The paper discusses algorithms for reconstructing curves with given Euclidean or affine curvatures.
problem Reconstructing planar curves with specified Euclidean or affine curvatures.
method The paper presents algorithms for curve reconstruction under the special Euclidean and equi-affine groups.
result The reconstructed curves are close to the original curves in terms of the specified curvatures.
4-manifolds with non-positive curvature are essentially Euclidean.
problem Understanding the structure of 4-manifolds with specific curvature properties.
method Proving homeomorphism to Euclidean space using globally non-positive curvature.
result CAT(0) 4-manifolds are homeomorphic to Euclidean space.
Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.
problem Approximating local extrinsic curvature on discrete manifolds.
method Constructing discrete curvature forms on piecewise flat manifolds, using weighted sums of hinge angles.
result Converges to smooth curvature values as mesh refinement occurs, favorably comparing with other discrete approaches.
The study characterizes canal hypersurfaces in Euclidean spaces and their curvature properties.
problem Characterizing canal hypersurfaces in Euclidean spaces.
method Analyzing canal hypersurfaces in Euclidean n-space, focusing on E4, computing curvature properties, and proving specific cases.
result Flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones, and minimal canal hypersurfaces are only generalized catenoids.
The study characterizes round spheres in Euclidean space based on r-mean curvature conditions.
problem Characterizing round spheres in Euclidean space under specific curvature conditions.
method Characterization based on r-mean curvature conditions.
result Characterizes round spheres in Euclidean space under suitable r-mean curvature conditions.
Constructs hyperspheres with prescribed mean curvature in Euclidean space.
problem Creating hyperspheres with a specific curvature in Euclidean space.
method Constructs families of smooth functions to fill Euclidean space with hyperspheres of prescribed mean curvature.
result Euclidean space can be filled with hyperspheres of prescribed mean curvature.
Study on hypersurfaces in pseudo-Euclidean space with constant curvature or rotational properties.
problem Characterizing hypersurfaces in pseudo-Euclidean space.
method Defined and studied warped product hypersurfaces with constant sectional curvature or rotational properties.
result Hypersurfaces in pseudo-Euclidean space either have constant curvature or are contained in rotational hypersurfaces.
The paper classifies special solitons and shrinkers in Euclidean space.
problem Characterizing special solitons and shrinkers in Euclidean space.
method Analyzing λ-translating solitons and λ-shrinkers with constant mean curvature. result Planes, spheres, and circular cylinders are the only λ-shrinkers and λ-translating solitons with constant mean curvature. Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem.
problem Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. method Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. result Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. Study classifies graphs in Euclidean and non-Euclidean spaces with specific curvature conditions.
problem Classifying graphs with prescribed curvature in various spaces.
method Proves rigidity and classification results for graphs in Riemannian manifolds, focusing on R2 and R3. result Provides general splitting theorems for graphs in these settings.
Investigates differences in solving mean curvature problems in Euclidean and Lorentz-Minkowski spaces.
problem Solving the Dirichlet problem for mean curvature equations in different spacetimes.
method Compares techniques and conditions for solvability in Euclidean and Lorentz-Minkowski spaces.
result Lorentz-Minkowski spacelike condition allows dropping mean convexity hypothesis.
Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.
problem Investigating curvature measures in spherical, hyperbolic, and de Sitter spaces.
method Establishing a unifying framework for curvature measures in real-analytic spaces of constant curvature.
result Floating bodies and duality in non-Euclidean spaces are connected to curvature measures in Euclidean space.
Study on minimal submanifolds with finite curvature in Euclidean space.
problem Finite diffeomorphism types of complete immersed minimal submanifolds with finite total curvature.
method Adapted method from Chodosh, Ketover, and Maximo for hypersurfaces to submanifolds of arbitrary codimension.
result Proved finite diffeomorphism types for complete immersed minimal submanifolds with finite total curvature.
Flat space for manifolds with tiny curvature.
problem Understanding manifolds with curvature concentration.
method Analyzing non-compact manifolds with non-negative Ricci curvature and small curvature concentration.
result Manifolds with curvature concentration are flat.
