Characterizes graphs with Lin-Lu-Yau curvature at least one and explores bone-idle graphs.
problem Characterizing graphs with specific curvature properties.
method Study of Ollivier-Ricci curvature and Lin-Lu-Yau curvature, exploration of regular graphs, and exact formula derivation.
result Characterizes edges that are bone-idle in regular graphs and provides a complete characterization of 4-regular bone-idle graphs.
Curvature formulas on regular graphs identified bone idle edges and graphs.
problem Understanding curvature in regular graphs and identifying bone idle edges.
method Explicit formulas for Lin-Lu-Yau and Ollivier-Ricci curvatures derived from graph parameters.
result Equality condition on regular graphs for Ollivier-Ricci curvature and characterization of bone idle edges.
Study on the regularity of p-Gauss curvature flow near flat interfaces.
problem Regularity of p-Gauss curvature flow near flat interfaces. method Analysis of convex hypersurface near the interface.
result Regularity of the convex hypersurface near the interface.
Global regularity proved for 4D Ricci flow with scalar curvature integral bound.
problem Global regularity of 4D Ricci flow with integral scalar curvature bound.
method Extended Ge-Jiang's result to include integral bound on scalar curvature.
result Global ε-regularity for 4D Ricci flow with integral scalar curvature bound. Derives spacetime regularity under specific curvature conditions.
problem Ensuring smoothness in spacetime models with given curvature constraints.
method General regularity estimate for 4-d spacetimes, using Ricci curvature and Lie derivatives.
result Establishes conditions for smoothness in spacetime models.
Study distance maps on spaces with curvature bound, proving regularity and sphere theorem.
problem Regularity of distance maps on geodesically complete spaces with curvature bound above.
method Define and prove regularity of distance maps as Hurewicz fibrations.
result Sphere theorem for geodesically complete CAT(1) spaces.
Study on Lin-Lu-Yau curvature and diameter of amply regular graphs.
problem Lower bounds of Lin-Lu-Yau curvature in amply regular graphs.
method Application of Hall's marriage theorem and geometric proof.
result Conference graphs have positive Lin-Lu-Yau curvature.
Curvature regularization prevents distortion in graph embeddings.
problem Graph topology patterns distort in Euclidean space, making detection difficult.
method Proposes curvature regularization to enforce flatness in embedding manifolds.
result Significant improvements in five embedding methods on open graph datasets.
Sharp bounds on diameter and eigenvalues for amply regular graphs.
problem Finding bounds for amply regular graphs' diameter and eigenvalues.
method New ideas relating discrete Ricci curvature to local matching properties, including a novel construction of a regular bipartite graph.
result Sharp diameter and eigenvalue bounds for amply regular graphs.
Proves optimal regularity for sphere minimizers in 3-sphere.
problem Finding optimal regularity for sphere minimizers.
method Proves C1,1 regularity for minimizers of prescribed mean curvature over isotopy classes. result Proves optimal C1,1 regularity for minimizers. The paper proves diameter bounds and finiteness for amply regular graphs.
problem Proving diameter bounds and finiteness for amply regular graphs.
method Improved curvature estimates and new Bakry-Émery curvature estimates.
result There are only finitely many amply regular graphs with specific parameters.
2-regular points found in spaces with lower Ricci curvature bound.
problem Characterizing points in spaces with lower Ricci curvature bound.
method Analyzing measured Gromov-Hausdorff limits of Riemannian manifolds.
result 2-regular points in interior geodesics of limit spaces are 2-rectifiable.
Convex solutions to a specific equation are smooth when the phase is smooth enough.
problem Regularity of solutions to the Lagrangian mean curvature equation.
method Showed regularity for convex solutions under Hölder continuity conditions on the phase.
result Convex viscosity solutions are regular if the Lagrangian phase is Hölder continuous.
Proves higher regularity for anisotropic inverse mean curvature flow.
problem Higher regularity of solutions to anisotropic inverse mean curvature flow.
method Proves Harnack estimate and constructs smooth solutions from C1 initial sets. result Smooth solutions become smooth outside a compact set.
