New curvature condition for variable curvature spaces, proving geometric properties.
problem Understanding variable curvature spaces and their geometric properties.
method Introducing a new curvature-dimension condition CD(k,N) for metric measure spaces with variable lower curvature bounds.
result Sharp geometric properties such as volume growth comparison and Bonnet-Myers theorem.
Develops methods to construct harmonic and wave maps into variable-curvature surfaces.
problem Limited explicit constructions for harmonic and wave maps in variable-curvature settings.
method Reduction framework for pseudo-Riemannian surfaces, geometric ansatz, first-order ODEs.
result Constructs explicit harmonic and wave maps into ellipsoids, hyperboloids, and Schwarzschild exterior.
This note surveys and compares results on the separation of variables construction for soliton solutions of curvature equations including the Kähler-Ricci flow and the Lagrangian mean curvature flow. In the last section, we propose some new generalizations in the Lagrangian mean curvature flow case.
Constructs orthogonal coordinates in curved spaces.
problem Separating variables in curved spaces.
method Explicit construction of orthogonal coordinates and transformations.
result Explicit formulas for Killing tensors and Stäckel matrices.
Study curvature bounds on metric spaces using variational inequalities and Wasserstein control.
problem Analyzing curvature bounds on metric measure spaces with variable curvature.
method Evolution variational inequality, entropic curvature-dimension condition, Riemannian curvature-dimension condition.
result Established equivalence between curvature-dimension conditions and stability under Gromov convergence.
Study on elastic curves with variable stiffness, derived from bending energy.
problem Modeling elastic wires with varying thickness.
method Derive Euler-Lagrange equations for curves with variable bending stiffness.
result Characterizations of elastic curves with variable stiffness.
Calculates moments of sectional curvature on Riemannian manifolds.
problem Understanding the distribution and moments of sectional curvature.
method Integrating local Riemannian invariants and analyzing sectional curvature on Grassmann bundles.
result Proves a weak version of the Hitchin-Thorpe Inequality.
The paper classifies curves in dual affine and Lorentz-Minkowski planes with constant curvature.
problem Classifying curves with constant curvature in dual affine and Lorentz-Minkowski planes.
method Investigation of invariants under equiaffine transformations and explicit equations for curves with constant curvature.
result Curves with constant curvature in dual affine and Lorentz-Minkowski planes are classified.
Unified approach to various energy conditions in spacetime geometry.
problem Synthetic quantification of energy conditions in spacetime.
method Introducing entropic timelike curvature dimension condition with variable Ricci curvature bounds.
result Unified approach to various energy conditions including strong, weak, and null energy conditions.
Using a new method we give elementary estimates for the capacity of non-contractible annuli on cylinders and provide examples, where these inequalities are sharp. Here the lower bound depends only on the area of the annulus. In the case of constant curvature this lower bound is obtained with the help of a symmetrizatio…
Study on diagonal and separating coordinates for symmetric spaces of rank 1.
problem Existence and nonexistence of diagonal and separating coordinates for symmetric spaces of rank 1.
method Generalization of results by Gauduchon and Moroianu, 2020, and analysis of constant sectional curvature and orthogonal separation of variables.
result Diagonal coordinates exist if and only if the symmetric space has constant sectional curvature.
The paper analyzes graph Laplacians on manifolds with curvature bounds and applies to non-collapsed spaces.
problem Analyzing spectral properties of graph Laplacians on manifolds with curvature constraints.
method Quantitative bounds on eigenvalues and eigenfunctions of graph Laplacians constructed from random variables on manifolds with uniform lower Ricci curvature bounds.
result Spectral convergence of graph Laplacians on manifolds with curvature bounds and in non-collapsed spaces.
Researchers classify and describe Kα-translators in Euclidean space.
problem Classifying and describing Kα-translators in Euclidean space. method Rotationally symmetric and helicoidal motions.
result For each α, there is a Kα-translator intersecting orthogonally the rotation axis. Study of curves in hyperbolic plane with variable curvature.
problem Finding curves with prescribed almost constant curvature in hyperbolic plane.
method Analyzing closed and embedded curves with geodesic curvature.
result Existence of curves with specified curvature variations.
