No theta-shaped networks shrink similarly under curvature.
problem Existence of theta-shaped self-similarly shrinking networks.
method Proof by contradiction and curvature motion analysis.
result No theta-shaped networks with specified angles exist.
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arXiv research
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169,341 papers · 148 categories
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48 results for “shrinking networks”
We consider convex symmetric lens-shaped networks in R^2 that evolve under curve shortening flow. We show that the enclosed convex domain shrinks to a point in finite time. Furthermore, after appropriate rescaling the evolving networks converge to a self-similarly shrinking network, which we prove to be unique in an ap…
Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutionsmath.AP
Study hexagonal network evolution under curvature flow.
problem Understanding hexagonal network evolution under curvature flow.
method Proved local existence of classical solutions and classified homothetically shrinking solutions.
result Provided an example of network shrinking to a segment with multiplicity two.
MorphNet automates neural network structure design for resource constraints.
problem Designing efficient neural network structures for resource-limited devices.
method Iteratively shrinks and expands networks using resource-weighted sparsifying regularizers and uniform multiplicative factors.
result Discover novel structures that improve performance while respecting resource constraints.
Hidden cost: Smoothing shrinks decision boundaries, affecting class-wise accuracy.
problem The fragility of machine learning models and the need for robustness verification.
method Randomized smoothing approach to achieve statistical robustness.
result Smoothed classifiers' decision boundaries shrink, leading to class-wise accuracy disparity.
Symmetries in shrinking Ricci solitons spread outward.
problem Understanding symmetries in shrinking Ricci solitons.
method Propagating approximate symmetries to larger scales.
result Symmetries in shrinking Ricci solitons spread outward.
Perelman proves compact and complete noncompact shrinking breather properties.
problem Proving properties of shrinking breathers in both compact and noncompact settings.
method Using Perelman's entropy formula and L-geometry, constructing ancient solutions.
result Complete noncompact case proven without technical assumptions.
Study proves shrinking doughnuts via modified curvature flow.
problem Existence of self-shrinking tori under mean curvature flow.
method Variational methods and modified curvature flow.
result Established an upper bound for the weighted energy of shrinking doughnuts.
In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton or the round cyl…
The study shows no non-constant positive -harmonic functions on complete gradient shrinking Ricci solitons.
problem Existence of non-constant positive -harmonic functions on gradient shrinking Ricci solitons.
method Proved the non-existence of non-constant positive -harmonic functions using Liouville-type theorems.
result No non-constant positive -harmonic functions on complete gradient shrinking Ricci solitons.
Study on shrinking solitons of generalized Ricci flow.
problem Characterizing shrinking solitons in generalized Ricci flow.
method Analyzing gradient shrinking solitons and pluriclosed solitons on compact manifolds.
result First non-trivial shrinking generalized soliton constructed.
R2D2-Net improves Bayesian neural networks by preventing over-shrinkage of important weights.
problem Bayesian neural networks struggle with choosing appropriate priors, leading to over-shrinkage or poor predictive performance.
method Proposes R2D2-Net with an R^2-induced Dirichlet Decomposition prior and variational Gibbs inference algorithm.
result R2D2-Net effectively shrinks irrelevant coefficients while preventing key features from over-shrinkage.
Sharp Li-Yau equality proven for shrinking Ricci solitons without curvature assumptions.
problem Classifying shrinking Ricci solitons without curvature or volume restrictions.
method Proving the sharp Li-Yau equality for conjugate heat kernel on shrinking Ricci solitons.
result Several estimates and classification of four-dimensional, non-compact shrinking Ricci solitons.
Two new proofs show Ricci flow breathers are special solutions.
problem Characterize solutions to Ricci flow on closed manifolds.
method Use singularity models and ancient solutions to show they are gradient Ricci solitons.
result Ricci flow breathers are gradient Ricci solitons.
Paper classifies 3D breathers and generalizes Ricci soliton results.
problem Classifying and understanding Ricci solitons and breathers.
method Developed a condition for the existence of asymptotic shrinking gradient Ricci solitons.
result Every complete shrinking Ricci soliton with bounded Ricci curvature is gradient.
5D shrinking solitons with bounded curvature are rigid.
problem Characterizing 5D shrinking gradient Ricci solitons.
method Proving rigidity for solitons with bounded curvature.
result 5D shrinking gradient Ricci solitons with bounded curvature are rigid.