Study proves existence of weak mean curvature flow with contact angle.
problem Existence of weak mean curvature flow with prescribed contact angle.
method Compactness theorem for varifolds and Ilmanen's regularization extended to capillarity.
result Existence of weak mean curvature flow with contact angle for general θ. Flow adjusts curvature to avoid a fixed region, proving bounds and regularity.
problem Adjusting curvature flow to avoid a fixed region.
method Flow by powers of Gauss curvature, proving optimal curvature bounds and regularity.
result Proves optimal curvature bounds and long time existence for all dimensions and powers.
Study proves a new flow method for mean curvature with volume change analysis.
problem Existence of BV flow via mean curvature flow.
method Elliptic regularization to prove existence of generalized BV flow.
result Proves existence of generalized BV flow with volume change expression.
The study examines the regularity of branched immersions using special coordinate systems.
problem Understanding the regularity of branched immersions and their fundamental elements.
method Development and use of special coordinate systems to express maps with branch points, proving existence and regularity conditions for mean curvature vectors.
result Characterization and existence of special coordinate systems for branch immersions, proving regularity conditions for mean curvature vectors.
Classifies regularity for Lagrangian mean curvature type equations.
problem Classifying regularity for Lagrangian mean curvature type equations.
method Generalized constant rank theorem for Legendre transform, constructed convex solutions, and showed regularity conditions.
result Optimal regularity conditions for Lagrangian mean curvature type equations.
Introduces non-regular spacetime geometry without smooth calculus.
problem Defining gravity without smooth spacetime geometry.
method Discusses non-regular spacetime geometry and curvature without differential calculus.
result Curvature and gravity can be defined without smooth spacetime calculus.
Discuss folklore statements about manifolds with curvature bounds.
problem Distance functions in manifolds with curvature bounds.
method Regularity, subsets of positive reach, and cut locus.
result Folklore statements about manifolds with curvature bounds are discussed.
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.
We study a modified notion of Ollivier's coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau in [11]. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the Ricci curvature is strictly greater than one. We then derive explicit Ricci curv…
Optimally regularizes boundaries in the Heisenberg group with prescribed curvature.
problem Optimizing boundaries with prescribed sub-Finsler mean curvature in the Heisenberg group.
method Analyzes critical sets of the prescribed mean curvature functional in the Heisenberg group.
result Characteristic curves of critical sets are C2-regular, optimal in the Heisenberg group. New relation between curvature bounds and spacetime inextendibility.
problem Inextendibility of spacetimes under low regularity conditions.
method Synthetic curvature and causal character analysis.
result Low-regularity spacetimes with unbounded curvature.
New bounds on geodesic dimension and curvature exponent in Carnot groups.
problem Characterizing geodesic dimension and curvature exponent in Carnot groups.
method Characterization and lower bound calculation for geodesic dimension and curvature exponent.
result Found an example where curvature exponent is greater than geodesic dimension.
Proves curvature bounds for close to 1 Perelman's reduced volume.
problem Curvature bounds for Ricci flow with close to 1 reduced volume.
method ε-regularity theorem for Perelman's reduced volume.
result Curvature radius cannot be too small if reduced volume is close to 1.
New relation between curvature bounds and spacetime inextendibility.
problem Inextendibility of spacetimes under low regularity conditions.
method Synthetic curvature and causal character maximizers.
result Low-regularity inextendibility linked to unbounded curvature.
The paper studies steady solitons with curvature decay and proves their smoothness.
problem Analyzing the properties of steady solitons with curvature decay.
method Bootstrap regularity in harmonic coordinates using the soliton equation.
result Steady gradient Ricci solitons are asymptotically cylindrical under certain curvature decay conditions.
In this paper, we investigate a regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are minimal regularizable submanifolds. We prove that, if the initial invariant hypersurface satisfies a certai…
Survey on geodesics on tetrahedra in curved spaces.
problem Understanding geodesics on tetrahedra in curved spaces.
method Analyzing geodesics on regular tetrahedra in spaces of constant curvature.
result Results on the behavior of simple closed geodesics.