New curvature measure on graphs improves diameter and eigenvalue bounds.
problem Improving curvature bounds on graph structures.
method Hybrid curvature definition on variable neighborhoods.
result Gradient estimates and curvature bounds proven.
The Newman-Penrose-Perjes formalism is applied to Sasakian 3-manifolds and the local form of the metric and contact structure is presented. The local moduli space can be parameterised by a single function of two variables and it is shown that, given any smooth function of two variables, there exists locally a Sasakian …
The scalar curvature for the noncommutative four torus TΘ4, where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3-sphere. This method is more convenient since it does not require the rearrangement le…
Study finds new factorable surfaces with non-zero curvature in pseudo-Galilean space.
problem Classifying surfaces with non-zero curvature in pseudo-Galilean space.
method Analyzing factorable surfaces as graphs of product functions.
result New classification results for factorable surfaces with non-zero Gaussian and mean curvature.
The study classifies Riemannian manifolds with curvature nullity.
problem Classifying Riemannian manifolds with nontrivial curvature nullity.
method Classification theorems based on curvature nullity, scalar curvature, and quotient existence.
result New classification theorems and revisited previous results.
We show that there is an infinite group of special automorphisms of the deformed group of diffeomorphisms, which describes parallel transports in Riemannian spaces of any variable curvature. Generators of translations of such group contain covariant derivatives, and structure functions - the curvature tensor.
Proves CLT for Brownian paths on pinched negative curvature manifolds.
problem Distribution of Brownian paths on pinched negative curvature manifolds.
method Proof of central limit theorem for distances and Green functions.
result Central limit theorem holds for Brownian paths in pinched negative curvature.
In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds with variable negative curvature. Applying results from dynamics on these spaces,…
Let T be the standard torus of revolution in R^3 with radii b and 1, 0<b<1. Let αbe a (p,q) torus curve on T. We show that there are points of zero curvature on αfor only one value of the variable radius of T, b=p^2/(p^2+q^2). The curve αhas non-vanishing curvature for all other values of b. Moreover, for this value of…
We prove a finiteness theorem for the class of complete finite volume Riemannian manifolds with pinched negative sectional curvature, fixed fundamental group, and of dimension >2. One of the key ingredients is that the fundamental group of such a manifold does not admit a small nontrivial action on an R-tree.
The study identifies unique Delaunay surfaces with constant mean curvature.
problem Characterizing surfaces with constant mean curvature.
method Analyzing surfaces defined by specific implicit equations.
result Only the plane and catenoid are Delaunay surfaces with nonzero constant mean curvature.
In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R^3. By means of a connection of the differential equations for these curvature lines and real Ricc…
Study Finsler metrics with Killing fields on constant flag curvature surfaces.
problem Characterize Finsler metrics with Killing fields on surfaces of constant flag curvature.
method Developed a normal form and method to calculate functions for spherically symmetric Finsler surfaces.
result Obtained the normal form of the Funk metric on the unit disk D^2.
The paper proves inequalities under Bakry-Émery-Ricci curvature bounds.
problem Proving functional inequalities under lower Bakry-Émery-Ricci curvature bounds.
method Lower m-Bakry-Émery-Ricci curvature bounds with ε-range. result Proves Cheng type inequality and local Sobolev inequality.
We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations F(x,u,du,d2u)=0 defined on a finite-dimensional Riemannian manifold M. Finest results (with hypothesis that require the function F to be degenerate ell…
A new method finds a subconscious point on curved surfaces.
problem Finding a point on variable curvature surfaces that minimizes distances.
method New variational method based on geodesic arc length.
result Solution involves a 'subconscious' point with positive curvature.
The paper studies constant curvature holomorphic two-spheres in complex Grassmann manifold.
problem Investigating constant curvature holomorphic two-spheres in complex Grassmann manifold.
method Exploring the theory of functions of one complex variable to determine curvature distribution and construct examples.
result Explicit characterization and construction of non-homogeneous constantly curved holomorphic two-spheres.
