New method controls surface extrinsic diameter for positive scalar curvature metrics.
problem Preventing complete metrics with positive scalar curvature on surfaces within manifolds.
method Interior control for extrinsic diameter of surfaces with positive scalar curvature.
result Closed aspherical manifolds cannot have complete metrics with positive scalar curvature when subsets are removed.
The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.
problem Understanding the structure of compact Riemannian manifolds with positive Ricci curvature.
method Establishing the extrinsic Bonnet-Myers theorem and showing almost rigidity for hypersurfaces.
result Proven the extrinsic Bonnet-Myers theorem for positive Ricci curvature manifolds and demonstrated almost rigidity for hypersurfaces.
The diameter of a disc filling a loop in the universal covering of a Riemannian manifold may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram filling a word that represents the i…
We show that the extrinsic diameter of immersed flat tori in the 3-sphere is π under a certain topological condition for the projection of their asymptotic curves with respect to the Hopf fibration.
New bounds on knot distortion and Seifert surface properties.
problem Understanding the distortion of knots and properties of Seifert surfaces.
method Analyzing embeddings of Seifert surfaces and using properties of monodromy maps.
result Bounds on the distortion of certain knots and properties of Seifert surfaces.
In this paper, we study the complete bounded λ-hypersurfaces in weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with ∣A∣≤α and get some applications of the volume comparison theorem. Secondly, we consider the relation amo…
Characterizes submanifolds with minimum ratio of diameter to focal radius.
problem Finding submanifolds with the minimum ratio of extrinsic diameter to focal radius.
method Combining K. Sakamoto's classification of submanifolds with planar geodesics and A. Schur's Bow Lemma for space curves.
result Essentially round spheres or Veronese embeddings of projective spaces achieve the minimum ratio.
This paper is a survey of some of the developments in coarse extrinsic geometry since its inception in the work of Gromov. Distortion, as measured by comparing the diameter of balls relative to different metrics, can be regarded as one of the simplist extrinsic notions. Results and examples concerning distorted subgrou…
The paper provides estimates for Steklov eigenvalues of surfaces with boundary.
problem Estimating Steklov eigenvalues of surfaces with boundary components.
method Computable lower bounds for the first non-zero Steklov eigenvalue using geometric quantities specific to manifolds with boundary.
result The geometry of the manifold away from the boundary affects the Steklov eigenvalue.
We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the gradient of harmonic functions.
Exact diameter found for some Riemann surfaces.
problem Calculating the diameter of compact Riemann surfaces exactly.
method Proved for a specific class of surfaces (generalized Bolza surfaces).
result Diameters of generalized Bolza surfaces are equal to their fundamental polygon radii.
Sharp bound on smallest diameter of hyperbolic surfaces.
problem Finding the smallest possible diameter of hyperbolic surfaces.
method Proved a specific formula for the minimal diameter.
result Minimal diameter is log(g)+25loglog(g)+O(1). Short proof shows infinite diameter for surface diffeomorphisms.
problem Infinite diameter of surface diffeomorphisms group.
method Short proof using Lp-diameter concept. result Infinite Lp-diameter of Diff0(S,area) group. Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.
problem Understanding the relationship between intrinsic and extrinsic metrics and surface area.
method Examined surfaces within a unit ball in R3, provided lower bounds on the ratio in terms of area, and showed non-existence of global lower bounds.
result Found that the ratio of intrinsic to extrinsic metrics has a lower bound in terms of surface area, but no global lower bound exists.
We study the topology of (properly) immersed complete minimal surfaces P2 in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see \cite{Pa}). We present an alternative and partially unified proof of…
The study defines and characterizes extrinsic catenaries in hyperbolic space.
problem Understanding catenaries in hyperbolic geometry.
method Defined extrinsic catenaries in hyperbolic plane, characterized them, and proved their relation to minimal surfaces.
result Extrinsic catenaries in hyperbolic space are critical points of a potential functional and generating curves of minimal surfaces.
Study systoles and diameters on hyperbolic surfaces, finding an upper bound for their ratio.
problem Understanding the relationship between systoles and diameters on hyperbolic surfaces.
method Exploring the inequality between systoles and diameters, deducing an upper bound for their ratio.
result The ratio of systoles and diameters has a genus-dependent upper bound.
