Proves uniqueness of Ricci flow with scaling invariant estimates.
problem Proving uniqueness of Ricci flow with scaling invariant curvature bound.
method Solving Ricci-harmonic map heat flow in unbounded curvature background.
result Complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique in dimension three.
Critical points of scale-invariant curvature energies in 4D are analytic.
problem Analyzing critical points of curvature energies in 4D manifolds.
method Applying Noether's theorem to identify conservation laws and lower order elliptic system of PDEs, then using integrability by compensation and interpolation theory.
result Critical points of scale-invariant curvature energies in 4D are analytic.
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, …
Study on curvature in finitely generated groups, showing positive curvature in specific cases.
problem Understanding curvature in finitely generated groups.
method Analyzing dead-end elements and related elements to find curvature, studying effect of radius.
result Examples of positive curvature for arbitrary radius in lamplighter and Houghton's group.
This paper tackles denoising of complex measures using optimal transport and curvature analysis.
problem Denoising of complex, possibly non-log-concave measures.
method Score function and optimal transport theory to revert Langevin diffusion chains.
result The difficulty of denoising depends on the curvature complexity of the initial measure at specific SNR scales.
Proves conditions for Willmore surfaces to have finite ends or finite total curvature.
problem Conditions for Willmore surfaces to have finite ends or finite total curvature.
method Analyzes scale-invariant second fundamental form near infinity.
result Proves conditions for Willmore surfaces to have finite ends or finite total curvature.
The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
problem Finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
method Discrete uniformization theorem, combinatorial α-Yamabe flow, combinatorial α-Calabi flow, edge flipping surgery.
result Longtime existence and convergence of combinatorial α-Yamabe flow and combinatorial α-Calabi flow with surgery.
Proves long-time Ricci flow existence and topological rigidity for pinched integral curvature manifolds.
problem Proving long-time existence and topological rigidity for manifolds with pinched scale-invariant integral curvature.
method Proves long-time existence of Ricci flow for manifolds with bounded curvature and pinched scale-invariant integral curvature, converging to a flat metric.
result Flow converges to a flat metric, implying topological rigidity of the manifold.
Proves planarity and convexity for ancient solutions of mean curvature flow.
problem Ancient solutions of mean curvature flow in higher codimension.
method Parabolically scale-invariant variation of planarity estimate, convexity proof for pinched solutions.
result Characterizes certain pinched complete ancient solutions and shrinkers in higher codimension.
Study shows one-dimensional location-scale-shape models are flat in Wasserstein geometry.
problem Investigating curvature in location-scale-shape models under Wasserstein metric.
method Introduced location-scale-shape model and investigated its geometry.
result Location-scale-shape model is intrinsically flat but extrinsically curved in Wasserstein geometry.
Modeling curvature-sensitive cells in visual cortex using manifold geometry.
problem Understanding how curvature influences cell function in the visual cortex.
method Developed a 4D manifold with canonical Engel structure to represent orientation, position, curvature, and scale.
result Characterized curvature-sensitive receptive profiles using left-invariant generators of the Engel structure.
Study shows how near crushing singularities, Kasner-like regions can exist.
problem Understanding spatial volume densities near crushing singularities.
method Relates existence of Kasner-like regions to asymptotics of spatial volume densities under scale-invariant curvature bounds.
result Kasner-like regions can exist near crushing singularities under certain curvature conditions.
We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Thus we extend well known notions of discrete pseudospherical surfaces and smooth pseudosperical surfaces on more exotic domains (e.g, the Cantor set). In particular, we present a new expression for the discrete Gauss…
Study on scalar curvature bounds and manifold topological complexity.
problem Understanding the topological complexity of manifolds with scalar curvature constraints.
method Introduced a small scale index theorem to establish bounds for Gromov's simplicial norm.
result Upper bound for Gromov's simplicial norm established in terms of scalar curvature, volume, and injectivity radius.
We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps from the spaces assigned to finer scales to those assigned to coarser scales play …
Study on curve diffusion flows with scale-critical curvature term.
problem Analyzing stability of curve diffusion flows with scale-critical curvature.
method Introduced and studied a one-parameter family of curve diffusion flows with a scale-critical cubic curvature term. Analyzed dynamical stability of homothetic circles using variational methods.
result Established that any small perturbation of an ω-fold circle monotonically approaches the unit ω-circle after rescaling, translation, and reparametrisation. We consider inverse curvature flows in hyperbolic space with starshaped initial hypersurface, driven by positive powers of a homogeneous curvature function. The solutions exist for all time and, after rescaling, converge to a sphere.
