Proves a new inequality for certain complex surfaces.
problem Analyzes compact Kähler surfaces with positive scalar curvature.
method Uses properties of holomorphic maps and ruled surfaces.
result Proves a 2-systolic inequality on these surfaces.
Constructs symplectic surface bundles with positive signatures.
problem Symplectic surface bundles over surfaces with positive signatures.
method Constructs symplectic surface bundles with specific genera.
result Determines commutator lengths of new mapping classes.
Study k-positive surface group representations and their degenerations.
problem Understanding the behavior of surface group representations under degenerations.
method Introduced k-positive representations and studied their degenerations using a limit theorem for positively ratioed representations.
result Degenerations of k-positive representations can lead to limits that are at least (k-3)-positive and irreducible limits are (k-1)-positive.
The study classifies graphs on surfaces with positive curvature properties.
problem Classifying graphs on surfaces with specific curvature properties.
method Using medial graphs and classification techniques.
result Complete classification of graphs on surfaces with positive Forman curvature and corner curvature.
Positive representations on surfaces have positive cross-ratios and satisfy a collar lemma.
problem Characterizing representations of surface groups with positive properties.
method Proving a collar lemma and showing positivity of cross-ratios for Θ-positive representations. result Closed subsets of representation varieties are characterized by Θ-positive representations. Positive factorization for pseudoperiodic homeomorphisms on surfaces.
problem Factorization of pseudoperiodic homeomorphisms on surfaces.
method Generalization of classical results on smooth germs of surfaces, topological characterization of monodromies, and use of positive factorization criteria.
result Pseudoperiodic homeomorphisms on surfaces with positive fractional Dehn twist coefficients and screw numbers admit a positive factorization.
We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splitability and the primeness of positive knots and links can be seen from their positive diagrams.
The paper finds infinite knots with surfaces of any positive genus.
problem Finding knots with surfaces of arbitrary positive genus.
method Constructing knot exteriors with longitudinal essential surfaces of any positive genus.
result Existence of infinitely many knots with surfaces of any positive genus.
Abby Thompson proved that if a link K is in thin position but not in bridge position then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that the if a link is in thin…
Nearly spherical, positively curved surfaces are mapped from a sphere.
problem Mapping nearly spherical, positively curved surfaces from a sphere.
method Combines Ricci flow, Kim-Milman construction, and Bakry-Émery criterion.
result Every nearly spherical, positively curved surface is the contractive image of a round sphere.
Proves a quasi-order on fibre surfaces of positive braid links.
problem Ordering fibre surfaces of positive braid links.
method Analyzes quasipositive surfaces and their inclusions.
result Induced order on fibre surfaces is a well-quasi-order under certain conditions.
Found a minimal surface in curved space.
problem Minimal surfaces in curved spaces.
method Constructed a metric with a specific surface.
result Found a minimal surface of index 1.
Positive factorization found for a specific map on surfaces.
problem Balanced superelliptic rotation on surfaces.
method Positive factorization approach.
result Positive factorization for balanced superelliptic rotation.
This paper studies non-compact Ricci surfaces with catenoidal ends.
problem Characterizing non-compact Ricci surfaces with catenoidal ends.
method Using Weierstrass data and minimal immersion techniques.
result Classification and existence results for positive genus Ricci surfaces.
Positive representations of surface groups in PO(p,q) form connected components of character varieties.
problem Characterizing representations of surface groups in special orthogonal groups PO(p,q).
method Using Anosov representations and root versus weight collar lemmas.
result Connected components of character varieties are formed by Θ-positive Anosov representations. The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.
problem Counting minimal surfaces in 3-manifolds with positive Ricci curvature.
method An enumerative min-max theorem linking surface counts to topological properties.
result Every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.
Sharp estimate for 2-systole on Kähler surfaces with positive scalar curvature.
problem Estimating the 2-systole on compact Kähler surfaces with positive scalar curvature.
method Combining classification of positive scalar curvature Kähler surfaces with Stern's level set method adapted to Kähler setting.
result Proved the sharp estimate minXS(ω)⋅sys2(ω)≤12π. The study classifies quasi-minimal surfaces in 4D pseudo-Riemannian space-forms with positive nullity.
problem Characterizing surfaces in pseudo-Riemannian space forms with positive nullity.
method Analyzing the relative null space and classifying quasi-minimal surfaces.
result Classifications of quasi-minimal surfaces with positive relative nullity.
New Θ-positive representations of surface groups discovered.
problem Generalizing Lusztig's total positivity to surface groups.
method Introducing Θ-positivity and proving properties of Θ-positive representations. result Discrete and faithful Θ-positive representations exist and form open sets in representation varieties. Paper proves rigidity of CMC surfaces in curved 3-manifolds.
problem Rigidity of CMC surfaces in positive curved 3-manifolds.
method Assumptions of surface being approximately round or invariant under even symmetry, and use of Hawking mass.
result Rigidity results for stable CMC surfaces with zero Hawking mass.
Symplectic Torelli groups of positive rational surfaces are trivial or sphere braid groups.
problem Understanding symplectic Torelli groups of rational surfaces.
method Using positivity condition, type of cohomology class, and Lagrangian spherical classes.
result Symplectic Torelli groups of positive rational surfaces are trivial or sphere braid groups.
