Classifies surfaces with no Gaussian curvature.
problem Classifying surfaces with vanishing Gaussian curvature.
method Analyzes Willmore surfaces, studies Willmore cones, gives a Bernstein-type theorem.
result Classifies simply-connected, complete Willmore surfaces with vanishing Gaussian curvature.
Study of Ricci flow on discrete surfaces of revolution with constant Gaussian curvature.
problem Understanding Ricci flow on discrete surfaces of revolution.
method Explicit parametrizations and Ricci flow analysis for discrete surfaces of revolution.
result Discrete surfaces of revolution approach constant Gaussian curvature under Ricci flow.
Study on discrete Gaussian curvature for polyhedral surfaces.
problem Discretization of Gaussian curvature for polyhedral surfaces.
method Generalization of discrete conformal equivalence to define discrete Gaussian curvature and classify polyhedral surfaces.
result Existence of polyhedral surfaces with constant discrete Gaussian curvature in every discrete conformal class.
Classifies surfaces with constant Gaussian curvature in Euclidean 3-space.
problem Classifying surfaces with constant Gaussian curvature in Euclidean 3-space.
method Analyzing surfaces as implicit equations and proving properties based on Gaussian curvature.
result Surfaces with constant Gaussian curvature are either surfaces of revolution, cylindrical surfaces, conical surfaces, or have specific forms.
Study geodesic curvature of logarithmic spirals on curved surfaces.
problem Understanding geodesic curvature on curved surfaces.
method Computed geodesic curvature of logarithmic spirals on surfaces of constant Gaussian curvature.
result Asymptotic behavior of geodesic curvature is independent of the ambient surface's curvature.
The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.
problem Prescribing discrete Gaussian curvature on polyhedral surfaces.
method Discrete conformal theory and variational principles with constraints.
result Proves Kazdan-Warner type theorems for polyhedral surfaces.
The paper shows Gaussian fluctuations in eigenvalue statistics of random hyperbolic surfaces.
problem Understanding fluctuations in Laplace eigenvalues of random hyperbolic surfaces.
method Analyzing fluctuations of linear statistics of Laplace eigenvalues over moduli space of surfaces of large genus.
result The distribution of linear statistics tends to a Gaussian as the genus of surfaces increases.
Flat surfaces in Lie groups with constant curvature are flat.
problem Characterizing surfaces in Lie groups with constant Gaussian curvature.
method Analyzing surfaces as products of curves and using bi-invariant metrics.
result All surfaces of constant curvature in 3D Lie groups are flat.
A set of control points can determine a Bezier surface and a triangulated surface simultaneously. We prove that the triangulated surface becomes homeomorphic and ambient isotopic to the Bezier surface via subdivision. We also show that the total Gaussian curvature of the triangulated surface converges to the total Gaus…
In the 3-dimensional Lorentz-Minkowski space we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also investigate the behavior of the Gaussian curvature near singular points o…
Paper solves long-standing Gaussian curvature conjecture for minimal graphs.
problem Gaussian curvature of minimal graphs over the unit disk.
method Complex-analytic methods, conformal harmonic parameterization.
result Sharp estimate for Gaussian curvature at the origin of minimal graphs.
Classifies surfaces in hyperbolic space with constant Gaussian curvature.
problem Classifying surfaces in hyperbolic space with specific curvature.
method Loop group method, spectral parameter deformation, holomorphic quadratic differentials.
result Weakly complete constant Gaussian curvature surfaces are in one-to-one correspondence with holomorphic quadratic differentials.
The paper introduces a new discretization of Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with conic singularities.
method Discrete conformal theory and variational principles with constraints.
result Established a discrete uniformization theorem for surfaces with non-positive Euler number.
A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this paper, we investigate the beh…
We prove that the only surfaces in 3-dimensional Euclidean space R3 with constant Gaussian curvature K and constructed by the sum of two space curves are cylindrical surfaces, in particular, K=0.
There are examples of complete spacelike surfaces in the Lorentzian product H2×R1 with constant Gaussian curvature K≤−1. In this paper, we show that there exists no complete spacelike surface in H2×R1 with constant Gaussian curvature K>−1.
