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
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48 results for “hyperbolic area”
A Sharp Isoperimetric Property of the Renormalized Area of a Minimal Surface in Hyperbolic Spacemath.DG
Minimal surfaces in hyperbolic space have a sharp area bound.
problem Bounding the renormalized area of minimal surfaces.
method Proving an inequality using conformal length of ideal boundary.
result Sharp isoperimetric property of renormalized area.
Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.
problem Finding area constraints for surfaces with constant mean curvature in hyperbolic 3-manifolds.
method Established an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and genus bounds.
result Area bound implies compactness for such surfaces, with specific proportional bounds for Bryant surfaces.
Geodesics on hyperbolic surfaces become evenly spread over time.
problem Distribution of geodesics on hyperbolic surfaces.
method Equidistribution analysis of geodesics with length less than T as T approaches infinity.
result Closed geodesics on hyperbolic surfaces of finite area become evenly spread over time.
The study finds the minimum average area ratio on hyperbolic manifolds and its relation to scalar curvature.
problem Finding the minimum average area ratio on hyperbolic manifolds.
method Analyzing the average area ratio and normalized total scalar curvature for hyperbolic n-manifolds.
result The average area ratio attains a local minimum of 1 at the hyperbolic metric.
Study of hyperbolic polyhedral surfaces with regular faces.
problem Understanding the properties of hyperbolic polyhedral surfaces with regular faces.
method Combinatorial and geometric analysis of hyperbolic polyhedral surfaces with regular faces.
result There is a gap between the areas of non-smooth hyperbolic polyhedral surfaces and smooth hyperbolic surfaces.
New bounds found for minimal surfaces in hyperbolic 3-manifolds.
problem Bounding the maximum principal curvatures of minimal surfaces in hyperbolic 3-manifolds.
method Short argument using work of Uhlenbeck and families of fibered hyperbolic 3-manifolds.
result Uniform lower bound for maximum principal curvatures greater than one.
Study on geodesic surfaces in hyperbolic 3-orbifolds, proving overlaps in area sets.
problem Understanding overlaps in geometric genus spectra of non-commensurable hyperbolic 3-orbifolds.
method Defined geometric genus spectrum and totally geodesic area set, proving results on overlaps.
result Arithmetic hyperbolic 3-orbifolds can have large overlaps in their totally geodesic area sets.
Study calculates the renormalized area of catenoids in hyperbolic spaces.
problem Calculating the renormalized area of catenoids in hyperbolic spaces.
method Variational characterization and Chern--Gauss--Bonnet formulas for locally conformally flat manifolds.
result Renormalized area of catenoids varies continuously from negative infinity to twice the area of totally geodesic hypersurfaces.
Motivated by classical theorems on minimal surface theory in compact hyperbolic three-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic three-manifold of finite volume. We prove any closed immersed incompressible surface can be deform…
We show that if P is an embedded least area (area minimizing) plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded.
We prove that there are finite area flat surfaces whose Veech group is an infinite cyclic group consisting of hyperbolic elements
Study counts minimal surfaces in curved 3D spaces, finding hyperbolic space minimizes area.
problem Counting minimal surfaces in negatively curved 3-manifolds.
method Introduced an asymptotic quantity to count area-minimizing surfaces and showed minimization by hyperbolic metric.
result Hyperbolic metric minimizes the quantity of area-minimizing surfaces in negatively curved 3-manifolds.
The paper proves inequalities for hyperbolic sets and curves.
problem Proving inequalities for sets and curves in hyperbolic geometry.
method Defining horocyclic Minkowski sums and proving inequalities for hyperbolic areas.
result Horocyclic Brunn-Minkowski inequality holds for hyperbolic sets.
Formula for renormalized area of hypersurfaces in hyperbolic spaces.
problem Calculating the renormalized area of asymptotically minimal hypersurfaces in hyperbolic spaces.
method Combining Chen's conformal invariant quantity and Chern-Gauss-Bonnet formulas.
result Extension of renormalized area formulas to higher dimensions and non-minimal cases.
