Optimizes bounds for multiple T-singularities on surfaces.
problem Bounding T-singularities on non-rational projective surfaces with many singularities.
method Analyzes combinatorial configurations and classifies them to find optimal bounds.
result Classifies all combinatorial configurations leading to high bounds, proving their non-existence gives optimal bounds.
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.
Toric quasifolds extend toric geometry to non-rational polytopes.
problem Extending toric geometry to non-rational polytopes.
method Introduced toric quasifolds to solve symplectic extension problem.
result Illustrated toric quasifolds and their atlases.
New proof for weak mixing in polygonal billiards.
problem Proving weak mixing in polygonal billiards.
method Using Baire category and eigenvalue analysis.
result Billiard flow is weakly mixing for non-rational polygons.
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. Study on stock price formation on trees with multi-population and non-rational agents.
problem Equilibrium price formation for risky stock with multi-population and non-rational agents.
method Combining mean-field game theory with binomial tree framework, proving existence of unique equilibrium, deriving explicit formula for transition probabilities.
result Existence of unique mean-field market-clearing equilibrium with explicit analytic formula for stock price transition probabilities.
Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence, Kollár formulated the algebraic version of the Montgomery-Yang problem: every projective surface S with the second Betti number $b_2…
For linear actions of real reductive Lie groups we prove the Kempf-Ness Theorem about closed orbits and the Kirwan-Ness Stratification Theorem of the null cone. Since our completely self-contained proof focuses strongly on geometric and analytic methods, essentially avoiding any deep algebraic result, it applies also t…
Monopole invariant studies contact structures on 3-manifolds.
problem Topology of contact structures on 3-manifolds.
method Generalized Kronheimer--Mrowka--Ozsváth--Szabó contact invariant for families of contact structures.
result Obstructs the existence of sections and detects exotic loops of contact structures.
Customer behavior is often assumed to follow weak rationality, which implies that adding a product to an assortment will not increase the choice probability of another product in that assortment. However, an increasing amount of research has revealed that customers are not necessarily rational when making decisions. In…
In this article we consider a generalization of manifolds and orbifolds which we call quasifolds; quasifolds of dimension k are locally isomorphic to the quotient of R^k by the action of a discrete group - tipically they are not Hausdorff topological spaces. The analogue of a torus in this geometry is a quasitorus. We …
We call complex quasifold of dimension k a space that is locally isomorphic to the quotient of an open subset of the space C^k by the holomorphic action of a discrete group; the analogue of a complex torus in this setting is called a complex quasitorus. We associate to each simple polytope, rational or not, a family of…
Associated with a smooth, d-closed (1,1)-form α of possibly non-rational De Rham cohomology class on a compact complex manifold X is a sequence of asymptotically holomorphic complex line bundles Lk on X equipped with (0,1)-connections ∂ˉk for which ∂ˉk2=0. Their study was…
Analyzes new economic paradigm for non-independent consumer choices.
problem Non-independent consumer choices due to firm supply and consumer information.
method Develops a new mathematical framework for economic systems.
result New paradigm for economic system description is necessary.
Study extends complex sections on non-holomorphic objects on Kähler manifolds.
problem Extension of smooth sections on non-holomorphic objects on Kähler manifolds.
method Use of asymptotically holomorphic line bundles, two twisted Laplace-type operators, and Bochner-Kodaira-Nakano-type inequalities.
result Extensions of smooth sections with control of their L2-norms for non-integrable objects. Study of cuspidal edges on focal surfaces of regular surfaces.
problem Clarifying the sign of singular curvature at cuspidal edges.
method Investigation using singularities of parallel surfaces.
result Clarification of the sign of singular curvature at cuspidal edges.
New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.
problem Limiting the outcomes of gluing Scherk surfaces into minimal surfaces.
method Constructing minimal surfaces by stacking and gluing doubly periodic Scherk surfaces.
result Except for special cases, gluing more Scherk surfaces results in known minimal surfaces.
Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.
problem Investigate differential geometry properties of focal surfaces of lightcone framed surfaces.
method Introduced lightcone frame to define lightcone framed surfaces, then investigated their differential geometry properties.
result Investigated differential geometry properties of focal surfaces of lightcone framed surfaces.
Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
The paper studies special surfaces with a new type of support function.
problem Characterizing surfaces with a specific quadratic support function.
method Developed a Weierstrass type representation involving holomorphic functions.
result Classified surfaces of rotation with this new type of support function.
The paper studies knitted surfaces and surface-links, showing their isotopy and closure properties.
problem Understanding the isotopy and closure properties of knitted surfaces and surface-links.
method Analyzing the structure and closure of knitted surfaces and surface-links in R4. result Any surface-link is ambient isotopic to the closure of a 2-dimensional knit.
We investigate surfaces with constant harmonic-mean curvature one (HMC-1 surfaces) in hyperbolic three-space. We allow them to have certain kinds of singularities, and discuss some global properties. As well as flat surfaces and surfaces with constant mean curvature one (CMC-1 surfaces), HMC-1 surfaces belong to a cert…
Unstable minimal surfaces in n-space link to hyperbolic products.
problem Characterizing unstable minimal surfaces in Rn and their product counterparts. method Lifting to R-trees, deforming to hyperbolic products, and proving instability equivalence. result Unstable minimal surfaces in Rn imply unstable surfaces in product hyperbolic spaces. Researchers generalize Ribaucour-type surfaces with new mathematical representation.
problem Defining and characterizing new geometric surfaces.
method Developed a new mathematical representation for GRT-surfaces involving holomorphic functions and a real function.
result Explicit examples and classification of GRT-surfaces of rotation.
Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.
problem Understanding automorphisms of fine curve graphs on surfaces.
method Analyzing vertices and edges of fine curve graphs to match with surface homeomorphisms.
result Automorphism group of fine curve graphs is naturally isomorphic to the homeomorphism group of boundaryless planar surfaces with at least 7 punctures.
This paper connects Laguerre minimal surfaces to Weierstrass representations.
problem Understanding the relationship between Laguerre minimal surfaces and Weierstrass representations.
method Defining spherical mean curvature and providing Weierstrass-type representations for two classes of surfaces.
result Laguerre minimal surfaces are related to H2-surfaces, providing a new Weierstrass-type representation. New surfaces in 4-ball constructed from knits, described by charts.
problem Constructing surfaces in 4-ball from knits.
method Introducing knitted surfaces, describing them with BMW charts.
result Every compact surface in 4-ball is ambiently isotopic to a knitted surface.
Study on singular points of translation surfaces under linearly dependent conditions.
problem Investigate singular points of translation surfaces under linearly dependent conditions.
method Use theories of generalised framed surfaces and framed surfaces.
result Introduce translation generalised framed surfaces and investigate their singular points.
Crochet patterns for minimal surfaces created using trigonometry.
problem Creating crochet patterns for minimal surfaces.
method Using trigonometric identities to calculate arc lengths.
result Crochet instructions for Enneper's surface.
New surfaces generalize Dini surfaces in 4D.
problem None explicitly stated; focuses on surface generalization.
method Introducing a new family of surfaces in 4D.
result Generalized Dini surfaces exist in 4D.
Here, we focus on focal surfaces of a tubular surface in Euclidean 3-space E^3: Firstly, we give the tubular surfaces with respect to Frenet and Darboux frames. Then, we define focal surfaces of these tubular surfaces. We get some results for these types of surfaces to become flat and we show that there is no minimal f…
New method classifies HCMU surfaces in 3D space forms as Weingarten surfaces.
problem Classifying HCMU surfaces in 3D space forms as Weingarten surfaces.
method Totally different method from previous work.
result Criteria for Weingarten surfaces that are also HCMU surfaces.
Minimal surfaces are the only biharmonic in Sol3.
problem Characterizing biharmonic surfaces in Sol3.
method Found local equations for biconservative surfaces and showed all biharmonic surfaces are minimal.
result All biharmonic surfaces in Sol3 are minimal.