The paper finds surfaces closest to being flat that span a given contour.
problem Finding surfaces in R3 that are as flat as possible while spanning a given contour. method The approach involves minimizing the total Gaussian curvature squared and solving a system of PDEs.
result The optimal surface is shown to be controlled by a biharmonic equation with specific boundary conditions.
This article is an application of the author's paper about a construction method for discrete constant negative Gaussian curvature surfaces, the nonlinear d'Alembert formula. The heart of this formula is the Birkhoff decomposition, and we give a simple algorithm for the Birkhoff decomposition. As an application, we dra…
In this paper we consider Lorentzian surfaces in the 4-dimensional pseudo-Riemannian sphere S24(1) with index 2 of curvature one. We obtain the complete classification of minimal Lorentzian surfaces S24(1) whose Gaussian and normal curvatures are constants. We conclude that such surfaces have th…
Single linear solve combines surface reconstruction and uncertainty quantification.
problem Reconstructing surfaces from partial point clouds with uncertainty.
method Geometric Gaussian processes for stochastic surface reconstruction.
result Single linear solve for surface reconstruction with probabilistic capabilities.
In this paper, we show that the constant property of the Gaussian curvature of surfaces of revolution in both R4 and R14 depend only on the radius of rotation. We then give necessary and sufficient conditions for the Gaussian curvature of the general rotational surfaces whose meridians lie in two…
Factorable surfaces, i.e. graphs associated with the product of two functions of one variable, constitute a wide class of surfaces. Such surfaces in the pseudo-Galilean space with zero Gaussian and mean curvature were obtained in [1]. In this study, we provide new classification results relating to the factorable surfa…
The paper examines conditions for Lagrangian surfaces in Kähler-Einstein manifolds.
problem Characterizing Hamiltonian stationary Lagrangian surfaces with non-negative Gaussian curvature.
method Simple conditions and characterization of surfaces in Kähler-Einstein manifolds.
result Conditions for surfaces to have Euclidean factors or be fiber bundles over circles.
Study on surfaces with conical singularities and geodesic boundaries, deriving existence results.
problem Existence of conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundaries.
method Variational argument to derive existence results for surfaces with at least two boundary components.
result First result in this setting for surfaces with conical singularities of both positive and negative orders.
Study modular surfaces in Lorentz-Minkowski 3-space, classifying and analyzing their curvature and applications.
problem Understanding the curvature properties of modular surfaces in Lorentz-Minkowski space.
method Analyzing the sign of Gaussian and mean curvature, classifying surfaces, and applying to conformal field theories.
result Complete classification of zero Gaussian curvature modular surfaces and non-existence of non-planar maximal modular surfaces.
New model uses heteroscedastic Gaussian process for alkenone SST proxy.
problem Restoring historical sea surface temperatures using proxies.
method Heteroscedastic Gaussian process regression method.
result Nonparametric approach handles variable noise patterns and outliers.
Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.
problem Distribution of zeros of Gaussian sections on semipositive line bundles.
method Analysis of Bergman kernels and random zeros in high tensor powers.
result Equidistribution, large deviation estimates, central limit theorem, and number variances for zeros in the semi-classical limit.
Study focal surfaces of wave fronts with unbounded curvatures.
problem Characterizing singularities of focal surfaces near non-degenerate singular points.
method Characterizations based on types of singularities and geometrical properties of initial fronts.
result Investigation of Gaussian curvature behavior of focal surfaces.
In this paper we classify compact minimal surfaces in S5 with non-negative Gaussian curvature using the notion of a contact angle.
This paper concerns the global theory of properly embedded spacelike surfaces in three-dimensional Minkowski space in relation to their Gaussian curvature. We prove that every regular domain which is not a wedge is uniquely foliated by properly embedded convex surfaces of constant Gaussian curvature. This is a conseque…
The study classifies surfaces with specific curvature properties.
problem Classifying surfaces with a particular curvature equation.
method Analyzing surfaces in 3D Euclidean space with a specific curvature equation.
result A one-parameter family of surfaces meeting the unit ball orthogonally.
In this paper, we study the timelike tubular Weingarten surfaces in 3-dimensional Minkowski space IR13.We have obtained some conditions for being (KII,H), (KII,K), timelike tubular Weingarten surfaces where are the second Gaussian curvature the Gaussian curvature and the mean curvature, respectively.
