The study of orthospectrum and simple orthospectrum of hyperbolic surfaces.
problem Characterizing hyperbolic surfaces through their orthospectrum and simple orthospectrum.
method Analyzing the orthospectrum and simple orthospectrum of compact hyperbolic surfaces with geodesic boundary.
result Generic surfaces are determined by their orthospectrum and simple orthospectrum, and there are finitely many surfaces sharing the same orthospectrum and simple orthospectrum.
Rotation systems can't always be drawn in surfaces.
problem Rotation systems and simple drawings in surfaces.
method Extended the plane result to all fixed surfaces.
result Existence of rotation systems not arising from simple drawings in any fixed surface.
Simple constructions of semi-discrete and discrete surfaces using Jacobi elliptic functions.
problem Constructing semi-discrete and discrete surfaces explicitly.
method Using Jacobi elliptic functions and τ-functions.
result Explicit constructions and periodicities of semi-discrete and discrete surfaces.
The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.
problem Characterizing hyperbolic surfaces by their simple length spectra.
method Proving rigidity of length identities over Teichmüller spaces.
result Simple length spectra can be used as moduli for generic hyperbolic surfaces.
The length of shortest non-simple geodesics grows logarithmically with surface genus.
problem Understanding the behavior of shortest non-simple closed geodesics on hyperbolic surfaces.
method Investigation of asymptotic behavior on random hyperbolic surfaces using the Weil-Petersson measure.
result The non-simple systole behaves like log(g) as g goes to infinity.
Simple lifts of non-simple curves on surfaces.
problem Finding simple lifts of non-simple closed curves on surfaces.
method Constructing finite covers and curves on surfaces.
result Explicit non-simple closed curves with simple lifts on degree n covers.
The study examines Riemannian surfaces with simple singularities.
problem Geometry of Riemannian surfaces with discrete singular points.
method Local and global description using divisors, Gauss-Bonnet formula, classifications theorem.
result Classification of flat metrics with simple singularities on compact surfaces.
The aim of this paper is to give a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a d…
A version of Dehn's algorithm for simple diagrams on a once punctured surface representing simple diagrams on a closed surface is presented
The paper characterizes simple closed curves on surfaces using profinite rigidity.
problem Characterizing simple closed curves on surfaces using profinite rigidity.
method Proving that elements with the same images under all finite groups are simple closed curves.
result The set of simple closed curves is closed in the profinite topology of the surface group.
Shortest non-simple closed geodesics on hyperbolic surfaces found.
problem Finding the shortest non-simple closed geodesics on hyperbolic surfaces.
method Analyzing closed geodesics with at least k self-intersections on hyperbolic surfaces.
result The shortest non-simple closed geodesics lie on an ideal pair of pants and have length $2\arccosh(2k+1)$.
Smooth surfaces in simply connected 4-manifolds yield groups with non-trivial homology.
problem Finding groups with non-trivial fundamental groups in the complements of surfaces in 4-manifolds.
method Combining topological constructions with homological properties of simple groups.
result The complement of a surface in a simply connected 4-manifold can have a non-trivial fundamental group.
New proof shows surfaces can have identical length spectra but not simple ones.
problem Identifying when two covers of a surface have identical length spectra but not simple ones.
method Characterized isomorphism of covers and constructed surfaces with identical spectra but different simple length spectra.
result Found surfaces with identical length spectra but not simple length isospectral covers.
Simple criteria for codimension two surface singularities.
problem Identifying singularities in surfaces of codimension two.
method Provided criteria for singularities in surfaces of codimension less than or equal to two.
result Conditions for codimension two singularities in ruled surfaces and center maps.
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…
Study proper sampling for X-ray transforms on simple surfaces.
problem Proper discretizing and sampling issues related to geodesic X-ray transforms on simple surfaces.
method Provide minimal sampling rates for faithful reconstruction, quantify sampling quality, and predict artifacts.
result Minimal sampling rates and artifact prediction for geodesic X-ray transforms on simple surfaces.
Simple Ricci flow proof for Riemann surfaces.
problem Uniformization theorem of Riemann surfaces
method Ricci flow and Hamilton's isoperimetric estimate
result Simple proof of uniformization theorem
Study finds counterexamples to simple loop conjecture in higher dimensions.
problem Simple loop conjecture for surfaces to manifolds of dimensions at least four.
method Provided specific counterexamples for every g≥2 and n≥4. result No simple loop is contained in the kernel of the map for certain surfaces and manifolds.
We study complex spatial quartic surfaces with simple singularities up to equisingular deformations; as a first step, give a complete equisingular deformation classification of the so-called non-special simple quartic surfaces.
Conditions for simple closed curves in surface covers.
problem Conditions for simple closed curves in surface covers.
method Necessary and sufficient conditions for integral homology classes.
result Conditions for existence of simple closed curves in covers.
We discuss some applications of an intrinsic multipication in the space of simple loops in a surface.
It is shown that various questions about the existence of simple closed curves in normal subgroups of surface groups are undecidable.
We determine the largest (i.e. smallest index) characteristic subgroup of surface groups not containing any simple loops.