It is shown that any handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link up to equivalences, so that every stable-ribbon surface-link is a ribbon surface-link. This is a generalization of a previously observed result for a stably trivial surface-link. Two observations are give…
This paper proves compactness of conformal Chern-minimal surfaces in Hermitian surfaces.
problem Compactness of conformal Chern-minimal surfaces in Hermitian surfaces.
method Proves compactness through bubble tree limit analysis.
result Compactness of conformal Chern-minimal surfaces is established with bounded area.
Study proves surfaces with constant curvature are simple shapes.
problem Characterizing singular minimal surfaces with constant curvature.
method Proved geometric properties of surfaces with constant curvature.
result Singular minimal surfaces with constant curvature are planes, spheres, and cylindrical surfaces.
New surface class defined using osculating circles.
problem Defining a new surface class in Euclidean space.
method Using osculating circles of curves and classification of specific types.
result Classification of canal and Weingarten surfaces.
The article constructs Bolza-like surfaces for infinitely many genera and studies their properties.
problem Maximizing systole functions in Teichmüller spaces for genus two and higher.
method Defining and constructing Bolza-like surfaces with specific triangulations and properties.
result Global maximal surfaces can be constructed using Bolza-like surfaces, and systolic geodesics intersect at even points.
The Enneper surface and helix surfaces are unique in their geometric properties.
problem Characterizing surfaces with specific geometric properties.
method Analyzing isogonal lines and pseudo-geodesic lines in 3D Euclidean space.
result Helix surfaces and Enneper surface are the only surfaces with isogonal lines as generalized helices and pseudo-geodesic lines.
Study on automorphisms of K3 and Enriques surfaces, proving entropy gaps and achirality.
problem Entropy norms and achirality of automorphisms on K3 and Enriques surfaces.
method Proves gap theorems for entropy norms and studies achirality in terms of genus-one fibrations.
result Entropy gaps and achirality results for automorphisms of K3 and Enriques surfaces.
We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the…
The paper connects least area surfaces to quasi-normal surfaces in 3-manifolds.
problem Understanding the properties of least area surfaces in 3-manifolds.
method Introducing quasi-normal surfaces and showing their relationship to least area surfaces in fine triangulations.
result Least area surfaces in 3-manifolds are quasi-normal with respect to fine triangulations, and this quasi-normality leads to piecewise flat approximations.
New concept of quasi-ribbon surface-links simplifies complex surface-links.
problem Complexity in surface-links of trivial components.
method Introducing quasi-ribbon surface-links as a generalization of ribbon surface-links.
result Every F-link of trivial components on a surface F with at most one aspheric component is a quasi-ribbon surface-link.
Study of timelike surfaces in Minkowski space with specific geometric properties.
problem Characterizing geometric properties of timelike surfaces in Minkowski space.
method Analytical study of two types of timelike general rotational surfaces.
result Explicit descriptions of minimal and surfaces with specific curvature properties.
The paper explores Kα-translators on parallel and canal surfaces in 3D space.
problem Investigating conditions for Kα-translators on parallel and canal surfaces. method Analyzing the conditions for Kα-translators on parallel surfaces and canal surfaces, proving their properties and existence. result No Kα-translators exist on the parallel surface of a rotational surface obtained from a canal surface with the same speed w, while the rotational surface itself is a Kα-translator. Researchers describe isometric deformations of T-hedra and T-surfaces.
problem Understanding the isometric deformations of discrete and smooth T-surfaces.
method Synthetic and analytic descriptions of T-hedra and T-surfaces, providing parametrizations of isometric deformations.
result Explicit parametrization of isometric deformations of T-hedra and T-surfaces.
The height function of various surfaces decomposes into finite sums of scaled and translated versions of itself.
problem Decomposing the height function of different types of surfaces into simpler components.
method Using Euler-Ramanujan identities and Weierstrass-Enneper representation to decompose height functions of minimal, maximal, timelike minimal, and Born-Infeld surfaces.
result The height function of various surfaces can be expressed as a finite sum of scaled and translated versions of itself.
Defines spherical type surfaces via support function and classifies them.
problem Characterizing surfaces via support function.
method Weierstrass type representation for SS-surfaces with prescribed Gauss map.
result Every compact and connected SS-surface is the sphere.