Study curve shortening flow in high dimensions with boundary constraints.
problem Understanding the behavior of curves in high-dimensional spaces with boundary conditions.
method Used curvature and higher-derivative estimates, Stahl-type maximum principle, and blow-up analysis.
result Flow converges to a shrinking semicircle model or has only semicircle boundary singularities in low entropy regimes.
The paper studies neural networks with wide layers and finds a deformed semicircle law.
problem Investigating spectral distributions of neural networks in the ultra-wide regime.
method Analyzes empirical kernel matrices, proves deformed semicircle law, provides nonlinear Hanson-Wright inequality.
result Emergence of a deformed semicircle law in the ultra-wide neural network regime.
A curve around a sphere must be at least 4π long.
problem Finding the shortest closed curve that encloses a sphere.
method Analyzing curves in Euclidean 3-space and comparing their lengths.
result The shortest curve is composed of 4 semicircles arranged like a baseball seam.
Study of correlated Wigner matrices with BBP transitions.
problem Understanding spectral transitions in correlated Wigner matrices.
method Analyzes a Wigner-type matrix with row/column correlations, decomposes into bulk and outliers, and uses integral operators to model transitions.
result Correlated Wigner matrices exhibit multiple BBP transitions at critical points.
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.
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.
We give two new proofs of Perelman's theorem that shrinking breathers of Ricci flow on closed manifolds are gradient Ricci solitons, using the fact that the singularity models of type I solutions are shrinking gradient Ricci solitons and the fact that non-collapsed type I ancient solutions have rescaled limits being sh…
Recent years have seen a growing interest in understanding deep neural networks from an optimization perspective. It is understood now that converging to low-cost local minima is sufficient for such models to become effective in practice. However, in this work, we propose a new hypothesis based on recent theoretical fi…
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.
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 R4 or the round cyl…
BS-NAS broadens and shrinks search space for optimal neural architectures.
problem Suboptimal channel numbers and model averaging effects in One-Shot NAS methods.
method Broadening with spring block for channel search, shrinking with underperforming operations removal, evolutionary algorithm for optimal architecture search.
result BS-NAS achieves state-of-the-art performance on ImageNet.
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.
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.
The main result in this paper is that the space of all smooth links in Euclidean 3-space isotopic to the trivial link of n components has the same homotopy type as its finite-dimensional subspace consisting of configurations of n unlinked Euclidean circles (the "rings" in the title). There is also an analogous result f…
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…
We first show that a Kähler cone appears as the tangent cone of a complete expanding gradient Kähler-Ricci soliton with quadratic curvature decay with derivatives if and only if it has a smooth canonical model (on which the soliton lives). This allows us to classify two-dimensional complete expanding gradient Kähler-Ri…
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.
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 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.
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.
We use variational methods and a modified curvature flow to give an alternative proof of the existence of a self-shrinking torus under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.
Let (Y,d) 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, Y 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.
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.
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.
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.
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.
We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in Rn. In addtion, we partially give analogous rigidity results of the Caffarelli-Kohn-Nirenberg inequalities on shrinking Ricci solitons.
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.
We prove that there does not exist non-constant positive f-harmonic function on the complete gradient shrinking Ricci solitons. We also prove the Lp(p≥1 or 0<p≤1) Liouville theorems on the complete gradient shrinking Ricci solitons.
We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the 4-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton $\…
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.
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.
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.
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.
Estimates heat equation on shrinking Ricci solitons with uniform bounds.
problem Analyzing heat equation on shrinking Ricci solitons.
method Proved L2 estimate with time-dependent Gaussian weight. result Uniform bounds for heat equation along Ricci flow.
We prove that all entire smooth strictly convex self-shrinking solutions on Rn to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean curvature flow in pseudo-Euclidean space due to Ding-Xin \cite{DX}. Moreover, we sh…
As of today, there are very few known complete shrinking Ricci solitons in dimension 4, and all examples discovered so far are Kähler and/or Einstein. In this note, we prove that any four dimensional J-invariant gradient shrinking Ricci solitons satisfy a differential form identity relating Kählerity annd Einstein-ness…
In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional…
In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of F-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabil…
Alternative proof for 4D shrinking Ricci solitons with constant scalar curvature.
problem Proving the structure of four-dimensional shrinking gradient Ricci solitons with constant scalar curvature.
method Analyzing the asymptotic geometry at infinity.
result Alternative proof that such solitons are finite quotients of R^2 x S^2.
The study shows ends of shrinking gradient ρ-Einstein solitons are non-parabolic.
problem Characterizing the ends of shrinking gradient ρ-Einstein solitons. method Proving non-parabolicity of ends and connectivity at infinity for specific conditions.
result Gradient shrinking ρ-Einstein solitons have non-parabolic ends under certain conditions. Paper proves properties of minimal hypersurfaces in specific solitons.
problem Characterizing minimal hypersurfaces in shrinking gradient Ricci solitons.
method Analyzes stable minimal hypersurfaces with specific curvature conditions.
result Minimal hypersurfaces in these solitons have zero second fundamental form and normal Ricci curvature.
Paper shows perturbed Taub-Bolt metric becomes singularity under Ricci flow.
problem Analyzing stability of Taub-Bolt metric under Ricci flow.
method Box argument and construction of Ricci flows on compact manifolds.
result Compact perturbation of Taub-Bolt metric evolves into finite time singularity.