We give a simple proof for the rotational symmetry of ancient solutions of Ricci flow on surfaces. As a consequence we obtain a simple proof of some results of P.Daskalopoulos, R.Hamilton and N.Sesum on the a priori estimates for the ancient solutions of Ricci flow on surfaces. We also give a simple proof for the solut…
In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in the boundary of M. Our main result is that for any g>=0…
Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.
problem Computing shortest non-separating simple closed curves on non-orientable surfaces.
method Developed tools for computing shortest curves, proving NP-hardness and tractability.
result Proved NP-hardness and fixed-parameter tractability for computing shortest orienting curves, and polynomial-time algorithm for non-orienting curves.
The paper provides infinite presentations for surface groups.
problem Understanding fundamental groups of surfaces.
method Infinite presentations of fundamental groups using simple loops.
result Infinite presentations for fundamental groups of surfaces and non-orientable surfaces.
A simple surface amalgam is the union of a finite collection of surfaces with precisely one boundary component each and which have their boundary curves identified. We prove if two fundamental groups of simple surface amalgams act properly and cocompactly by isometries on the same proper geodesic metric space, then the…
We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.
Solves open problem on simple surfaces with novel twistor correspondence.
problem Existence of nontrivial holomorphic vector bundles on simple surfaces.
method Novel twistor correspondence, Nash-Moser inverse function theorem, and microlocal analysis.
result Simple surface twistor space supports no nontrivial holomorphic vector bundles.
We show that the attenuated geodesic ray transform on two dimensional simple surfaces is injective. Moreover we give a stability estimate and develop a reconstruction procedure.
Simplified presentations for non-orientable surface mapping class groups.
problem Presenting the mapping class group of non-orientable surfaces.
method Provided simpler infinite presentations.
result Simplified presentations for the group.
New representations for surface groups expand known Anosov classes.
problem Understanding new types of representations for surface groups.
method Introducing and studying simple Anosov representations of closed hyperbolic surface groups.
result Simple Anosov representations strictly contain Anosov representations.
Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.
problem Distribution of shapes of complementary subsurfaces in moduli space.
method Study of shapes of complementary subsurfaces in moduli space as boundary lengths go to infinity.
result Random subsurfaces look like random ribbon graphs.
We produce a sequence of finite dimensional representations of the fundamental group π1(S) of a closed surface where all simple closed curves act with finite order, but where each non--simple closed curve eventually acts with infinite order. As a consequence, we obtain a representation theoretic algorithm which deci…
Infinite type surfaces can be perfectly divided into triangles.
problem Triangulating surfaces of infinite type.
method Showed arcs can be completed into triangulations if they intersect curves a finite number of times.
result Any surface of infinite type admits an ideal triangulation.
Automorphisms of fine 1-curve graph linked to surface homeomorphisms.
problem Understanding automorphisms of fine 1-curve graphs.
method Isomorphic mapping to surface homeomorphisms.
result Automorphism group is isomorphic to homeomorphism group of a surface.
We consider collections of disjoint simple closed curves in a compact orientable surface which decompose the surface into pairs of pants. The isotopy classes of such curve systems form the vertices of a 2-complex, whose edges correspond to certain simple moves in which only one curve changes, and whose 2-cells correspo…
Unified view of integrable systems linking CMC, isothermic, and Willmore surfaces.
problem Understanding the relationships between different types of surfaces and their integrable systems.
method Unified view through families of flat connections and parallel sections.
result Complete description of links between different surface types and their dressing transformations.
Biholomorphisms of transport twistor spaces are rigid under certain conditions.
problem Rigidity of biholomorphisms in transport twistor spaces.
method Proof of rigidity for biholomorphisms between transport twistor spaces of simple or Anosov surfaces.
result Biholomorphisms are rigid, up to constant rescaling and the antipodal map, being lifts of orientation-preserving isometries.
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.
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.
New representations for surface groups in PU(2,1) are stable and larger than convex cocompact ones.
problem Characterizing representations of surface groups in PU(2,1).
method Introducing simple-stable representations and proving their properties.
result The set of conjugacy classes of simple-stable representations is a domain of discontinuity strictly larger than convex cocompact representations.
Simple properties of the Gauss map characterise important classes of surfaces in $\Rq$: R-surfaces, the real version of plane complex curves; Lagrangean surfaces; isoclinic surfaces.
In this note, we give an explicit counterexample to the simple loop conjecture for representations of surface groups into PSL(2,R). Specifically, we show that for any surface with negative Euler characteristic and genus at least 1, there are uncountably many non-conjugate, non-injective homomorphisms of its fundamental…
In this paper, we study the geometry of surfaces with the generalised simple lift property. This work generalises previous results by Bernstein and Tinaglia, and it is motivated by the fact that leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the generalised s…
We give a new proof of McShane's classification of simple cuspidal geodesics, using simple equivariant methods in the hyperbolic plane.
We show that the number of simple closed geodesics of length bounded by L on a hyperbolic surface of genus g with c cusps and b boundary components grows roughly like L^{6g+2b+2c-6}. This has been conjectured for some time.