The paper proves the existence of capillary geodesics on Riemannian 2-disks.
problem Existence of capillary geodesics on Riemannian 2-disks with specific conditions.
method Analytical proof and examples.
result Existence of capillary geodesics with contact angle θ ∈ (0, π/2).
Minimal surfaces in hyperbolic space have a renormalized area criterion.
problem Minimal surfaces in hyperbolic space
method Renormalized area criterion
result Y must be a totally geodesic disk
We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These res…
Inverse mean curvature flow converges to a disk in hyperbolic space.
problem Understanding flow behavior in hyperbolic geometry.
method Inverse mean curvature flow with free boundary on geodesic spheres.
result Flow converges to a totally geodesic disk.
This paper shows that every totally-geodesic isometry from the unit disk to a finite-dimensional Teichmüller space for the intrinsic Kobayashi metric is either holomorphic or anti-holomorphic; in particular, it is a Teichmüller disk. Additionally, a similar result is proved for a large class of disk-rigid domains, whic…
The study finds minimal surfaces in complex space forms are often totally geodesic.
problem Characterizing minimal surfaces with specific geometric properties in complex space forms.
method Analyzing free-boundary minimal surfaces in geodesic balls of complex space forms.
result Minimal surfaces in certain complex space forms are either totally geodesic or superminimal.
Paper bounds total geodesic curvature using boundary data in hyperbolic gravity.
problem Bounding total geodesic curvature in a hyperbolic setting.
method Derives an upper bound for total geodesic curvature in terms of boundary data.
result Upper bound for total geodesic curvature expressed solely in terms of boundary data.
Sharp inequalities for curved surfaces and cones.
problem Optimizing areas in nonpositively curved spaces.
method Proving inequalities for disks and triangles in cones.
result Minimal area properties for specific shapes in cones.
The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincaré disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain Ω must necessarily be asymptotically totally geodesic. A…
In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the …
A Zoll metric is a Riemannian metric whose geodesics are all circles of equal length. Via the twistor correspondence of LeBrun and Mason, a Zoll metric on the 2 dimensional sphere corresponds to a family of holomorphic disks in CP_2 with boundary in a totally real submanifold P. In this paper, we show that for a fixed …
The study proves inequalities for area and boundary length of disks in convex manifolds.
problem Proving inequalities for area and boundary length of disks in convex manifolds.
method Analyzing disks with homotopically non-trivial boundaries in manifolds with convex mean curvature boundary and positive scalar curvature.
result Proves an inequality involving area and boundary length of disks.
We develop a global twistor correspondence for pseudo-Riemannian conformal structures of signature (++--) with self-dual Weyl curvature. Near the conformal class of the standard indefinite product metric on S^2 x S^2, there is an infinite-dimensional moduli space of such conformal structures, and each of these has the …
Loewner inequality proven for curved surfaces.
problem Proving Loewner's inequality for nonpositively curved surfaces.
method Combining Gauss-Bonnet formula with averaging argument using geodesic flow invariance.
result Found a disk with large total curvature around its center, leading to large area.
The study finds at least two short, simple geodesic chords on a disk with convex boundary.
problem Existence of short, simple geodesic chords on a 2-disk with convex boundary.
method Proof of existence using Riemannian geometry and bounds on lengths.
result Existence of at least two short, simple orthogonal geodesic chords on a 2-disk with convex boundary.
We show for a non homogeneous boundary value problem for the Ricci flow on the disk that when the initial metric has positive curvature and the boundary is convex then the initial metric is deformed, via the normalized flow and along sequences of times, to a metric of constant curvature and totally geodesic boundary. W…
We determine all Finsler metrics of Randers type for which the Riemannian part is a scalar multiple of the Euclidean metric, on an open subset of the Euclidean plane, whose geodesics are circles. We show that the Riemannian part must be of constant Gaussian curvature, and that for every such Riemannian metric there is …
The paper finds at least N orthogonal Finsler geodesic chords in a disk-like manifold.
problem Existence and multiplicity of orthogonal Finsler geodesic chords in a disk-like manifold.
method Study of Finsler geodesic chords under reversibility assumption.
result At least N orthogonal Finsler geodesic chords found in a disk-like manifold.