Tight embeddings of 2-tori in 3D space contain short loops.
problem Finding the shortest non-contractible loops in twisted 2-tori.
method Proving systolic inequalities for T2 embeddings in R3. result Highly twisted 2-tori contain non-contractible loops of small diameter.
Sharp upper bound found for stable minimal surfaces.
problem Bounding the diameter of stable minimal surfaces.
method Analyzing three-dimensional Riemannian manifolds with specific curvature conditions.
result Sharp upper bound for the diameter of stable minimal surfaces.
Paper bounds surface diameter and solves Plateau-Douglas problem.
problem Bounding the diameter of compact surfaces and solving the Plateau-Douglas problem.
method Geometric argument based on Topping's diameter bound for closed surfaces.
result Explicit nonexistence criterion for the Plateau-Douglas problem.
Unified study of surfaces using Clifford algebras.
problem Classifying immersed surfaces in various manifolds.
method Using Clifford algebras to construct formalism for immersed bilegendrian surfaces.
result Full classifications of immersed bilegendrian surfaces in the unit tangent bundle of the 3-sphere.
The study characterizes constant curvature manifolds using ruled surfaces.
problem Characterizing manifolds of constant curvature using ruled surfaces.
method Investigating ruled surfaces in 3d Riemannian manifolds, finding stiction curve, distribution parameter, and fundamental forms.
result Identifies necessary and sufficient conditions for extrinsically flat surfaces to be ruled and proves manifold properties.
Study on biconservative surfaces in 4D hyperbolic space, providing extrinsic descriptions.
problem Characterizing biconservative surfaces in hyperbolic space.
method Local extrinsic description through normal flow and analysis of vector fields.
result Classification of non-CMC, PNMC biconservative surfaces in H4 into three cases. Geodesic spheres are the only quasicomplete surfaces in 3-space-forms.
problem Classifying quasicomplete surfaces in 3-space-forms.
method Using quasicompleteness as a weaker form of completeness, the global geometry of surfaces is determined.
result Geodesic spheres are the only quasicomplete surfaces of constant extrinsic curvature in 3-space-forms.
Paper proves a Penrose inequality in extrinsic geometry.
problem Proving a Penrose inequality in extrinsic geometry.
method Analyzing minimal capillary surfaces and their free energy.
result Established an extrinsic Penrose inequality.
In this article, we give the integrability conditions for the existence of an isometric immersion from an orientable simply connected surface having prescribed Gauss map and positive extrinsic curvature into some unimodular Lie groups. In particular, we discuss the case when the Lie group is the euclidean unit sphere $…
The paper estimates surface diameter in conformal spaces.
problem Estimating the diameter of surfaces in conformally flat spaces.
method Using mean curvature and boundary length, the paper gives an upper bound for the intrinsic diameter.
result The result provides an a priori estimate for connected solutions of Plateau's problem and a necessary condition for the existence of such solutions.
New model calculates logarithmic surface diameter.
problem Calculating diameter of random hyperbolic surfaces.
method Exploration process inspired by graph breadth-first search.
result Diameter is logarithmic in surface genus.
We determine the asymptotic growth rate of the diameter of the random hyperbolic surfaces constructed by Brooks and Makover. This model consists of a uniform gluing of 2n hyperbolic ideal triangles along their sides followed by a compactification to get a random hyperbolic surface of genus roughly n/2. We show that…
This note shows surfaces in stable 3D data are bounded by area and diameter.
problem Bounding stable surfaces in 3D initial data sets.
method Demonstrating area and diameter bounds for stable, marginally outer trapped surfaces.
result Stable surfaces in 3D data sets are bounded by both area and diameter.
Metric surfaces can be divided into small triangles.
problem Decomposing metric surfaces into triangles.
method Proving any metric space homeomorphic to a surface can be divided into non-overlapping convex triangles of small diameter.
result Metric surfaces can be decomposed into triangles of arbitrarily small diameter.
We prove that every complete connected immersed surface with positive extrinsic curvature K in H2×R must be properly embedded, homeomorphic to a sphere or a plane and, in the latter case, study the behavior of the end. Then, we focus our attention on surfaces with positive constant extrinsic curvature (K−s…
We provide a quantitative obstruction to collapsing surfaces of genus at least 2 under a lower curvature bound and an upper diameter bound. Keywords: curvature; diameter; volume; filling radius; systole; Gromov-Hausdorff distance
Given a 2-dimensional surface M and a constant C we construct a Riemannian metric g, so that diameter diam(M,g)=1 and every 1-cycle dividing M into two regions of equal area has length >C. It follows that there exists no universal inequality bounding 1-width of M in terms of its diameter. This answers a question of Ste…
Quaternionic reformulation simplifies surface curvature theory.
problem Prescribed extrinsic curvature of surfaces.
method Quaternionic reformulation of Labourie's theory.
result Simpler proofs and higher-dimensional generalization.