The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.
problem Prescribing discrete Gaussian curvature on polyhedral surfaces.
method Discrete conformal theory and variational principles with constraints.
result Proves Kazdan-Warner type theorems for polyhedral surfaces.
Study on heat content for domains with fractal boundaries.
problem Analyzing short-time asymptotics of heat content for domains with fractal boundaries.
method Developing mathematical analysis on de Gennes' hypothesis and exploring fractal curvatures.
result Fractal curvatures and their scaling exponents may emerge in the short-time heat content asymptotics of domains with fractal boundaries.
We prove a relation between the scaling hβ of the elastic energies of shrinking non-Euclidean bodies Sh of thickness h→0, and the curvature along their mid-surface S. This extends and generalizes similar results for plates [BLS16, LRR] to any dimension and co-dimension. In particular, it proves that the na…
Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.
problem Integral of scalar curvature on manifolds with a pole.
method Asymptotic scaling invariant integral of scalar curvature equals a term determined by asymptotic volume ratio.
result Integral of scalar curvature equals a volume ratio term.
We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a…
Study on ancient Ricci flows with positive curvature, proving noncollapsedness.
problem Characterizing ancient Ricci flows with positive sectional curvature.
method Analyzing complete and noncompact Type I ancient Ricci flows with positive sectional curvature.
result Ancient solutions are noncollapsed on all scales in complete and noncompact cases, and in even-dimensional closed cases.
This paper argues that a class of Riemannian metrics, called warped metrics, plays a fundamental role in statistical problems involving location-scale models. The paper reports three new results : i) the Rao-Fisher metric of any location-scale model is a warped metric, provided that this model satisfies a natural invar…
Using the monotonicity formulas of Colding and Minicozzi, we prove that on any complete, non-parabolic Riemannian manifold (M3,g) with non-negative Ricci curvature, the asymptotic weighted scaling invariant integral of scalar curvature has an explicit bound in form of asymptotic volume ratio.
Ricci flow controls curvature on manifolds with bounds.
problem Controlling curvature on manifolds with given bounds.
method Ricci flow with curvature bounds and entropy controls.
result Global curvature control at positive times for manifolds.
In this paper we consider closed non-collapsed ancient solutions to the mean curvature flow (n≥2) which are uniformly two-convex. We prove that any two such ancient solutions are the same up to translations and scaling. In particular, they must coincide up to translations and scaling with the rotationally symmetr…
We analyze the Hessian spectra of large models up to 100B parameters.
problem Accurate Hessian spectra of large foundation models are difficult to obtain.
method We use shard-local finite-difference Hessian vector products and stochastic Lanczos quadrature.
result We produce the first large-scale spectral density estimates of foundation models.
In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in Rn+1 starting from any n-dimensional (ε,R)-Reifenberg flat set with ε sufficiently small. More precisely, we show that the level set flow in such a situation is non-fattening and …
We define a hybrid between Ollvier and Bakry Emery curvature on graphs with dependence on a variable neighborhood. The hexagonal lattice is non-negatively curved under this new curvature notion. Bonnet-Myers diameter bounds and Lichnerowicz eigenvalue estimates follow from the standard arguments. We prove gradient esti…
Three-manifolds with non-negative pinched Ricci curvature have complete Ricci flows.
problem Proving Hamilton's pinching conjecture for three-manifolds.
method Ricci flow with scale-invariant curvature decay and pinching preservation.
result Hamilton's pinching conjecture is proven without additional hypotheses.
A new geometric method approximates slow invariant manifolds without explicit time-scale separation.
problem Approximating slow invariant manifolds in systems with multiple time-scales.
method Geodesic Stretching and Flow Curvature methods translated into tensorial constructions of Riemannian geometry.
result The method approximates normally attracting invariant manifolds without requiring explicit time-scale separation.