Geodesic disks maximize the first non-trivial Neumann eigenvalue on spheres.
problem Maximizing the first non-trivial Neumann eigenvalue on spheres.
method Proving maximizers are geodesic disks.
result Geodesic disks maximize the first non-trivial Neumann eigenvalue.
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 goal of the article is to provide different explicit quantifications of the non density of simple closed geodesics on hyperbolic surfaces. In particular, we show that within any embedded metric disk on a surface, lies a disk of radius only depending on the topology of the surface (and the size of the first embedded…
New Fuchsian groups found with special embedding properties.
problem Finding new Fuchsian groups with specific embedding properties.
method Using period domains and properties of complex hyperbolic surfaces.
result First cocompact nonarithmetic Fuchsian groups with modular embedding not commensurable with triangle groups.
Geodesics on polygons in a unit disk are studied with unique metric properties.
problem Characterizing geodesics on polygons within a unit disk.
method Defining a metric on polygons such that geodesics are curves in the family C.
result The constructed metric space is not isometric to any convex domain in R^2.
Paper finds conformal metrics on a disk with specific curvatures.
problem Existence of conformal metrics with prescribed Gaussian and geodesic curvatures.
method Computation of Leray-Schauder degree in a compact setting.
result Existence results under conditions involving both curvatures.
The paper proves ideal triangulations and disk unfolding for singular flat surfaces.
problem Proving ideal triangulations and disk unfolding for singular flat surfaces.
method Using geodesic triangulation and finite geodesic connections.
result Each singular flat surface has an ideal triangulation and can be unfolded into a flat disk.
We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetr…
Following ideas of Choe and Fernandez-do Carmo, we give sufficient conditions for a disk type surface, with piecewise smooth boundary, to be totally umbilical for a given Coddazi pair. As a consequence, we obtain rigidity results for surfaces in space forms and in homogeneous product spaces that generalizes some known …
The paper solves a problem in metric geometry for disks with negative curvature.
problem Prescribing negative Gaussian curvature on the disk and boundary geodesic curvature.
method Variational approach and refined blow-up analysis for approximated problems.
result Existence of solutions under natural curvature assumptions.
Rare Teichmüller disks converge to small limit sets.
problem Understanding limit sets of Teichmüller disks.
method Analyzing Thurston boundary of Teichmüller space.
result Teichmüller disks with smallest limit sets are exceptional.
Study on cylindrical handlebody-knots with symmetry and rigidity properties.
problem Characterizing symmetry groups of cylindrical handlebody-knots of genus two.
method Classification of essential annuli and analysis of symmetry groups based on Koda-Ozawa theorem.
result Most exteriors of genus two cylindrical handlebody-knots contain no essential disks or tori, and type 3-3 annuli are often unique up to isotopy. We prove that the unique least-perimeter way of partitioning the unit 2-dimensional disk into three regions of prescribed areas is by means of the standard graph consisting in three balanced constant geodesic curvature curves meeting themselves at 120 degrees, and reaching orthogonally the boundary of the disk.
Classifies totally geodesic submanifolds in Hopf-Berger spheres.
problem Classifying totally geodesic submanifolds in Hopf-Berger spheres.
method Investigates Hopf-Berger spheres, a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations.
result Discovered intriguing examples of totally geodesic submanifolds, including isometric submanifolds of real projective spaces and non-congruent submanifolds.
Unique geodesics selected by energy minimization in Teichmüller space.
problem Finding a unique geodesic between points in Teichmüller space.
method Energy minimization of harmonic map rays, extending Thurston boundary.
result Selection of a unique Thurston geodesic through points in Teichmüller space.