Minimal surfaces and average area ratio found to be maximized by hyperbolic metrics.
problem Finding sharp relations between minimal surface entropy and average area ratio.
method Ricci flow with surgery and invariant measures.
result Minimal surface entropy maximized by hyperbolic metrics among metrics with scalar curvature ≥ -6.
Minimal area of Teichmüller curves in genus two is found to be 3π/5.
problem Finding the minimum hyperbolic area of Teichmüller curves in genus two.
method Combining small-area classification of orbifolds and affine descent construction for quadratic differentials, excluding patterns by arithmetic, marked-point, and covering obstructions.
result The minimum hyperbolic area of Teichmüller curves in genus two is 3π/5.
Study area minimizing currents in conformal cones, solving Dirichlet problems.
problem Area minimizing currents in conformal cones.
method Minimizing problem of area functionals, Dirichlet problem of minimal surface equations.
result Existence and uniqueness of area minimizing currents in a wide class of conformal manifolds.
New tiling algorithm for hyperbolic 3-manifolds, characterizing cusp areas.
problem Characterizing and computing maximal cusp areas in hyperbolic 3-manifolds.
method Developed a new tiling algorithm and provided simpler expressions for distances.
result Completely characterized the space of cusp neighborhoods and found the Epstein-Penner decomposition.
We prove, in the context of Hilbert geometry, the equivalence between the existence of an upper bound on the area of ideal triangles and the Gromov-hyperbolicity.
The paper examines flows that preserve area and length in hyperbolic geometry.
problem Preserving area and length in hyperbolic geometry.
method Inverse curvature flows for convex curves in hyperbolic plane.
result The flows converge to geodesic circles under certain conditions.
Sharp area estimates for minimal submanifolds in curved spaces.
problem Estimating the area of minimal submanifolds passing through a specific point.
method Proving sharp area estimates in hyperbolic and spherical spaces.
result Sharp area estimates analogous to Euclidean settings.
We consider properly immersed finite topology minimal surfaces S in complete finite volume hyperbolic 3-manifolds N, and in M x S(1), where M is a complete hyperbolic surface of finite area. We prove S has finite total curvature equal to 2πtimes the Euler characteristic of S, and we describe the geometry of the ends of…
Estimates area and spectrum of stable minimal surfaces in Euclidean and hyperbolic spaces.
problem Estimating the growth of area and spectrum of stable minimal surfaces.
method Elementary argument and stability inequality for Euclidean space; explicit area growth estimate for hyperbolic space; scalar curvature lower bound for spectrum.
result Minimal surfaces in Euclidean space grow like the Euclidean plane, and in hyperbolic space, explicit area growth estimates are derived.
Study counts minimal Lagrangians in hyperbolic surfaces with precise growth rate.
problem Counting minimal Lagrangians in hyperbolic surfaces.
method Uses Mirzakhani functions to show growth rate and proves rigidity of area spectrum.
result Number of minimal Lagrangians grows as .
Two curves in hyperbolic space share a bounded distance, leading to a minimizing surface.
problem Finding a surface minimizing area between two disjoint curves in hyperbolic space.
method Analyzing the asymptotic boundary of hyperbolic 3-space, applying Definition 1.8 for distance bounds, and proving Theorems 1.7 and 1.11.
result Existence of an area-minimizing surface between two disjoint curves with bounded distance.
We prove prove a bridge principle at infinity for area-minimizing surfaces in the hyperbolic space , and we use it to prove that any open, connected, orientable surface can be properly embedded in as an area-minimizing surface. Moreover, the embedding can be constructed in such a way that t…
This is an expository paper on Mom-technology, describing the recent work of the authors in this area (found in arXiv:math/0606072, arXiv:0705.4325, and arXiv:0809.0346) concerning the use of Mom-technology to find the minimum-volume compact hyperbolic 3-manifold and the 10 smallest cusped hyperbolic 3-manifolds. In ad…
We study the number and the length of systoles on complete finite area orientable hyperbolic surfaces. In particular, we prove upper bounds on the number of systoles that a surface can have (the so-called kissing number for hyperbolic surfaces). Our main result is a bound which only depends on the topology of the surfa…
Study Brownian motion on Grassmann manifold using matrix stochastic calculus.
problem Understanding Brownian motion on non-compact Grassmann manifold.
method Realize Brownian motion as matrix diffusion process, use matrix stochastic calculus, and hyperbolic Stiefel fibration.
result Connection to generalized Maass Laplacian of complex hyperbolic space.
Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries. Results on weighted best and random approximation and the new approach to floati…
New graph-based complexity measure for hyperbolic 3-manifolds.
problem Defining a complexity measure for hyperbolic 3-manifolds.
method Using Morse functions to graphs and Gromov area.
result Linear relationship between graph adjacency and metric complexity.
The paper extends the isoperimetric inequality to disconnected regions.
problem Generalizing the isoperimetric inequality to regions that can split area.
method Analyzes the inequality in Euclidean, spherical, and hyperbolic geometries, providing conditions for multiple regions.
result Necessary and sufficient conditions for the inequality to hold for multiple regions in different geometries.
Surveying tools for estimating geometric properties of hyperbolic knots.
problem Determining geometric properties of hyperbolic knots.
method Analyzing link diagrams to estimate geometric invariants.
result Estimating volume, cusp shape, and cusp area of hyperbolic knots.
Scharlemann and Thompson define a numerical complexity for a 3-manifold using handle decompositions of the manifold. We show that for compact hyperbolic 3-manifolds this is linearly related to a definition of metric complexity in terms of the areas of level sets of Morse functions.
New translations defined; curve shortening flow solved in hyperbolic plane.
problem Solving curve shortening flow in hyperbolic geometry.
method Introduced new translations, solved equations, analyzed ancient solutions.
result Explicit solutions and area estimates for ancient solutions.
Study uses renormalized area to determine metric expansion from minimal surfaces.
problem Recovering the expansion of asymptotically hyperbolic metrics from minimal surfaces.
method Uses renormalized area functional on minimal submanifolds to recover metric expansion.
result Proves rigidity for log-analytic metrics and determines obstruction tensor.
Study on high-codimensional minimal surfaces in hyperbolic space.
problem Understanding high-codimensional minimal surfaces in hyperbolic space.
method Investigating asymptotic behavior and boundary regularity of area-minimizing currents.
result Established boundary regularity results for high-codimensional minimal surfaces near their asymptotic boundaries.
The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.
problem Finding the minimum perimeter for polygons with a fixed area in hyperbolic geometry.
method Proving analogues of the discrete isoperimetric inequality for cyclic and tangential polygons in hyperbolic geometry, considering both single and multiple polygons.
result Established two versions of the isoperimetric inequality for multiple polygons in hyperbolic geometry with certain area or perimeter restrictions.
This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to expl…
Smooth cones can be approximated by smooth hypersurfaces.
problem Approximating cones with smooth surfaces.
method Hyperbolic unfoldings and Jacobi field operator potential theory.
result Every area minimizing cone can be approximated by smooth hypersurfaces.
We show that for a generic simple closed curve C in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exist a unique least area plane P in X with asymptotic boundary C. This result has interesting topological applications for constructions of canonical 2-dimensional objects in 3-mani…
Improved bound on the product of first Laplacian eigenvalue and area for genus three surfaces.
problem Bounding the product of the first eigenvalue of the Laplacian and the area for compact surfaces of genus three.
method Improved the bound established by Yang and Yau, using numerical computations for the hyperbolic Klein quartic surface.
result Showed that the product of the first eigenvalue of the Laplacian and the area is bounded above by approximately 21.668π.
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.
In this paper, we prove uniform lower bounds on the volume growth of balls in the universal covers of Riemannian surfaces and graphs. More precisely, there exists a constant such that if is a closed hyperbolic surface and another metric on with $\area(M,h)\leq δ\area(M,hyp)$ then for every radiu…
In this work we investigate the following isoperimetric problem: to find the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (the area of the part of the boundary contained in the horospheres is not included). We reduce the problem to the study of rotationa…
The paper defines a new metric space for convex bodies using hyperbolic geometry.
problem Defining a proper geodesic distance for convex bodies.
method Using intrinsic area to define a distance on convex bodies in hyperbolic space.
result The space of similarity classes of convex bodies has curvature bounded from below by -1.
We introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientation-preserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce v…