arXiv research

BS-NAS broadens and shrinks search space for optimal neural architectures.

Paper shows perturbed Taub-Bolt metric becomes singularity under Ricci flow.

We shrink confidence sets for equivalent discrete distributions using permutation equivalence.

Following work of Colding-Minicozzi, we define a notion of entropy for connections over $\mathbb R^n$ which has shrinking Yang-Mills solitons as critical points. As in Colding-Minicozzi, this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stabilit…

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 $R^4$ or the round cyl…

Study on shrinking solitons of generalized Ricci flow.

Sharp Li-Yau equality proven for shrinking Ricci solitons without curvature assumptions.

Paper classifies 3D breathers and generalizes Ricci soliton results.

5D shrinking solitons with bounded curvature are rigid.

Paper proves rigidity for Ricci solitons with specific conditions.

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…

New proof of shrinking gradient Ricci soliton rigidity.

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…

Complete shrinking soliton found on a specific complex surface.

The study proves compactness and existence of entropy minimizers for self-shrinking surfaces.

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.

The paper proves rigidity for self-shrinking spacelike graphs in pseudo-Euclidean space.

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.

A new method shrinks a complex structure without much change.

Sharp upper diameter limit found for Ricci solitons.

The study proves rotationally symmetric property of certain shrinking gradient Yamabe solitons.

Sharp Gaussian bounds derived for Schrödinger kernel on Ricci solitons.

The paper proves manifold isometries for certain gradient Ricci solitons.

R2D2-Net improves Bayesian neural networks by preventing over-shrinkage of important weights.

The study examines constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.

We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $\mathbb{R}^n$. In addtion, we partially give analogous rigidity results of the Caffarelli-Kohn-Nirenberg inequalities on shrinking Ricci solitons.

Simply-connected shrinking Kähler-Ricci solitons are proven.

We prove that there does not exist non-constant positive $f$-harmonic function on the complete gradient shrinking Ricci solitons. We also prove the $L^{p}(p\geq 1 \ {\rm or}\ 0<p\leq 1)$ 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.

Paper proves genus of surfaces decreases in mean curvature flow.

Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.

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.

5D shrinking Ricci solitons with constant scalar curvature are rigid.

Estimates heat equation on shrinking Ricci solitons with uniform bounds.

We prove that all entire smooth strictly convex self-shrinking solutions on $\mathbb{R}^n$ 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 $\mathcal{F}$-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.

The study shows ends of shrinking gradient $ρ$-Einstein solitons are non-parabolic.

Paper proves properties of minimal hypersurfaces in specific solitons.

Uniqueness proven for specific types of geometric structures.

In this paper we prove that any asymptotically cylindrical gradient shrinking Ricci soliton is isometric to a cylinder.

In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.