Study proves finiteness for distance functions on curved surfaces with controlled curvature.
problem Understanding distance functions on curved surfaces with Hölder continuous curvature.
method Proves a finiteness principle using Whitney extension theory for geodesics and points on Riemannian surfaces with Hölder continuous curvature.
result Establishes a finiteness principle for isometric embedding of metric spaces into Riemannian surfaces with controlled curvature.
Unified approach to totally ramified values in various surface theories.
problem Totally ramified values in value distribution theory, normal family theory, and Gauss maps of surfaces.
method Bloch--Ros principle applied to various surface theories.
result Unified approach to phenomena concerning totally ramified values.
We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss/Bonnet theorem and the mean-curvature force balance equation.
Proposes a Gaussian process model for constrained dynamics learning.
problem Challenges in identifying constrained dynamics of mechanical systems.
method Combines analytical mechanics with Gaussian process regression.
result Improves data efficiency and constraint integrity in predictions.
We show the uniqueness of strictly convex closed smooth self-similar solutions to the α-Gauss curvature flow with (1/n)<α<1+(1/n). We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the α-Gauss c…
The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is considered in this paper through the Gauss-Codazzi system for the second fundam…
We study some basic problems of translating solitons: the volume growth, generalized maximum principle, Gauss maps and certain functions related to the Gauss maps, finally we carry out point-wise estimates and integral estimates for the squared norm of the second fundamental form. Those estimates give rigidity theorems…
We consider flows with normal velocities equal to powers strictly larger than one of the Gauss curvature. Under such flows closed strictly convex surfaces converge to points. In his work on the square of the norm of the second fundamental form, Schnürer proposes criteria for selecting quantities that are suitable for p…
Study curve shortening flows on specific surfaces, proving properties and existence.
problem Analyzing curve shortening flows on rotational surfaces with negative Gauss curvatures.
method Assume negative Gauss curvatures and conditions on Gauss curvature and curve curvature. Prove curve remains a graph and establish flow properties.
result Prove the curve remains a graph over parallels and establish long-time existence of the flow.
Both analytic and geometric forms of an optimal monotone principle for Lp-integral of the Green function of a simply-connected planar domain Ω with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geome…
We consider geometric flow equations for contracting and expanding normal velocities, including powers of the Gauss curvature, of the mean curvature, and of the norm of the second fundamental form, and ask whether - after appropriate rescaling - closed strictly convex surfaces converge to spheres. To prove this, many a…
We derive a parabolic version of Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on ℓ-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in euclidean spaces with uniform bounded second fundamenta…
Directly proves Brioschi formula for Gaussian curvature.
problem Express Gaussian curvature in terms of local coordinates.
method Elementary proof without Christoffel symbols.
result Directly derived Brioschi formula for Gaussian curvature.
A new method for 1-bit matrix completion that is faster and more accurate.
problem Estimating a low-rank matrix from binary observations.
method Majorization-Minimization Gauss-Newton (MMGN) method.
result MMGN outperforms existing methods in accuracy and speed.
New framework allows selective removal of stale data in option calibration.
problem Inability to remove old data from calibrated option pricing models without full retraining.
method Introduces operator-theoretic Gauss-Newton framework for selective forgetting.
result Provides stability guarantees and perturbation bounds for selective data removal.
Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate "Measurement of circle". The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygon…
The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric (0,2)-tensors. More precisely, we construct a family of generalized curvature measures attached to such manifolds, extending the Riemannian Lipschitz-Killing curvature mea…
Paper generalizes Bloch-Ros principle to various surface classes.
problem Understanding the relationship between normal family theory, value distribution theory, and surface theory.
method Formulation and generalization of Bloch-Ros principle to different surface classes.
result Effective criterion for determining Gaussian curvature estimates for various surface classes.