The paper studies essential spectra of submanifolds in Euclidean spaces.
problem Investigating the essential spectrum of submanifolds under geometric conditions.
method Analyzing submanifolds in Euclidean spaces with various geometric constraints.
result The essential spectrum of a complete non-compact submanifold is [0,+∞) if the second fundamental form satisfies certain Lp norms. 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.
The paper studies Finsler manifolds with a new curvature concept.
problem Understanding Finsler manifolds with positive weighted flag curvature.
method Introducing a new curvature concept based on the flag curvature and a non-Riemannian quantity, T-curvature.
result Positive weighted flag curvature implies the manifold is diffeomorphic to Euclidean space.
Compactness theorem for manifolds with scalar curvature and entropy bounds.
problem Understanding the structure of manifolds with specific curvature and entropy bounds.
method Using volume upper bounds to prove Gromov-Hausdorff closeness to Euclidean balls.
result Unit balls in such manifolds are bi-Hölder and bi-W1,p homeomorphic to Euclidean balls. Paper defines curvature equivalence for Legendre curves in a plane.
problem No specific problem stated; focuses on Legendre curves.
method Introduced curvature equivalence relation for Legendre curves.
result Local and global classifications of Legendre curves under curvature equivalence.
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.
Proving geodesic triangulation spaces are Euclidean.
problem Proving spaces of geodesic triangulations are homeomorphic to Euclidean spaces.
method Proposing an approach to prove homeomorphism using negative curvature surfaces.
result Spaces of geodesic triangulations are homeomorphic to Euclidean spaces.
Radial graphs with constant mean curvature found in Euclidean space.
problem Existence of hypersurfaces with constant mean curvature.
method Radial graphs over domains of the unit sphere, Dirichlet problem.
result Existence of hypersurfaces with positive constant mean curvature.
The paper proves pseudolocality theorems for Ricci flows on incomplete manifolds.
problem Pseudolocality of Ricci flows on incomplete manifolds.
method Proves pseudolocality theorems for Ricci flows under specific curvature and isoperimetric conditions.
result Constructs solutions of Ricci flow in balls with pseudolocality property.
The paper classifies hypersurfaces with constant curvature in Euclidean spaces.
problem Classifying separable hypersurfaces with constant sectional curvature.
method Analytical proof and classification of hypersurfaces in Euclidean spaces.
result Hyperspheres are the only separable hypersurfaces with nonzero constant sectional curvature.
Develops weak formulation for spacelike flows in pseudo-Euclidean space.
problem Weak formulation of spacelike mean curvature flow in pseudo-Euclidean space.
method Based on spacelike integer rectifiable varifolds and pseudo-Euclidean first variation.
result Existence and compactness of spacelike Brakke flows with fixed boundary.
Curvature conditions distinguish Euclidean space and disks in contractible manifolds.
problem Distinguish Euclidean space and disks among contractible manifolds.
method Investigate curvature conditions on open and compact contractible manifolds with boundary.
result Stronger curvature conditions can distinguish disks from Euclidean spaces.
The paper proves a theorem about mean curvature in Euclidean and hyperbolic spaces.
problem Proving a theorem about mean curvature in Euclidean and hyperbolic spaces.
method Analyzing connected mean convex regions with at least two components in Rn+1 and hyperbolic space. result Connected mean convex regions in Rn+1 with at least two components cannot have strictly positive mean curvature. The paper explores rigidity of hypersurfaces with constant curvature in Euclidean spaces.
problem Rigidity of hypersurfaces with constant mean and scalar curvature.
method Characterizations and rigidity results under various conditions of Gaussian-Kronecker and r-th mean curvatures. result Rigidity theorems for hypersurfaces in dimensions 4, 5, and 6, and general dimensions under pinching conditions.
Optimal bounds found for torus curvatures in high dimensions.
problem Finding optimal bounds on normal curvatures of tori.
method Analyzing immersed n-torus in a Euclidean ball of large dimension.
result Optimal bounds on normal curvatures of tori established.
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.
The study characterizes compact submanifolds with pinched Ricci curvature in Euclidean and spherical space forms.
problem Characterizing compact submanifolds with specific Ricci curvature bounds.
method Proving rigidity results for submanifolds with Ricci curvature bounded below by a function of mean curvature.
result Submanifolds are either isometric to the Einstein Clifford torus or have vanishing homology groups up to a certain degree.
No regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces are found.
problem Existence of regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces.
method Analyzing polynomials defining hypersurfaces of various degrees and shapes.
result Hyperspheres and round cylinders are the only such hypersurfaces defined by polynomials of degree ≤3.
The study proves rigidity for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.
problem Proving rigidity for Poincaré-Einstein manifolds with specific conformal infinity.
method Analyzing curvature tensors over level sets of a boundary defining function.
result Rigidity theorem for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.
In this paper, we give a complete description of all translation hypersurfaces with constant r-curvature Sr, in the Euclidean space.
This paper extends a 3D result to higher dimensions for manifolds with positive curvature.
problem Proving higher-dimensional manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to the Euclidean disk.
method Using the positive curvature operator and strictly convex boundary conditions to deduce the manifold's diffeomorphism to the Euclidean disk.
result Compact n-manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to the standard n-dimensional Euclidean disk.
New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.
problem Finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows with surgery for inversive distance circle packings.
result Longtime existence and global convergence of combinatorial curvature flows with surgery.
Local smoothing of metrics with small curvature, removing Ricci curvature condition.
problem Establishing local smoothing of metrics with curvature concentration.
method Local mollification, removing Ricci curvature condition, Sobolev constants and volume growth.
result Compactness of manifolds with small curvature concentration under Ahlfors regularity and Sobolev constant.
Compact gradient ρ-Einstein solitons are isometric to Euclidean spheres.
problem Characterizing gradient ρ-Einstein solitons in Riemannian manifolds.
method Proved isometry by showing constant scalar curvature for compact cases and vanishing scalar curvature for non-compact cases with integral conditions.
result Compact gradient ρ-Einstein solitons are isometric to Euclidean spheres.
The paper defines and calculates fourth fundamental form and i-th curvatures for hypersurfaces in 4D Euclidean space.
problem Calculating curvatures for hypersurfaces in 4D Euclidean space.
method Defining fourth fundamental form and i-th curvatures for hypersurfaces, calculating them on rotational hypersurface, and studying hypersurfaces satisfying a specific differential equation.
result Fourth fundamental form and i-th curvatures are defined and calculated for hypersurfaces in 4D Euclidean space.
Open manifolds with nonnegative Ricci curvature and Euclidean volume growth have finitely generated fundamental groups.
problem Understanding fundamental groups of open manifolds with specific curvature properties.
method Analyzing universal covers with Euclidean volume growth and applying topological group theory.
result Fundamental groups are finitely generated and virtually abelian under given conditions.
New methods create full discretized isothermic tori in Euclidean spaces.
problem Creating full discretized isothermic tori in Euclidean spaces.
method Using Darboux transformations and periodic curvature line systems.
result Discrete and semi-discrete k-dimensional isothermic tori in n-dimensional Euclidean space.
Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.
problem Rigidity of Kähler manifolds with nonnegative Ricci curvature.
method Analysis of Kähler manifolds with specific properties.
result Complete noncompact Kähler surface with nonnegative Ricci curvature, Euclidean volume growth, and quadratic curvature decay is biholomorphic to the resolution of an affine algebraic variety.
Examines discrete curvature's relation to smooth curvature in 3 spaces.
problem Understanding how discrete curvature relates to smooth curvature in different spaces.
method Using specific triangular tilings of 3 types of spaces to examine curvatures.
result Discrete curvature can sense the smooth curvature of ambient space forms.
We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yam…
We classify all Hamiltonian stationary Lagrangian surfaces in complex Euclidean plane which are self-similar solutions of the mean curvature flow.
Proves inequalities on curved spaces with positive curvature.
problem Proving inequalities on manifolds with nonnegative Ricci curvature.
method Analyzes manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Proves Heisenberg-Pauli-Weyl, Hardy-Sobolev, and Caffarelli-Kohn-Nirenberg inequalities.
We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curva…
We prove generalized lower Ricci bounds for Euclidean and spherical cones over compact Riemannian manifolds. These cones are regarded as complete metric measure spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-di…