New geometric invariant from min-max width of spheres on Riemannian 2-spheres.
problem Understanding the min-max width of spheres associated to distance functions.
method Application of min-max methods to pairs of points on Riemannian 2-spheres.
result The min-max width does not always equal half the length of a simple closed geodesic.
For large genus hyperbolic surfaces, this paper proves eigenvalue conditions and diameter bounds.
problem Eigenvalue and diameter bounds for large genus hyperbolic surfaces.
method Analysis of moduli space of hyperbolic surfaces with Weil-Petersson metric.
result Generic hyperbolic surfaces of large genus have first eigenvalues greater than 3/16 - ε.
The paper proves a linear diameter bound for hyperbolic knot complexes.
problem Understanding the diameter of Kakimizu complexes for hyperbolic knots.
method Defined a complex ISℓ(K) to study incompressible Seifert surfaces and proved its diameter has a linear upper bound. result The diameter of the Kakimizu complex for hyperbolic knots grows linearly with genus, confirming a conjecture.
Extrinsic Geometric Flow (EGF) for a codimension-one foliation has been recently introduced by authors as deformations of Riemannian metrics subject to quantities expressed in terms of its second fundamental form. In the paper we introduce soliton solutions to EGF and study their geometry for totally umbilical foliatio…
The study examines flip-graphs of non-orientable surfaces and their diameters.
problem Understanding the structure and diameter of flip-graphs of non-orientable surfaces.
method Constructing triangulations of non-orientable surfaces, quotienting by homeomorphisms, and analyzing the resulting flip-graphs.
result Bounds on the diameter of flip-graphs of non-orientable surfaces, with specific growth rates for Möbius strips.
Extends diameter bounds for submanifolds with boundary and minor curvature restrictions.
problem Bounding the diameter of submanifolds with boundary and minor curvature restrictions.
method Applies bounds dependent on mean curvature and area to minimal, constant mean curvature, and prescribed mean curvature surfaces.
result Diameter bounds for submanifolds with boundary and minor curvature restrictions.
Find simple geodesics in hyperbolic surfaces with bounded diameter.
problem Finding efficient pants decompositions in hyperbolic surfaces.
method Use Uryson width to bound the diameter of curves in a pants decomposition.
result Each curve in the pants decomposition is contained in a ball of bounded diameter.
We prove that the minimal diameter of a hyperbolic compact orientable surface of genus g is asymptotic to logg as g→∞. The proof relies on a random construction, which we analyse using lattice point counting theory and the exploration of random trivalent graphs.
Infinite diameter proved for contractible loops space.
problem Infinite diameter of contractible loops space in a compact surface.
method New functionals on normed groups, more general than quasi-morphisms.
result Proved infinite diameter of contractible loops space.
Uniform diameter bounds for Calabi-Yau fibrations with singular fibers.
problem Bounding the diameter of Calabi-Yau fibrations near singular fibers.
method Uniform diameter bound proof for Calabi-Yau fibrations with canonical singular fibers.
result Uniform diameter bounds for all fibres in suitable rescaling.
This is an English translation of the following paper, published several years ago: Nikonorov Yu.G. On the geodesic diameter of surfaces with involutive isometry (Russian), Tr. Rubtsovsk. Ind. Inst., 2001, V. 9, 62-65, Zbl. 1015.53041. All inserted footnotes provide additional information related to the mentioned probl…
An extrinsic representation of a Ricci flow on a differentiable n-manifold M is a family of submanifolds S(t), each smoothly embedded in R^{n+k}, evolving as a function of time t such that the metrics induced on the submanifolds S(t) by the ambient Euclidean metric yield the Ricci flow on M. When does such a representa…
The paper calculates the size of origami orbit graphs in complex surfaces.
problem Calculating the size of origami orbit graphs in complex surfaces.
method Classification of SL(2,Z)-orbits of primitive origamis and reuse of machinery for Prym eigenforms. result Diameter bounds of O(N2/3logN) for orbit graphs in H(2) and H(4), H(6).