The paper proves inequalities for closed surfaces involving mean curvature.
problem Proving geometric inequalities for closed surfaces in Euclidean space.
method Verification of inequalities for convex surfaces and addressing Topping's conjecture.
result Optimal scaling law between Willmore energy and isoperimetric ratio for convex surfaces.
We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Two special cases, namely dicrete pseudospherical surfaces and smooth pseudosperical surfaces are consistent with this description. In particular, we define the Gaussian curvature in the discrete case.
We introduce a scalable measure of curvature for analyzing training dynamics of large language models.
problem Analyzing the training dynamics of large language models due to high computational cost of measuring Hessian sharpness.
method We introduce critical sharpness and relative critical sharpness as computationally efficient measures capturing Hessian sharpness phenomena.
result We provide the first demonstration of sharpness phenomena at scale up to 7B parameters.
This note revisits the inverse mean curvature flow in the 3-dimensional hyperbolic space. In particular, we show that the limiting shape is not necessarily round after scaling, thus resolving an inconsistency in the literature.
In this paper, we introduce a new combinatorial curvature on two and three dimensional triangulated manifolds, which transforms in the same way as that of the smooth scalar curvature under scaling of the metric and could be used to approximate the Gauss curvature on two dimensional manifolds. Then we use the flow metho…
For collapsing sequences of Riemannian manifolds which satisfy a uniform lower Ricci curvature bound it is shown that there is a sequence of scales such that for a set of good base points of large measure the pointed rescaled manifolds subconverge to a product of a Euclidean and a compact space. All Euclidean factors h…
Sharp estimates for p-capacity on manifolds with Ricci curvature bounds.
problem Estimating p-capacity on manifolds with Ricci curvature constraints.
method Sharp comparison inequalities, warped-product model ends, and scale-invariant quantities.
result Characterization of equality cases and optimal ranges for normalization parameters.
Existence proved for Ricci curvature on sphere product.
problem Existence of metrics with prescribed Ricci curvature on product spheres.
method Proved existence for certain doubly warped product metrics.
result Existence of metrics with prescribed Ricci curvature on Sd1+1imesSd2. New method improves curvature estimates for stable surfaces.
problem Curvature estimates for stable surfaces in Rn+1. method Replacing Young's inequality with Hölder's inequality simplifies and improves curvature estimates.
result The new method yields a strictly smaller constant and a natural extension to CMC settings.
Study of splitting maps in Type I Ricci flows for understanding singular set structure.
problem Understanding the structure of the singular set in non-collapsed Ricci limit spaces.
method Construction and investigation of almost splitting maps on Ricci flows that are almost self-similar.
result Sharp splitting maps remain splitting maps at smaller scales under certain conditions.
In this paper we prove that an embedded and simply connected constant mean curvature surface with curvature large at a point contains a multi-valued graph around that point on the scale of ∣A∣2, where ∣A∣2 is the norm squared of the second fundamental form. This generalizes Colding and Minicozzi's result for mini…
Generalizes rigidity of scalar curvature for convex domains.
problem Rigidity of scalar curvature for convex domains.
method Harmonic spinors on convex domains with boundary conditions constructed by Brendle.
result Rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain.
In the study manifolds of Ricci curvature bounded below, a stumbling obstruction is the lack of links between large-scale geometry and small-scale geometry at a fixed reference point. There have been few links (volume, dimension) when the unit ball at the point is not collapsed, that is, vol(B1(p))≥v>0. …
The paper reduces normal curvature and enhances homology recovery via embedded submanifolds.
problem Recovering the homology of submanifolds with narrow cycles.
method Embedding submanifolds into scaled oriented Grassmannian bundles to reduce normal curvature and stabilize Čech persistent homology.
result The Čech persistent homology is stable with respect to the interleaving distance and provides lower bounds on scales for homology recovery.
The paper extends Hawking--Page solutions to various spacetimes with singularities.
problem Understanding the extensions of Hawking--Page solutions with different types of singularities.
method Kaluza--Klein reduction and Christodoulou's methods.
result Extensions of Lorentzian Hawking--Page solutions with null, spacelike singularities, and Cauchy horizons of Taub--NUT type are proven.
No breathers found for mean curvature flow in noncompact space.
problem Existence of breathers in mean curvature flow.
method Proved no breather theorems for mean curvature flow in noncompact Euclidean space.
result Breathers must be solitonic solutions under certain conditions.