The Ekeland variational principle implies what can be regarded as a strong version, in the C1 category, of the Yau minimum principle: under the appropriate hypotheses {\it every} minimizing sequence admits a {\it good shadow}, a second minimizing sequence that has good properties and is asymptotic to the original on…
The paper proves the existence of a unique circle packing on hyperbolic surfaces.
problem Proving the existence of a unique inversive distance circle packing on hyperbolic polyhedral surfaces.
method Deforming the surface by discrete Ricci flow, doing surgery by edge flipping, and using a variational principle of a convex Ricci potential.
result There exists a unique inversive distance circle packing that is discrete conformal to the original one.
Gauss diagrams' properties can change with Hamiltonian cycle choice.
problem The impact of Hamiltonian cycle choice on Gauss diagrams.
method Examined realizable and unrealizable Gauss diagrams, and proved preservation of realizability under certain Hamiltonian cycle changes.
result Properties of Gauss diagrams can vary with Hamiltonian cycle choice.
By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopically equivalent to the empty Gauss word, disproving a conjecture by Turaev. In fact, …
Algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
problem Recognizing and performing Reidemeister moves in Gauss diagrams.
method Simple algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
result Simple algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
We discuss Gauss codes of virtual diagrams and virtual doodles. The notion of a left canonical Gauss code is introduced and it is shown that oriented virtual doodles are uniquely presented by left canonical Gauss codes.
Incorrect parity-based descriptions of realizable Gauss diagrams found, but bipartite graphs provide a valid approach.
problem Incorrect descriptions of realizable Gauss diagrams using parity conditions.
method Used bipartite graphs to describe realizable Gauss diagrams.
result Realizable Gauss diagrams can be accurately described using bipartite graphs.
Study rotational surfaces with prescribed Gauss curvature in 3D space.
problem Classify and analyze rotational surfaces with prescribed Gauss curvature.
method Phase plane analysis and mild assumptions on the prescribed function.
result Existence of singular radial solutions intersecting orthogonally the axis of rotation.
In this paper we develop a global correspondence between immersed horospherically convex hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is injective and when an immersed horospherically convex hypersurface can be unfolded …
Paper solves Orlicz-Aleksandrov problem using Gauss curvature flow.
problem Orlicz-Aleksandrov problem
method Gauss curvature flow
result Existence of smooth solutions
Study Gauss maps of surfaces in Heisenberg group using hyperbolic geometry.
problem Understanding the geometry of surfaces in Heisenberg group.
method Using Gans model of hyperbolic plane, relate tension field of Gauss map to surface's mean curvature.
result Established a relationship between tension field and mean curvature.
Study on Gauss images of specific minimal surfaces with finite curvature.
problem Characterizing Gauss images of minimal surfaces with finite total curvature.
method Analyzing the number and weight of omitted and totally ramified values of Gauss maps.
result Construction of new minimal surfaces with specific Gauss map properties.
Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.
problem Exploring hypersurfaces with constant Gauss-Kronecker curvature.
method Solving ODE for generating curves and analyzing geometric properties.
result Discovery of non-compact rotational hypersurfaces with negative Gauss-Kronecker curvature and finite volume.
We give an estimate of the Gauss curvature for minimal surfaces in Rm whose Gauss map omits more than m(m+1)/2 hyperplanes in Pm−1(C).
The paper studies the space of Gauss maps of complete minimal surfaces and their homotopy types.
problem Understanding the space of Gauss maps of complete minimal surfaces and their homotopy types.
method Proves the Gauss map assignment is a Serre fibration and determines the homotopy type of the space of meromorphic functions.
result The space of meromorphic functions on M that are the Gauss map of a complete full conformal minimal immersion has the same homotopy type as the space of all continuous maps from M to the 2-sphere. New discretizations of principal curvature lines discovered.
problem Discretizing principal curvature line parametrizations.
method Generalization of polar pairs of line congruences in the Lie quadric.
result New discretizations of orthogonal and Gauss-orthogonal parametrizations.
