The paper studies Kropina metrics with a specific curvature property.
problem Characterizing Kropina metrics with isotropic scalar curvature.
method Tensor analysis to derive expressions and characterize metrics.
result Characterization of Kropina metrics with isotropic scalar curvature.
This paper reviews discrete curvature models for geometric data analysis.
problem Capturing intrinsic geometric structure in diverse data representations.
method Comprehensive review of discrete curvature models from Riemannian and metric geometry perspectives.
result Systematic pipeline for curvature-driven data analysis and learning.
Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
Deep Curvature Suite offers a PyTorch package for neural network curvature analysis.
problem Insufficient use of curvature information in neural networks.
method Implementation of Lanczos algorithm for neural network curvature analysis.
result Our package outperforms existing methods for similar purposes.
This paper connects graph curvature to community structure.
problem Understanding the relationship between network curvature and community formation.
method Defining curvature on networks and analyzing its relation to community structure.
result Apriori bounds on the curvature of intercommunity edges.
Local singularity analysis for Ricci flows with applications to bounded scalar curvature.
problem Understanding the nature of singularities in Ricci flows.
method Local singularity analysis, introducing Type I and Type II singular points, and proving curvature blow-up rates.
result Ricci curvature must blow up at least at a Type I rate near singular points of a Ricci flow.
Study finds solutions to curvature equation with boundary conditions.
problem Finding solutions to curvature equation with boundary conditions.
method Established local C2 estimates and used blow-up analysis. result Existence of conformal metrics with prescribed curvature and boundary conditions.
The paper shows that ancient noncollapsed mean curvature flows have a blowdown of at most n-2 dimensions.
problem Understanding the blowdown of ancient noncollapsed mean curvature flows.
method Fine cylindrical analysis and fine neck analysis generalization.
result The blowdown of ancient noncollapsed mean curvature flows is at most n-2 dimensional.
The paper constructs surfaces with prescribed mean curvature in a specific space.
problem Finding surfaces with a given mean curvature in a particular geometric space.
method Phase plane analysis to construct entire rotational graphs and catenoid-type surfaces.
result Classification result for surfaces with linearly prescribed mean curvature.
We have performed an empirical comparison of two distinct notions of discrete Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci curvature were developed based on different properties of the classical smooth notion…
Discussing curvature flows and their applications.
problem Analyzing expanding curvature flows.
method Classical aspects of expanding curvature flows.
result First applications of curvature flows.
The study analyzes weighted manifolds with curvature bounds, proving eigenvalue estimates and inequalities.
problem Analyzing geometric properties of weighted manifolds under Ricci curvature bounds.
method Develops geometric analysis techniques on weighted Riemannian manifolds with lower 0-weighted Ricci curvature bounds. result Proves eigenvalue estimates for Steklov and ABP inequalities on weighted manifolds.
Combines curvature descriptors with TDA for graph model evaluation.
problem Evaluating graph generative models efficiently and accurately.
method Combines graph curvature descriptors with topological data analysis.
result Robust, expressive descriptors for graph generative models.
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.
Ancient pancake solutions found for curvature flows.
problem Finding unique ancient solutions to curvature flows.
method Constructing and analyzing O(1)imesO(n)-invariant ancient solutions. result Unique O(n)-invariant ancient solutions found. Study biharmonic curves in warped product manifolds with curvature analysis.
problem Characterize biharmonic curves in warped product manifolds.
method Establish a main theorem, analyze four cases, construct examples.
result Reveal curvature-related characteristics of biharmonic curves.
The paper estimates curvature for a specific type of equations.
problem Estimating curvature for Hessian type equations.
method Establishing curvature estimates for a class of Hessian type equations.
result Curvature estimates for Hessian type equations have been successfully established.
The paper solves a problem in metric geometry for disks with negative curvature.
problem Prescribing negative Gaussian curvature on the disk and boundary geodesic curvature.
method Variational approach and refined blow-up analysis for approximated problems.
result Existence of solutions under natural curvature assumptions.
Principal Component Analysis can be performed over small domains of an embedded Riemannian manifold in order to relate the covariance analysis of the underlying point set with the local extrinsic and intrinsic curvature. We show that the volume of domains on a submanifold of general codimension, determined by the inter…
Study curvature and torsion from cross-ratios in discrete curves.
problem Define curvature and torsion for discrete curves using cross-ratios.
method Use Möbius invariant point-insertion-rule to construct circles and express torsion using cross-ratio.
result Discrete curvature and torsion defined using cross-ratios converge to smooth curvature and torsion as sampling density increases.
Three geometric analysis results on curve flows and Lie groups.
problem Analyzing geometric flows and Lie groups.
method Curve-shortening flow, point-wise curvature preserving flow, Lie group analysis.
result Interpolation between Sol and hyperbolic space in Lie groups.
Some analysis on the Lorentzian distance in a spacetime with controlled sectional (or Ricci) curvatures is done. In particular, we focus on the study of the restriction of such distance to a spacelike hypersurface satisfying the Omori-Yau maximum principle. As a consequence, and under appropriate hypotheses on the (sec…
We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanžurová are genuine generalizations of the ordinary notion of k-curvature homogeneity. The homothety group plays an essential role in the analysis.
There are two primary goals to this paper. In the first part of the paper we study smooth metric measure spaces (M^n,g,e^{-f}dv_g) and give several ways of characterizing bounds -Kg\leq \Ric+\nabla^2f\leq Kg on the Ricci curvature of the manifold. In particular, we see how bounded Ricci curvature on M controls the anal…
Lectures on surface evolution through singularities.
problem Analyzing the mean curvature flow of surfaces and their singularities.
method Analysis of neck and conical singularities, using monotonicity formulas, epsilon-regularity, weak solutions, and blowup techniques.
result Unique evolution through neck singularities, nonuniqueness through conical singularities.
