Constructs flow lines connecting unstable to stable self-expanders.
problem Existence of monotone Morse flow lines for expander functionals.
method Constructs a singular Morse flow line connecting unstable to stable self-expanders.
result Constructs a monotone flow line with a small singular set.
New expanders found using origami surfaces with spectral gap.
problem Constructing expanders with spectral gap on surfaces of arbitrary genus.
method Affine actions on origami surfaces to achieve spectral gap.
result New expanders distinct from classical ones.
The paper extends rigidity results for λ-self-expanders to hyperplanes, spheres, and cylinders.
problem Characterizing λ-self-expanders as hyperplanes, spheres, and cylinders. method Extending results on self-expanders to λ-self-expanders, proving rigidity results. result Characterizes hyperplanes, spheres, and cylinders as λ-self-expanders. New self-expander found between two given asymptotic ones.
problem Finding new self-expanders between given asymptotic ones.
method Developed a min-max theory for asymptotically conical self-expanders of mean curvature flow.
result Existence of a new asymptotically conical self-expander trapped between two given ones.
No expanding breathers found in noncompact Ricci flows with certain curvature conditions.
problem Finding expanding breathers in noncompact Ricci flows.
method Curvature positivity conditions (weak PIC-2 or nonnegative bisectional curvature).
result Complete noncompact expanding breathers are gradient solitons.
New expanders for mean curvature flow contradict genus-reduction conjecture.
problem Contradicting Ilmanen's genus-reduction conjecture for mean curvature flow.
method Construct new expanders asymptotic to cones arising from shrinkers.
result Existence of expanders of arbitrarily large genus.
New degree theory proves existence of solitons on 4D manifolds.
problem Existence of gradient expanding solitons on 4D manifolds.
method Developed new degree theory for 4D, asymptotically conical gradient expanding solitons.
result Existence of solitons asymptotic to any cone over S^3 with non-negative scalar curvature.
New expanding Ricci solitons found starting in dimension four.
problem Finding expanding Ricci solitons in specific dimensions.
method Constructing gradient expanding Ricci solitons asymptotic to cones and on trivial vector bundles.
result Continuous families of expanding Ricci solitons on products of Einstein manifolds.
Study on the spectrum of drift Laplacian on Ricci expanders.
problem Analyzing the spectrum of the drift Laplacian on Ricci expanders.
method Investigation of discrete spectrum under proper potential function, asymptotic behavior of potential function, and computation of eigenvalues.
result Discrete spectrum of the drift Laplacian on Ricci expanders with bounded Ricci curvature.
Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expand…
The paper classifies expanding gradient Yamabe solitons based on scalar curvature.
problem Classifying expanding gradient Yamabe solitons based on scalar curvature.
method Rigorous analysis of scalar curvature in both cases: greater than and less than the soliton constant.
result Complete classification of nontrivial complete expanding gradient Yamabe solitons.
Constructs self-expanders of positive genus for cones in R^3.
problem Creating self-expanders of positive genus for cones in R^3.
method Constructs self-expanders asymptotic to cones, uses mean curvature flow.
result Constructs self-expanders with unbounded genus asymptotic to a rotationally symmetric cone.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
Study finds unique self-expanders for mean curvature flow.
problem Finding solutions to mean curvature flow.
method Derived equation based on generalized Lawson-Osserman cone and modified equilibria theory.
result Existence and uniqueness of self-expanders proved.
We derive a sharp lower bound for the scalar curvature of non-flat and non-compact expanding gradient Ricci soliton provided that the scalar curvature is non-negative and the potential function is proper. We also give an upper bound for the scalar curvature of noncompact expander when the Ricci curvature is nonpositive…
Study cohomogeneity one expanding Ricci solitons on specific topologies.
problem Characterize and analyze cohomogeneity one expanding Ricci solitons.
method Analyze ODEs, define expander degree, calculate cohomogeneity one expander degree.
result Reconstruct and calculate cohomogeneity one expander degree for specific topologies.
We give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu's property A.
We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. …
The paper examines properties and rigidity of self-expanders in Euclidean space.
problem Characterizing and estimating properties of self-expanders in Euclidean space.
method Analyzing mean curvature flow, volume growths, and stability of self-expanders.
result Proves the uniqueness of certain self-expanders in 3D space.
The paper classifies 2D complete Lagrangian self-expanders in complex 2-space.
problem Classifying complete Lagrangian self-expanders in complex 2-space.
method Obtained a classification theorem.
result A classification of 2D complete Lagrangian self-expanders with constant squared norm of the second fundamental form.
Strict convexity proven for certain self-expanders in high dimensions.
problem Convexity of self-expanders in mean curvature flow.
method Investigation of convexity properties for asymptotically conical self-expanders.
result Strict convexity proven for n-dimensional self-expanders. We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all two-dimensional self-expanders of a fixed topological type and, in all dimensions, for the set of self-expa…
Study classifies 4D Ricci solitons with specific curvature conditions.
problem Classifying 4D gradient steady and expanding Ricci solitons with given curvature properties.
method Asymptotically cylindrical and conical assumptions; half-harmonic and half-nonnegative isotropic curvature conditions.
result Partial classification of 4D gradient expanding Ricci solitons with half-nonnegative isotropic curvature.