Classifies totally geodesic submanifolds in symmetric spaces.
problem Classifying submanifolds in symmetric spaces.
method Classification of totally geodesic submanifolds in products of rank one symmetric spaces.
result Infinitely many examples of irreducible totally geodesic submanifolds in Hermitian symmetric spaces.
The paper compares eigenvalues of Laplacians on fibred manifolds using symmetrization techniques.
problem Comparing eigenvalues of Laplacians on fibred Riemannian manifolds.
method Using fiberwise spherical and Euclidean symmetrization, the paper proves various comparison theorems.
result Eigenvalues of fibred manifolds are compared to their base manifolds under certain curvature conditions.
In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for…
We show that any totally geodesic submanifold of Teichmuller space of dimension greater than one covers a totally geodesic subvariety, and only finitely many totally geodesic subvarieties of dimension greater than one exist in each moduli space.
The paper proves the existence and properties of geodesics on convex surfaces.
problem Existence and properties of geodesics on convex surfaces with free boundaries.
method Free boundary curve shortening flow on closed surfaces with strictly convex boundary.
result Existence of two free boundary embedded geodesics and geodesics with Morse Index 1 and 2.
Totally geodesic surfaces found in knots and links.
problem Finding totally geodesic surfaces in knots and links.
method Constructing infinite families of knots and links with totally geodesic spanning surfaces in various 3-manifolds.
result Infinite families of knots and links with totally geodesic spanning surfaces in multiple 3-manifolds.
Finite totally geodesic hypersurfaces in curved manifolds proven.
problem Characterizing totally geodesic hypersurfaces in curved manifolds.
method Analytic Riemannian manifold analysis with negative sectional curvature.
result Closed manifolds with negative curvature have only finitely many totally geodesic hypersurfaces.
Classifies totally geodesic submanifolds in exceptional symmetric spaces.
problem Classifying totally geodesic submanifolds in exceptional symmetric spaces.
method Classification and introduction of an invariant (Dynkin index) for totally geodesic embeddings.
result Existence of a totally geodesic submanifold of minimal codimension with specific properties.
Paper solves degenerated circle pattern metric problem in spherical geometry.
problem Existence and rigidity of (degenerated) circle pattern metrics with prescribed total geodesic curvatures.
method Defined prescribed combinatorial Ricci flows and studied their convergence.
result First degenerated result for total geodesic curvatures in spherical background geometry.
Classifies totally geodesic submanifolds in specific geometric spaces.
problem Identifying totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
method Developed new techniques for studying totally geodesic submanifolds in analytic Riemannian manifolds, homogeneous spaces, and Riemannian cones.
result Obtained a classification of totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
On simple geodesic disks of constant curvature, we derive new functional relations for the geodesic X-ray transform, involving a certain class of elliptic differential operators whose ellipticity degenerates normally at the boundary. We then use these relations to derive sharp mapping properties for the X-ray transform…
In this note, we observe that if B is a ball in a Euclidean space with dimension n, n≥3, then a stable CMC hypersurface Σ with free boundary in B satisfies \[ nA\leq L\leq nA\left( \frac{1+\sqrt{1+4(n+1)H^2}}{2} \right)\,, \] where L, A and H denote the length of ∂Σ, the area of Σ and the…
The study finds totally geodesic surfaces in hyperbolic 3-manifolds and verifies a conjecture.
problem Finding totally geodesic surfaces in hyperbolic 3-manifolds.
method Developed algorithms to determine and verify the existence of totally geodesic surfaces.
result Discovered nine 3-manifolds with totally geodesic surfaces and verified Menasco-Reid's conjecture for knots up to 12 crossings.
Minimal hypersurfaces in S^5 with specific curvature properties are totally geodesic.
problem Characterizing minimal hypersurfaces in S^5 with certain curvature conditions.
method Analyzing hypersurfaces with constant scalar curvature and zero Gauss curvature.
result Minimal hypersurfaces in S^5 with these curvature properties are totally geodesic.
In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…