Ricci flows with bounded scalar curvature cannot develop Type I singular points.
problem Ricci flows with bounded scalar curvature
method Local singularity analysis
result Scalar curvature must blow up at a Type I rate at each Type I point
Study finds existence of Q-curvature metrics on even-dimensional manifolds with conical singularities.
problem Existence of Q-curvature metrics on manifolds with conical singularities. method Blow-up analysis of a 2mth-order PDE and variational min-max argument. result First existence result for supercritical conic manifolds (except spheres).
Simply connected surfaces with large constant mean curvature and free boundaries concentrate at critical points of the boundary's mean curvature.
problem Surfaces with large constant mean curvature and free boundaries.
method Proving concentration at critical points of the boundary's mean curvature.
result Simply connected H-surfaces concentrate at critical points of the boundary's mean curvature.
Study shows convergence for mean curvature flow on almost minimal totally real submanifolds.
problem Mean curvature flow of totally real submanifolds.
method Quantitative analysis of almost minimal submanifolds.
result Established convergence result for mean curvature flow.
The c-curvature of a complete surface with Gauss curvature close to 1 in C2 norm is almost-positive (in the sense of Kim--McCann). Our proof goes by a careful case by case analysis combined with perturbation arguments from the constant curvature case, keeping track of an estimate on the closeness curvature conditi…
Proposes CC-NMDF for analyzing manifold-valued data.
problem Nonlinear structure in manifold-valued data requires new analysis methods.
method Curvature-corrected nonnegative manifold data factorization (CC-NMDF) with an iterative algorithm.
result Demonstrates CC-NMDF on real-world diffusion tensor MRI data.
We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.
The study proves curvature rigidity for convex polytopes.
problem Proving curvature rigidity for convex polytopes.
method Using Fredholm theory for Dirac operators and a theorem of Fefferman and Phong.
result Scalar curvature rigidity theorem for convex polytopes proved.
Investigate scalar curvature under geometric flows
problem Behavior of scalar curvature under geometric flows
method Three specific cases: Ricci flow, Kähler-Ricci flow, Laplacian flow
result Long-time existence of flows
Estimates for eigenfunctions and quasimodes on compact manifolds.
problem Characterizing eigenfunctions and quasimodes on compact manifolds.
method Sharp Lq-estimates for log-quasimodes, focusing on small Lebesgue exponents. result No characterization possible for q>qc. We present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities.
Study shows different behaviors of noncompact hypersurfaces under mean curvature flow.
problem Analyzing stability of noncompact hypersurfaces with curvature blowup.
method Numerical overlap method to construct global solutions.
result Existence of near and far classes of initial data leading to distinct behaviors.
Compact metrics found with specific curvature properties on 3D surfaces.
problem Finding compact metrics with constant curvature on 3D surfaces.
method Blow-up analysis of Yamabe equation with critical Sobolev exponents.
result Proved the compactness of conformal metrics with constant scalar curvature and boundary mean curvature.
We draw elliptic regularity results for 4-manifolds with an elliptic system, without Sobolev constant control. Direct use of analysis is circumvented; the results come mainly through geometric and topological arguments. In contrast to our previous paper, which worked predominantly on the scale of the curvature radius, …
Rigidity theorem for scalar curvature on odd-dimensional singular manifolds.
problem Understanding scalar curvature on manifolds with cone-like singularities.
method Analysis of abstract cone operators, spectral flow argument, and twisted Dirac operators.
result Lipschitz rigidity for scalar curvature on Riemannian spin manifolds with cone-like singularities in odd dimensions.
Survey of nonnegative scalar curvature sequences and their limits.
problem Understanding sequences of manifolds with nonnegative scalar curvature.
method Analyzing sequences of manifolds with nonnegative scalar curvature and proving convergence.
result Proved the GH and SWIF convergence of an extreme example.
New proofs for curvature problems using a viscosity approach.
problem Constant rank theorems for curvature problems in compact and non-compact settings.
method Viscosity approach to prove constant rank theorems for curvature problems.
result Generalization of a differential inequality for subtrace.
Study submanifolds in curved spaces with specific curvature bounds.
problem Investigate submanifolds in curved spaces with Ricci pinched conditions.
method Analyze submanifolds in Riemannian space forms with a lower Ricci curvature bound.
result Eliminate the need for mean curvature vector field to be parallel.
Paper disproves potential singularity models for 3D hypersurfaces in R^4.
problem Noncollapsed wing-like flows as singularity models for mean curvature flow in R^4.
method Fine bubble-sheet analysis generalizing fine neck analysis.
result Ancient noncollapsed flows in R^4 are always simple geometric shapes, not wedges.
Survey solves curvature problems with hyperbolic spaces.
problem Singularities in hypersurface geometry.
method Hyperbolic unfolding correspondence linking hypersurfaces to Gromov hyperbolic spaces.
result Eliminates hypersurface singularities in scalar curvature geometry.
Proves a higher-order positive energy theorem for stationary solutions in fourth-order gravity.
problem Proving a positive energy theorem for fourth-order gravitational theories.
method Analyzes geometric analysis intersections and links to Q-curvature. result Establishes a positive energy theorem for stationary solutions in fourth-order gravity, similar to the classical ADM theorem.
We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our ex…
The present paper describes a way to relate Martin boundaries on spaces of varying topology. This enables us to approach some detailed inductive analysis of the eigenfunctions of conformal Laplacians on minimal hypersurfaces near their singularities. This can directly be used resp. translated to understand the way how …