It is shown that 3 disjoint sets with fixed Gaussian volumes that partition Rn with nearly minimum total Gaussian surface area must be close to adjacent 120 degree sectors, when n≥2. These same results hold for any number m≤n+1 of sets partitioning Rn, conditional on the solut…
Gaussian process regression loses locality in high dimensions, affecting molecular energy surface fitting.
problem Loss of locality in high-dimensional Gaussian process regression.
method Analysis of Matern family kernels and multi-zeta basis functions.
result The property of locality disappears in high dimensions, impacting regression quality.
Paper estimates Gaussian curvature of minimal graphs in a specific manifold.
problem Estimating Gaussian curvature of minimal graphs in MimesR. method Using Weierstrass representation via ℘−harmonic mappings and Schwarz lemma type results. result Proves Schwarz lemma type and Heinz type results for harmonic mappings.
Study classifies helix surfaces in Lorentzian Heisenberg group.
problem Classifying helix surfaces in Lorentzian Heisenberg group.
method Complete description of ambient space geometry, classification of minimal and CMC helix surfaces, investigation of constant angle surfaces.
result Explicit parametrizations of minimal and CMC helix surfaces in $\htt$.
Let h be a complete metric of Gaussian curvature K0 on a punctured Riemann surface of genus g≥1 (or the sphere with at least three punctures). Given a smooth negative function K with K=K0 in neighbourhoods of the punctures we prove that there exists a metric conformal to h which attains this function…
New proof shows special surfaces have finite type.
problem Characterizing surfaces with finite topological type.
method Shorter proof of Huber's theorem.
result Special surfaces have finite topological type.
We study the equation for improper (parabolic) affine spheres from the view point of contact geometry and provide the generic classification of singularities appearing in geometric solutions to the equation as well as their duals. We also show the results for surfaces of constant Gaussian curvatureand for developable s…
Study finds Scherk type surfaces as extremals for zero-curvature minimal graphs.
problem Finding extremal minimal graphs with zero Gaussian curvature at the center.
method Analyzing Scherk surfaces and their properties.
result Scherk type minimal surfaces are extremals for zero-curvature minimal graphs.
In this paper we consider the Gaussian thermostat ray transform on both closed Riemannian surfaces and compact Riemannian surfaces with boundary. We establish certain results on the injectivity of the thermostat ray transform and the surjectivity of its adjoint.
Study non-degenerate anisocurved surfaces in homogeneous 3-manifolds.
problem Compare and study surfaces with opposite Gaussian curvatures under two different metrics.
method Consider surfaces in homogeneous 3-manifolds with two metrics, impose extrinsic curvature conditions, and analyze Gaussian curvature functions.
result Identify and characterize anisocurved surfaces with opposite Gaussian curvatures under both metrics.
The paper proves a theorem for discretizing Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with nonpositive Euler number.
method Discrete conformal theory and variational principles with constraints.
result Each decorated piecewise Euclidean metric on surfaces with nonpositive Euler number is discrete conformal to a metric with a specific discrete curvature constant.
We study the canonical metric on a compact Riemann surface of genus at least two. While it is known that the canonical metric is of nonpositive curvature, we show that its Gaussian curvatures are not bounded away from zero nor negative infinity when the surface is close to the compactification divisor of Riemann's modu…
We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relations…
Improved agnostic learning time via Gaussian surface area analysis.
problem Learning polynomial threshold functions under Gaussian marginals.
method Improvement of polynomial degree required for approximation.
result Near optimal bounds on agnostic learning complexity.
Develops an algorithm to approximate non-Gaussian posterior distributions in Bayesian inference.
problem Sampling non-Gaussian posterior distributions in Bayesian inverse problems.
method Iterative construction of Gaussian Process (GP) augmented proposal distributions for MCMC sampling.
result Optimal selection of sampling points using maximum information gain from GP surface.
Helix surfaces in Anti-de Sitter space maintain constant Gaussian curvature.
problem Understanding helix surfaces in Anti-de Sitter space with Berger-like metrics.
method Proved helix surfaces have constant Gaussian curvature and described them explicitly.
result Explicit local description of helix surfaces in terms of isometries and curves.