The paper improves eigenvalue estimates for manifolds with Ricci curvature conditions.
problem Eigenvalue estimates for manifolds with Ricci curvature conditions.
method Proves eigenvalue estimates using a Kato condition on the negative part of Ricci curvature.
result Optimal eigenvalue estimates for Zhong-Yang type and Cheng-type bounds.
New equations reveal how cylinder power in progressive lenses depends on geodesic curvature.
problem Current understanding of cylinder power in progressive lenses is incomplete.
method Derived complete compatibility equations for spatially-varying curvature surfaces.
result Cylinder power depends on geodesic curvature, not just principal curvature.
The geometry of oscillatory integrals on manifolds with intermediate symmetry.
problem Classification of curvature conditions in Sogge's program.
method Proposing a classification of curvature conditions.
result No manifolds satisfy the chaotic curvature condition of order 1.
We give examples of isospectral non-isometric surfaces of genus 2 and 3 with variable curvatures and apply the result to construct isospectral potentials on Riemann surfaces of genus 2.
Study of long-time behavior of solutions on negatively curved manifolds.
problem Long-time behavior of solutions to the Porous Medium Equation on Cartan-Hadamard manifolds with negative curvature.
method Analysis of long-time behavior, proving existence and uniqueness of solutions, using comparison principles.
result Unexpected separate-variable behavior, reminiscent of Dirichlet problems on bounded Euclidean domains.
Investigates parallel spinors on Eguchi-Hanson metrics.
problem Analyzing parallel spinors on specific metrics.
method Investigated parallel spinors on Eguchi-Hanson metrics with harmonic spinors.
result Found complex 2-dimensional space of complex parallel spinors and solutions for metrics with zero scalar curvature.
To any completely integrable second-order system of real or complex partial differential equations in n > 1 independent variables and in one dependent variable, Mohsen Hachtroudi associated in 1937 a normal projective (Cartan) connection, and he computed its curvature. By means of a natural transfer of jet polynomials …
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.
The goal of this paper is twofold: we study metric measure spaces (X,d,m) with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function k:X→R we introduce the curvature-dimension condition CD(k,∞) which canonically ex…
Study of deformed Hermitian Yang-Mills equations with variable Kähler metrics.
problem Solving special Lagrangian type equations with variable metrics.
method Introducing extended gauge group to couple moment maps and scalar curvature.
result Solutions satisfy a mixture of K-stability and Bridgeland-type stability.
Note shows equivalence of recent NEC reformulation to classical NEC for C2-metrics.
problem Consistency of null energy condition in Lorentzian length spaces.
method Shows equivalence of recent reformulation of null energy condition to classical formulation for C2-metrics. result Equivalence of recent reformulation of null energy condition to classical formulation for C2-metrics. Study on dynamic curves with elastic energy and spontaneous curvature.
problem Modeling and analyzing dynamic planar curves with elastic energy.
method Gradient flow of inclination angle, nonlocal quasilinear system, local well-posedness, global existence, convergence.
result Local well-posedness, global existence, convergence of the flow for weak regularity initial data.
Study improves Poisson equation solutions on various manifolds.
problem Improving solutions to Poisson equation on different types of manifolds.
method Established L1 estimates for mixed boundary conditions on manifolds with specific curvature properties. result Generalized existing theorems to broader Riemannian settings.
We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé's system of equations. We show that the symmetry group of the Lamé's system, satisfying Guichard condition, is given by translations and dilations in the independent variables and dilations in the dep…
Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.
problem Preserving convexity and convergence of curves under curvature flows on pinched Hadamard surfaces.
method Area- and length-preserving curvature flows, refined comparison arguments, delicate curvature estimates.
result Convexity is preserved and curves converge to a geodesic circle under certain conditions.
Formula solves discrete pseudospherical surfaces in Euclidean space.
problem Discrete surfaces with constant negative curvature.
method Loop group decompositions and nonlinear d'Alembert formula.
result Computed examples of discrete pseudospherical surfaces.
In this paper we study the rigidity of infinite volume 3-manifolds with sectional curvature −b2≤K≤−1 and finitely generated fundamental group. In-particular, we generalize the Sullivan's quasi-conformal rigidity for finitely generated fundamental group with empty dissipative set to negative variable curvature …