Study space-like and time-like surfaces in Robertson-Walker space-times with positive nullity.
problem Characterize space-like and time-like surfaces in Robertson-Walker space-times with positive relative nullity.
method Provide necessary and sufficient conditions, local classification theorems, and analyze special spaces.
result Local classification theorems for space-like and time-like surfaces in L14(f,0) with positive relative nullity. This paper studies symplectic structures on elliptic surfaces with positive Euler number.
problem Determining symplectic representatives for cohomology classes on elliptic surfaces.
method Analyzes the symplectic cone for elliptic surfaces with positive Euler number.
result Characterizes the symplectic cone for elliptic surfaces with positive Euler number.
Minimal surfaces in spheres found for any genus.
problem Finding minimal surfaces with arbitrary genus in 3-spheres.
method Topological structure analysis and embedding theorem.
result Every positive Ricci curvature 3-sphere contains a genus g surface.
In this paper we prove a compactness theorem for constant mean curvature surfaces with area and genus bound in three manifold with positive Ricci curvature. As an application, we give a lower bound of first eigenvalue of constant mean curvature surfaces in three manifold with positive Ricci curvature.
Collar lemma proven for certain surface group representations.
problem Proving a collar lemma for specific surface group representations.
method Using partial hyperconvexity properties and Anosov representations.
result 'Positivity properties' hold for partially hyperconvex representations.
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.
We show that the twisted SL(2) skein algebra of a surface has a natural basis (the bracelets basis) that is positive, in the sense that the structure constants for multiplication are positive integers.
Is any positive knot the closure of a positive braid? No. But if we consider positivity in terms of the generators of the braid group due to Birman, Ko and Lee, then the answer is yes. In this paper we prove that the same occurs when considering homogeneity. In the way we prove that the plumbing of two surfaces is a BK…
Researchers parametrize spaces of positive representations for Lie groups.
problem Tackling spaces of positive representations for Lie groups.
method Generalizing Lusztig's total positivity, they introduce spaces of positive framed representations and parametrize them.
result The number of connected components of the space of framed positive representations agrees with the number of positive representations.
Classifies positive integral friezes on surfaces.
problem Classifying positive integral friezes on marked bordered surfaces.
method One-to-one correspondence with ideal triangulations and rescaling constants.
result Number of non-equivalent friezes on bordered surfaces is finite.
We show that given a trivalent graph in S3, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of Thompson to graphs. It follows that any graph complement always contains a useful p…
We compute the Dijkgraaf-Witten invariants of surfaces in terms of projective representations of groups. As an application we prove that the complex Dijkgraaf-Witten invariants of surfaces of positive genus are positive integers.
We show that the Chebyshev polynomials form a basic block of any positive basis of the Kauffman bracket skein algebras of surfaces.
New analysis of crushing surfaces of positive genus impacts triangulation complexity.
problem Crushing surfaces of positive genus can drastically change triangulation topology.
method Detailed analysis of crushing effects on closed essential surfaces of positive genus.
result Proved results about triangulation complexity and JSJ decompositions.
Study finds criteria for surfaces with specific curvature properties.
problem Understanding Kählerian or projective structures on surfaces with non-positive curvature.
method Established a criterion for compact Hermitian surfaces with non-positive second Chern-Ricci curvature.
result Found conditions for Kählerian or projective structures on surfaces with non-positive curvature.
Nontrivial infinitesimal bendings for a class of two-dimensional surfaces are constructed. The surfaces considered here are orientable; compact; with boundary; have positive curvature everywhere except at finitely many planar points; and have vanishing first homology group.As a consequence, a nonrigidity result for suc…
Positive holomorphic sectional curvature on rational surfaces is characterized by Kähler metrics.
problem Characterizing rational surfaces by the existence of a Kähler metric with positive holomorphic sectional curvature.
method Constructing Kähler metrics on projective manifolds obtained from toric manifolds.
result Every projective manifold obtained from a projective toric manifold by a finite sequence of blow-ups at points admits a Kähler metric with positive holomorphic sectional curvature.
New metrics connect surfaces with Anosov flows to those with negative curvature.
problem Creating metrics with Anosov flows on surfaces of positive curvature.
method Constructing metrics with Anosov geodesic flows and positive curvature regions, connecting to negative curvature metrics via smooth conformal deformations.
result Existence of a smooth curve of conformal deformations connecting Anosov metrics to metrics of negative curvature.
The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.
problem Creating minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature.
method Develops a min-max theory and shows deformability of surfaces in a generic metric.
result Establishes a theorem for producing minimal surfaces with prescribed genus.
A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in euclidean 3-space with a point of positive curvature and no skew loops are the quadrics. In particular, ellipsoids are the only closed surfaces without skew loops. We also prove results about skew loops on cylinders an…
We use Colding--Minicozzi lamination theory to study the systole of large genus minimal surfaces in an ambient three-manifold of positive Ricci curvature.
Paper finds minimal number of curves in surface systems.
problem Determining minimal number of curves in surface systems.
method Analyzes oriented surfaces of genus g for positive integers k and g.
result Exact minimal number of curves in filling k-systems.
We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension (2+1). The proof follows from polyhedral approximation.
Proves Khovanov homology functoriality and positivity for gl2 webs.
problem Proving functoriality and positivity of Khovanov homology.
method Categorical proof using linear complexes stability.
result Strong positivity result for surface skein algebras.
Study mapping class group actions on surface fundamental groups.
problem Understanding group actions on surface fundamental groups.
method Computing image of mapping class group generators as outer automorphisms.
result Computed images of mapping class group generators on fundamental groups.
Refined estimates for surfaces in curved spaces based on Willmore functional.
problem Estimating the position of surfaces in curved spaces accurately.
method Critical points of the Willmore functional, constrained area, refined geometric center of mass.
result Improved position estimates related to ambient scalar curvature.
We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension −∞. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although th…