds.
result Recent results on geodesic and Killing fields, rigidity and splitting theorems, etc.
Study geodesics in totally real submanifolds of Kähler-Einstein manifolds.
problem Geodesics in totally real submanifolds of Kähler-Einstein manifolds.
method Define a canonical connection and geodesics on the space of totally real submanifolds, and study their properties.
result Existence and uniqueness of geodesics in totally real submanifolds.
Study null geodesics on even-dimensional conformal manifolds, finding Einstein metrics and CR structures.
problem Investigate null geodesics and their geometric properties on conformal manifolds.
method Analyze the Weyl tensor and its effects on the geometry of null geodesic congruences.
result Find Einstein metrics and CR structures on the leaf space of null geodesic congruences.
If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct product SL(2,R)^m x SL(2,C)^n$, with m + n > 1. (In geometric terms, this can be in…
The Weil-Petersson metric's curvature vanishes for surfaces with short geodesics.
problem Understanding the vanishing rate of Weil-Petersson sectional curvatures.
method Analyzing the moduli space of Riemann surfaces, bounding curvature, and using product metrics.
result Bounding the sectional curvature away from zero in terms of geodesic lengths.
Maps converge to simpler structures under certain tension conditions.
problem Understanding convergence of maps under tension decay.
method Sharp criterion on tension decay ensures subconvergence to simpler structures.
result Maps subconverge to a structure made of harmonic maps.
Proves min-max theory for constant geodesic curvature curves on closed surfaces.
problem Prescribing mean curvature on surfaces with constant geodesic curvature.
method Min-max theory applied to classify blowups and ensure almost embedded solutions.
result Produces a solution with constant geodesic curvature c on closed surfaces. In this note we prove convexity, in the sense of Colding-Naber, of the regular set of solutions to some complex Monge-Ampere equations with conical singularities along simple normal crossing divisors. In particular, any two points in the regular set can be joined by a smooth minimal geodesic lying entirely in the regul…
The paper studies conformal generic submersions from almost Hermitian manifolds.
problem Exploring new types of submersions in almost Hermitian manifolds.
method Extending existing submersions to study conformal generic submersions.
result Characterizations and properties of conformal generic submersions are obtained.
The paper constructs a new Ricci-flat metric on almost abelian Lie groups.
problem Finding Lorentzian homogeneous Ricci-flat metrics on almost abelian Lie groups.
method Constructing left-invariant metrics on almost abelian Lie groups, focusing on dimensions four or higher.
result A new Ricci-flat metric that generalizes the Petrov solution to higher dimensions.
Study horocycle and geodesic orbits on hyperbolic surfaces.
problem Understanding the relationship between horocyclic and geodesic orbits on geometrically infinite surfaces.
method Analyzing the topological dynamics of the horocycle flow on hyperbolic surfaces, focusing on specific vectors and their orbits.
result The closure of the horocycle flow of a non-periodic vector intersects geodesic orbits along an unbounded sequence of points.
Constructs ε-splitting maps for geodesic balls with non-negative Ricci curvature.
problem Constructing ε-splitting maps for geodesic balls with non-negative Ricci curvature.
method Induction and stratified almost Gou-Gu Theorem for finding directional points; error estimates for projections.
result Constructs ε-splitting maps on concentric geodesic balls with uniformly small radius. Paper shows convergence types match for almost minimal sets.
problem Matching convergence types for almost minimal sets.
method Hausdorff and varifold convergence comparison on almost minimal sets.
result Hausdorff and varifold convergence coincide on almost minimal sets.
First-passage percolation affects graph properties like curvature and geodesics.
problem Effect of first-passage percolation on graph curvature and geodesics.
method Randomly perturbs the metric of a graph by assigning random edge lengths.
result Non-positive curvature and geodesic properties are not preserved by first-passage percolation.
We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichmüller space, if the quasicircle is sufficiently close to being the bou…
In an earlier paper we discussed soldered forms, multivector fields and Riemannian metrics. In particular, we showed that a Riemannian submanifold is totally geodesic iff the metric is soldered to the submanifold. In the present note we discuss general, soldered tensor fields. In particular, we prove that the almost co…
Study calculates first p-widths of unit disk.
problem Computing first p-widths of the unit disk. method Regularity result for integral 1-varifolds on compact 2-manifolds with convex boundary, applied to unit disk.
result Computed first p-widths for p=1,...,4. New polyhedron example disproves Durer's conjecture.
problem Durer's conjecture on unfolding convex polyhedra.
method Constructing a specific polyhedron with pseudo-edge graph.
result Example polyhedron disproves conjecture.
Unique tangent cones found for boundary points of 2D almost-minimizing currents.
problem Characterizing boundary points of two-dimensional almost-minimizing currents.
method Combining epiperimetric inequality and almost-monotonicity formula.
result Tangent cones at singular boundary points are unique.
Study on minimizing closed geodesics on polygons and disks.
problem Understanding and minimizing closed geodesics on polygons and disks.
method Developed new techniques to study minimizing properties of 1/k geodesics on doubled polygons.
result Demonstrated a sequence of doubled polygons whose closed geodesics exhibit unbounded minimizing properties.
Geodesics spiral around compact subsets in CAT(0) spaces.
problem Understanding geodesic spiraling in CAT(0) spaces.
method Logarithm law-type result for geodesics in quotients of rank one CAT(0) spaces.
result Proved logarithm law for geodesic spiraling in certain CAT(0) spaces.