The Gauss Image Measure uniquely identifies dual convex bodies up to dilation.
problem Identifying dual convex bodies based on their Gauss Image Measure.
method Analyzing the Gauss Image Measure and its properties to establish the uniqueness of dual bodies.
result Dual convex bodies are equal up to a dilation on each path-connected component of the support of the measure.
Study on multiple linking numbers, extending Gauss diagram formulas.
problem Extending link invariants to more complex diagrams.
method Investigate second Gauss diagram formula involving two arrows.
result Discover two types of multiple linking numbers, one Vassiliev invariant, the other sensitive to Reidemeister moves.
Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.
problem Estimating the Morse index of anisotropic minimal surfaces.
method Local analysis of Gauss map, conformal geometric techniques applied to the Gauss map.
result Upper and lower estimates for the Morse index of anisotropic minimal surfaces.
Study classifies hypersurfaces in 4D Lorentz-Minkowski space using specific operators.
problem Classifying tubular hypersurfaces in 4D Lorentz-Minkowski space.
method Analysis of Gauss map and linearized operators L1 and L2. result Classifications of hypersurfaces with specific types of Gauss maps.
We study the Gauss map of minimal surfaces in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane H2. Conversely, any nowhere antiholomorphic harmonic map into $\mathbb{…
New experiments show Gauss diagrams not all as simple as previously thought.
problem Determining which Gauss diagrams correspond to knots.
method Experimental mathematics using Prolog and constraint satisfaction.
result Found counterexamples showing existing criteria are incomplete.
The paper examines circle graphs of Gauss diagrams and finds counterexamples to previous descriptions.
problem Problems with previous descriptions of realizable Gauss diagrams.
method Experimental checking and formulation of new descriptions of realizable circle graphs.
result New descriptions of realizable circle graphs and an algorithm for checking realizability.
Classification of constant curvature surfaces in Berger spheres.
problem Identifying complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres.
method Complete classification through detailed analysis of Clifford tori and spheres.
result Rotationally invariant spheres with constant Gauss curvature are the only topological spheres in Berger spheres for K>KP. This foreword discusses the contributions of Bolyai, Gauss, and Lobachevsky to non-Euclidean geometry.
problem The development of non-Euclidean geometries by Bolyai, Gauss, and Lobachevsky.
method Historical review of the contributions of these mathematicians.
result The foundational work on non-Euclidean geometries by Bolyai, Gauss, and Lobachevsky.
From the point of view of index theory, we give a simple proof of a Gauss-Bonnet-Chern formula for all Finsler manifolds by the Cartan connection. Based on this, we establish a Gauss-Bonnet-Chern formula for any metric-compatible connection and also derive the Gauss-Bonnet-Chern formula of Lackey.
We study the biharmonic stress-energy tensor S2 of Gauss map. Adding few assumptions, the Gauss map with vanishing S2 would be harmonic.
Gauss map of complete minimal surfaces avoids certain hypersurfaces.
problem Characterizing the range of Gauss maps of minimal surfaces.
method Analyzing the degree of hypersurfaces omitted by the Gauss map.
result Gauss map can omit hypersurfaces of degree at most nn+2(n+1)n+2. We prove a global smooth isometric immersion for negatively curved surfaces with finite total curvature.
problem Finding a sufficient condition for a complete negatively curved surface to be isometrically embedded in R^3.
method Developed new techniques to overcome slow decay and oscillations of Gauss curvature, reformulating the Gauss-Codazzi equations as a symmetric hyperbolic system.
result Proved the global existence of a smooth solution to the Gauss-Codazzi system, achieving a global smooth isometric immersion of the surface into R^3.
Generalizes Gauss-Bonnet to metrics with logarithmic singularities.
problem Calculating curvature for metrics with singularities on compact surfaces.
method Proves a generalized Gauss-Bonnet formula under Lebesgue integrability condition.
result Establishes formula for special Kähler metrics with meromorphic cubic differentials.