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.
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.
Paper finds smooth convex solutions to curvature problem.
problem Finding smooth, convex solutions to curvature problems.
method Established existence of solutions through mathematical analysis.
result Smooth, origin-symmetric, strictly convex solutions found.
We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.
Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
Given a compact four dimensional smooth Riemannian manifold (M,g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero and the evolution equation by T-curvature at the boundary with the condition that the Q-curvature …
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.
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.
Extends curve theory to non-smooth data with finite curvature and torsion.
problem Applying classical curve theory to non-smooth data.
method Using distributional derivative measures of functions of bounded variation.
result Essentially unique non-smooth curve solution with finite total curvature and torsion.
Smooths metrics with nonnegative scalar curvature near singular sets.
problem Approximating metrics with nonnegative scalar curvature near singularities.
method Ricci-DeTurck flow to approximate metrics.
result Approximated metrics converge to the original metric in C∞ away from the singular set. Smooth solutions found for modified mean curvature flow in Riemannian manifolds.
problem Existence of smooth solutions for modified mean curvature flow.
method A priori estimates for modified mean curvature flow in Riemannian manifolds with Killing vector field.
result Existence of smooth, entire, longtime solutions for modified mean curvature flow with smooth initial data.
Proves existence of smooth metrics with specific curvature properties.
problem Existence of smooth metrics with prescribed negative Ricci curvature.
method Formulated and proved for general domains in Euclidean space.
result Existence of smooth complete conformal metrics with prescribed negative Ricci curvature.
Proves existence of smooth convex solutions to capillary curvature equations.
problem Proving existence of smooth convex solutions to capillary curvature equations.
method Gradient estimate for capillary curvature equations in half-space.
result Existence of even, smooth, strictly convex solutions for all 1<p<k+1 and θ∈(0,π/2). Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth embeddings of the ball with arbitrary small volume.
Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.
problem Smooth metrics on manifolds with curvature and injectivity constraints.
method Bi-Lipschitz smoothing with controlled smoothing and volume lower bounds.
result Proves existence of smooth metrics with curvature bounds and injectivity radius constraints.
We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds.
Formula derived for curvature on smooth manifolds.
problem Calculating curvature on smooth manifolds.
method Derived a formula for sectional curvature.
result Formula for sectional curvature on smooth manifolds.
Graphs can be smoothed or squashed too, study finds.
problem Graph Neural Networks struggle with over-smoothing and over-squashing issues.
method Unified framework using Ollivier-Ricci curvature to address both issues.
result Over-smoothing and over-squashing linked to positive and negative graph curvature respectively.
Smooth curves with specific curvature can be closely approximated.
problem Approximating smooth curves with prescribed curvature.
method Application of h-principle to C1-dense approximation of curves. result Existence of C∞ knots with prescribed curvature. Study on when smooth Ricci flow remains smooth at the start.
problem When does a smooth Ricci flow remain smooth down to the initial time?
method Curvature estimates and lower Ricci bounds in three dimensions.
result Positive results for flows with lower curvature bounds, negative for others.
Mean curvature flow shows singularities on smooth surfaces.
problem Understanding singularities in mean curvature flow.
method Analyzing spherical or nondegenerate neck pinches.
result First singular time has isolated singularities.
The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…
Proves smoothness of conical singularities in mean curvature flow.
problem Resolving singularities in mean curvature flow.
method Analyzes smooth hypersurfaces with isolated conical singularities.
result Smoothness of level set flow through asymptotically conical singularities.
Smooth solutions found for a curvature problem in hyperbolic space.
problem Existence of smooth complete hypersurfaces with prescribed curvature in hyperbolic space.
method Utilized Pogorelov type interior second order estimate.
result Affirmative answers for specific curvature cases in hyperbolic space.
Survey on preserving curvature bounds for non-smooth Ricci flow.
problem Preserving curvature bounds for non-smooth initial data in Ricci flow.
method Survey of various weak initial data and preservation of curvature bounds.
result Various curvature lower bounds preserved up to a constant for non-smooth initial data.
Paper proves minimizing movements match smooth droplet flow in 3D.
problem Consistency of minimizing movements with smooth mean curvature flow.
method Proved minimizing movements coincide with smooth droplet flow.
result Minimizing movements and smooth mean curvature flow are consistent in 3D.
This note is a continuation of the author's paper \cite{Li}. We prove that if the metric g of a 4-manifold has bounded Ricci curvature and the curvature has no local concentration everywhere, then it can be smoothed to a metric with bounded sectional curvature. Here we don't assume the bound for local Sobolev constan…
Curvature estimates prove existence of smooth hypersurfaces in hyperbolic space.
problem Existence of smooth complete hypersurfaces with constant curvature in hyperbolic space.
method Deriving curvature estimates to prove existence for all curvature values.
result Existence of smooth hypersurfaces for all possible curvature values.