We introduce a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space, through higher order equations. We prove that the regularized problems converge to the mean curvature flow for all times before the first singularity.
New bounds for low-regularity Riemannian metrics defined via distributional curvature.
problem Establishing curvature bounds for Riemannian metrics of low regularity.
method Introducing a distributional version of sectional curvature for C1 and C0 metrics. result New bounds for low-regularity metrics recover classical bounds in Alexandrov spaces.
The paper proves regularity for varifolds with bounded anisotropic mean curvature.
problem Regularity of varifolds with bounded anisotropic mean curvature.
method Local anisotropic regularity theorem and touching balls approach.
result Varifolds can be covered by countably many C2-regular submanifolds. We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White's local regularity theorem for mean curvature …
Proves long-term smoothness of curved surfaces evolving under specific curvature rules.
problem Long-term regularity of curved surfaces evolving under p-Gauss curvature flow. method Transformed the curvature flow into a Monge-Ampère equation and studied its asymptotic cone.
result Proved regularity of the interface in all dimensions for $p> rac1n$.
We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy types of spaces of its local minima. We consider properties of the total curvatu…
Positive mass theorem for asymptotically flat manifolds with non-negative distributional scalar curvature
problem Positive mass theorem
method Ricci flow smoothing
result Asymptotically flat manifolds with non-negative ADM mass
The paper proves rigidity and ε-regularity theorems for Ricci shrinkers.
problem Understanding the structure and behavior of Ricci shrinkers.
method Proving rigidity and ε-regularity theorems for Ricci shrinkers using entropy and curvature.
result Non-compact Ricci shrinkers are asymptotic to cones under certain curvature conditions.
Study of cuspidal edges on focal surfaces of regular surfaces.
problem Clarifying the sign of singular curvature at cuspidal edges.
method Investigation using singularities of parallel surfaces.
result Clarification of the sign of singular curvature at cuspidal edges.
Study proves existence of expanding solutions for multiphase surfaces with regular junctions.
problem Existence of self-similar expanding solutions for multiphase surfaces with regular junctions.
method Proves existence of solutions for a multiphase surface with regular junctions using mean curvature flow.
result Multiple self-similar expanding solutions exist for the initial condition of a multiphase surface with regular junctions.
CurvSSL improves SSL by aligning local manifold curvature.
problem Improving self-supervised learning by capturing local manifold geometry.
method CurvSSL augments Barlow Twins with a curvature-based regularizer to align and decorrelate embeddings across augmentations.
result Curvature-regularized SSL yields competitive or improved linear evaluation performance.
The study extends convergence theorems for Ricci-limit spaces with bounded curvature.
problem Understanding convergence properties of Ricci-limit spaces with bounded curvature.
method Establishing C1,α-regularities and applying Fukaya's fibration theorem. result Optimal generalization of Fukaya's fibration theorem to C1,α limit spaces. Convex surfaces derived from specific Riemannian manifolds with high regularity.
problem Proving convexity of surfaces derived from Riemannian manifolds.
method Analyzing solutions to the very weak Monge-Ampère equation.
result Proved convexity of weakly regular surfaces with nonnegative intrinsic curvature.
Introduces Lie group actions in smoothing processes for currents and spaces with curvature.
problem Regularization of currents and metrics on manifolds and spaces with curvature.
method Actions of compact Lie groups in De Rham approximation and smoothing of Riemannian metrics.
result Effective smoothing processes for currents and metrics on manifolds and spaces with curvature.
Proves existence and uniqueness of curvature motion for regular networks.
problem Existence and uniqueness of motion by curvature for regular networks.
method Proves existence and uniqueness using $W^{2- rac{2}{p}}_p$ initial data and investigates regularization effects.
result Proves existence and uniqueness of motion by curvature for regular networks.
In this paper, we investigate the regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are regularized minimal. We prove that, if the invariant hypersurface satisfies a certain kind of horizontall…
In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on a given Kähler manifold. The goal of this paper is to give an ε-regularity theorem for the line bundle mea…