Proves shrinking Ricci solitons near Kähler cones are Kähler.
problem Characterizing shrinking gradient Ricci solitons near Kähler cones.
method Analyzes asymptotic behavior and proves Kähler property.
result Proves shrinking gradient Ricci solitons are Kähler near Kähler cones.
Paper proves rigidity for Ricci solitons with specific conditions.
problem Understanding the properties of Ricci solitons under various conditions.
method Analyzes shrinking and expanding Ricci solitons with specific constraints.
result Compact shrinking Ricci solitons are Einstein if the potential function is controlled.
Compact shrinking solitons with positive curvature are proven to be compact.
problem Characterizing shrinking gradient Kähler-Ricci solitons with specific curvature properties.
method Analyzing properties of complete shrinking gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature.
result Compact shrinking solitons with positive curvature are proven to be compact.
New proof of shrinking gradient Ricci soliton rigidity.
problem Rigidity of shrinking gradient Ricci solitons.
method Maximum principle, maximum curvature condition.
result Shrinking gradient Ricci soliton with constant scalar curvature is isometric to a finite quotient of R^2 x S^2.
We prove the following: Let (M,g,X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M,g) is isometric to R^4 or a finite quotient of S^2xR^2 or S^3xR. In the process we also show that a complete shrinking soliton (M,g,X) with bounded curvature is gradient and k-nonc…
Compact shrinking Kähler-Ricci solitons with positive curvature are proven to be finite.
problem Characterizing shrinking Kähler-Ricci solitons with positive bisectional curvature.
method Alternative proof using Munteanu and Wang's argument.
result Compactness of shrinking gradient Kähler-Ricci solitons with positive bisectional curvature.
Study proves rigidity for solitons with uncertainty principle.
problem Rigidity of shrinking Ricci solitons with uncertainty principle.
method Proves rigidity theorems for shrinking gradient Ricci solitons.
result Proves rigidity theorems with sharp constant in R^n.
Complete shrinking soliton found on a specific complex surface.
problem Classifying complete shrinking gradient Kähler-Ricci solitons in two complex dimensions.
method Proved existence of a unique soliton with bounded scalar curvature on a specific blowup.
result Complete classification of such solitons in two complex dimensions.
Study Bernstein results for self-shrinking solutions in Lagrangian flow.
problem Understanding entire self-shrinking solutions in Lagrangian flow.
method Prior estimates and barriers construction.
result Showed Bernstein type results for self-shrinking solutions.
New findings on shrinking solitons with positive isotropic curvature.
problem Characterizing shrinking solitons with positive isotropic curvature.
method Analyzing properties of gradient shrinking solitons in dimensions 5 and above.
result Non-flat complete shrinking solitons with positive isotropic curvature are quotients of the round sphere or the cylinder.
Rigidity proven for specific gradient Ricci solitons.
problem Characterizing gradient shrinking Ricci solitons with certain tensor properties.
method Proving rigidity through fourth order divergence-free tensors and Weyl tensor conditions.
result Gradient shrinking Ricci solitons with specific tensor properties are rigid.
The study proves compactness and existence of entropy minimizers for self-shrinking surfaces.
problem Understanding entropy in higher-codimension mean curvature flow.
method Measure-theoretical techniques and rigidity results for self-shrinkers.
result Existence of entropy minimizers and improved rigidity results.
Isometries of cones embed in those of asymptotic shrinking Ricci solitons.
problem Characterizing isometries of asymptotically conical shrinking Ricci solitons.
method Analyzing the embedding of isometries from the cone's cross-section into the entire shrinker.
result Isometries of the cone's cross-section embed in the entire shrinker.
In this paper, we will give a local version of the Hamilton-Ivey type pinching estimate of the gradient shrinking soliton with vanishing Weyl tensor, and then give a complete classification on gradient shrinking solitons with vanishing Weyl tensor.
The paper proves various inequalities on gradient shrinking Ricci solitons.
problem Understanding geometric inequalities on gradient shrinking Ricci solitons.
method Proving multiple inequalities equivalent on complete gradient shrinking Ricci solitons.
result Various inequalities (Sobolev, logarithmic Sobolev, Schrödinger, etc.) are equivalent on gradient shrinking Ricci solitons.
The paper proves rigidity for self-shrinking spacelike graphs in pseudo-Euclidean space.
problem Understanding the rigidity of self-shrinking spacelike graphs in pseudo-Euclidean space.
method Volume growth estimate and Co-Area formula.
result Various rigidity results for spacelike entire self-shrinking graphs.