Smooth submetries between curved spaces are smooth.
problem Smoothness of submetries between curved spaces.
method Proving smoothness of submetries in a general setting, including Riemannian submersions and isometric actions.
result Smoothness of the base manifold is implied by the smoothness of the total manifold without curvature assumptions.
Smooth 3D flows from non-smooth starting points.
problem Creating smooth Ricci flows from non-smooth initial conditions.
method Generalized singular Ricci flow applied to 3D complete manifolds.
result Existence of smooth Ricci flows starting from non-smooth initial conditions.
New method uses curvature to improve graph neural networks.
problem Graph Neural Networks struggle with over-smoothing and over-squashing.
method Augmented Forman-Ricci curvature (AFRC) for scalable rewiring.
result AFRC effectively mitigates over-smoothing and over-squashing.
The paper proves smoothness of mean curvature flow for generic initial data in 3D and 4D.
problem Smoothness of mean curvature flow for generic initial data.
method Long-time existence and uniqueness result for ancient mean curvature flows.
result Smooth mean curvature flow until disappearance in a round point for low-entropy hypersurfaces in 4D.
Study on special Lagrangian curvature potential equation, proving existence and uniqueness of smooth solutions.
problem Second boundary value problem for special Lagrangian curvature potential equation.
method Method of continuity with a-priori estimate.
result Existence and uniqueness of smooth uniformly convex solutions.
We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space …
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.
Preserves scalar curvature bounds under weak convergence of 3-manifolds.
problem Preserving scalar curvature bounds under weak convergence of 3-manifolds.
method Comparison between μ-bubbles in M_k and M.
result Scalar curvature lower bounds are preserved under weak convergence.
New Einstein metrics found on manifolds with opposite curvature signs.
problem Finding Einstein metrics with opposite curvature signs on manifolds.
method Reviewing and extending previous work on high-dimensional smooth closed manifolds.
result Proved various related results, including new Einstein metrics.
The paper develops bounds and regularity for minimal boundaries in non-smooth spaces with Ricci curvature.
problem Minimal boundaries in non-smooth spaces with Ricci curvature.
method Intrinsic theory of Laplacian bounds, PDE principle, sharp Laplacian bounds on distance function, regularity theory for perimeter-minimizing boundaries.
result Sharp Laplacian bounds and regularity results for perimeter-minimizing boundaries.
The study extends curvature bounds to non-smooth spaces and proves stability of mean curvature.
problem Proving curvature bounds in non-smooth spaces.
method Extending results from smooth Riemannian manifolds to non-smooth RCD spaces.
result Stability of mean curvature bounds under uniform convergence.
Smooth convergence to an enveloping cylinder proved for mean curvature flow of complete graphical hypersurfaces.
problem Proving smooth convergence of mean curvature flow to an enveloping cylinder.
method Analyzing mean curvature flow of complete graphical hypersurfaces over domains Ωt, proving convergence under certain circumstances. result Smooth convergence of Mt−hen+1 to the enveloping cylinder under specific conditions. Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.
problem Approximating weak mean curvature flows with singularities using smooth flows.
method Combining Choi-Haslhofer-Hershkovits and Choi-Haslhofer-Hershkovits-White work on canonical neighbourhoods and barriers to flows with surgery.
result Smooth flows with surgery can approximate weak mean curvature flows with spherical and neck-pinch singularities.
New invariant for 4D hypersurfaces ensures smooth critical points.
problem Understanding smoothness of curvature energies on 4D hypersurfaces.
method Developed a new conformally invariant energy.
result Critical points of new energy are smooth.
Smooth curves from polygonal chains with vertex preservation and explicit curvature control.
problem Preserving vertices while smoothing polygonal chains to C∞ curves. method Directional mollification operator for polygonal chains.
result Smooth curves that intersect original vertices and maintain explicit curvature bounds.
Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.
problem Eigenvalue problems for Bi-drifted Laplacian on compact manifolds with boundary conditions.
method Obtained lower bounds using specific curvature conditions.
result Lower bounds for the first eigenvalue of Bi-drifted Laplacian.
Study smooth hypersurfaces with prescribed curvature in Minkowski space.
problem Existence of smooth spacelike hypersurfaces with prescribed curvature.
method Proving existence based on C2 estimates. result Existence of smooth spacelike hypersurfaces with prescribed curvature.
Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
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.
We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of discrete nets on the smooth surface one can obtain second-order approximation g…