A new method shrinks a complex structure without much change.
problem Understanding the geometry of Bing's wild involution.
method Producing a counterintuitive construction to shrink the Bing decomposition without much change.
result A method to shrink a complex structure (Bing's decomposition) without much change.
Let be a Gromov-Hausdorff limit of closed shrinking Ricci solitons with uniformly upper bounded diameter and lower bounded volume. We prove that off a closed subset of codimension at least 2, is a smooth manifold satisfying a shrinking Ricci soliton equation.
Sharp upper diameter limit found for Ricci solitons.
problem Bounding the diameter of compact shrinking Ricci solitons.
method Used a sharp logarithmic Sobolev inequality and Vitali-type covering argument.
result Sharp upper diameter bound established in terms of scalar curvature and entropy.
The paper studies geometric properties of self-shrinkers in shrinking Ricci solitons.
problem Understanding geometric properties of self-shrinkers in specific geometric settings.
method Proved spectral properties of drifted Laplacian and used them to derive geometric properties.
result Described domains in the ambient space that cannot contain self-shrinkers.
Sharp Gaussian bounds derived for Schrödinger kernel on Ricci solitons.
problem Analyzing Schrödinger heat kernel on gradient shrinking Ricci solitons.
method Deriving sharp Gaussian upper bounds for the Schrödinger heat kernel.
result Sharp upper and lower bounds for eigenvalues of the Schrödinger operator.
Rigidity theorem for convex solutions to Hessian quotient flows.
problem Proving rigidity of convex solutions to Hessian quotient flows.
method Analyzing entire smooth strictly convex self-shrinking solutions on .
result All entire smooth strictly convex self-shrinking solutions to Hessian quotient flows are quadratic.
The study proves rotationally symmetric property of certain shrinking gradient Yamabe solitons.
problem Understanding the rotational symmetry of specific shrinking gradient Yamabe solitons.
method Analyzing nontrivial complete shrinking gradient Yamabe solitons with bounded scalar curvature.
result The assumption of bounded scalar curvature and strict inequality at some point is necessary and sufficient for rotational symmetry.
The paper proves manifold isometries for certain gradient Ricci solitons.
problem Characterizing isometry of gradient shrinking Ricci solitons.
method Analyzing volume growth, scalar curvature, and potential function subharmonicity.
result Gradient shrinking Ricci solitons with specific properties are isometric to spheres or other specific manifolds.
The study examines constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
problem Characterizing constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
method Analyzing properties of hypersurfaces in specific ambient spaces (shrinking Ricci solitons).
result Conditions for a constant weighted mean curvature hypersurface to be a level set of the potential function.
Simply-connected shrinking Kähler-Ricci solitons are proven.
problem Simply-connectedness of shrinking Kähler-Ricci solitons.
method Using results by Sun-Zhang and Wylie.
result Shrinking Kähler-Ricci solitons are simply-connected.
The study examines four-dimensional gradient Ricci solitons and their properties.
problem Characterizing four-dimensional complete gradient shrinking Ricci solitons.
method Proving properties and providing curvature estimates for solitons under specific conditions.
result Conditions for four-dimensional complete gradient shrinking Ricci solitons.
Paper proves genus of surfaces decreases in mean curvature flow.
problem Understanding the evolution of surfaces under mean curvature flow.
method Analyzes mean curvature flow of compact surfaces in 3-manifolds with specific curvature conditions.
result Genus of the regular set decreases over time.
The study finds only spheres shrink self-similarly with quotient curvature speeds.
problem Characterizing self-similar shrinkers for quotient curvature speeds.
method Examined closed hypersurfaces shrinking with quotient curvature speeds.
result Only shrinking spheres are self-similar solutions.
Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Fourmath.DG
Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.
problem Finding topological obstructions to compact gradient shrinking Ricci solitons in dimension four.
method Discussion of background material, introduction of new problem, exploration of limitations of current results.
result Introduction of new problem and limitations of current results in extending Hitchin-Thorpe inequality.
It is shown that the diameter of a compact shrinking Ricci soliton has a universal lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with lower Ricci curvature bound to a twisted Laplacian on compact shrinking Ricci solitons.
Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvaturemath.DG
5D shrinking Ricci solitons with constant scalar curvature are rigid.
problem Characterizing 5D shrinking gradient Ricci solitons with constant scalar curvature.
method Proving rigidity by showing they are finite quotients of a known space.
result 5D shrinking gradient Ricci solitons with constant scalar curvature are rigid.