We show that the space of asymptotically conical self-expanders of the mean curvature flow is a smooth Banach manifold. An immediate consequence is that non-degenerate self-expanders -- that is, those self-expanders that admit no non-trivial normal Jacobi fields that fix the asymptotic cone -- are generic in a certain …
We define a way of approximating actions on measure spaces using finite graphs; we then show that in quite general settings these graphs form a family of expanders if and only if the action is expanding in measure. This provides a somewhat unified approach to construct expanders. We also show that the graphs we obtain …
In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to O(n)-invariant co…
Proves uniqueness of small entropy self-expanders.
problem Uniqueness of self-expanders with small entropy.
method Mountain-pass theorem and integer degree argument.
result Proves uniqueness result for self-expanders.
Study self-expanding solutions of mean curvature flow in various dimensions.
problem Characterize complete mean convex self-expanding hypersurfaces and their properties.
method Analyzing the function ∣A∣2/∣H∣2 and ∣Aξ∣2/∣H∣2 to understand the structure of self-expanders. result Complete mean convex self-expanders are products of self-expanding curves and flat subspaces under certain conditions.
We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that expanders with vanishing relative entropy are unique in a generic sense. This also impl…
3-manifolds with torsion homology expand in all dimensions.
problem Constructing 3-manifolds with good expansion properties.
method Constructing 3-manifolds with specific homological properties and demonstrating their expansion.
result 3-manifolds with torsion homology expand in all dimensions.
Unique continuation result for expanding Ricci solitons.
problem Unique continuation of expanding Ricci solitons.
method Optimal relative integral convergence rate, relative entropy.
result Well-defined relative entropy for expanding solitons.
Unique steady and expanding solitons with spherical links identified.
problem Characterizing steady and expanding Ricci solitons with specific asymptotic symmetries.
method Symmetry principle applied to asymptotically cylindrical and conical GRSs, proving uniqueness for Bryant solitons.
result Bryant steady and expanding solitons are the unique asymptotically cylindrical and conical GRSs with spherical links under certain conditions.
The paper classifies 3D self-expanders with specific properties.
problem Classifying self-expanders with constant curvature components.
method Complete classification of 3D self-expanders with specific curvature conditions.
result Completely classified 3D self-expanders with constant curvature components.
In this paper, we study self-expanders for mean curvature flows. First we show the discreteness of the spectrum of the drifted Laplacian on them. Next we give a universal lower bound of the bottom of the spectrum of the drifted Laplacian and prove that this lower bound is achieved if and only if the self-expander is th…
The paper proves convexity of certain solitons and expanders in high dimensions.
problem Proving convexity of specific solitons and expanders in Rn+1. method Inspired by Spruck-Xiao and Derdziński, the paper uses geometric analysis to prove convexity.
result The paper proves the convexity of complete 2-convex translating and expanding solitons and expanders in Rn+1 for n≥3. We first investigate the asymptotics of conical expanding gradient Ricci solitons by proving sharp decay rates to the asymptotic cone both in the generic and the asymptotically Ricci flat case. We then establish a compactness theorem concerning nonnegatively curved expanding gradient Ricci solitons.
The study classifies translating and self-expanding solitons in 3D space.
problem Characterizing the topology and index of solitons in mean curvature flow.
method Analyzing the spectrum and index of expanding and translating solitons in R3. result Translating and self-expanding solitons have finite topology under certain conditions.
New example of non-Kähler soliton with Kähler-like behavior at infinity.
problem Constructing non-Kähler expanding gradient Ricci solitons.
method Asymptotically conical (AC) construction with Kähler tangent cone at infinity.
result Example of a non-Kähler soliton with a Kähler-like behavior at infinity.
Constructs expanding gradient Ricci solitons with unique properties.
problem Creating expanding gradient Ricci solitons with specific characteristics.
method Combining previous work with localized maximum principle.
result Constructs various examples of expanding gradient Ricci solitons with positive curvature and exotic curvature decay.
The paper pinches curvature in expanding Ricci solitons.
problem Curvature pinching in expanding Ricci solitons.
method Hamilton-Ivey type curvature pinching estimates.
result Three-dimensional Hamilton-Ivey type curvature pinching theorem.
We establish Carleman inequalities for the weighted laplacian associated to an expanding gradient Ricci soliton. As a consequence, a unique continuation at infinity is proved for asymptotically Ricci flat Ricci expanders. The obstruction at infinity is a symmetric 2-tensor defined on the link of the corresponding asymp…
We investigate the possibility of desingularizing a positively curved metric cone by an expanding gradient Ricci soliton with positive curvature operator. This amounts to study the deformation of such geometric structures. As a consequence, we prove that the moduli space of conical positively curved gradient Ricci expa…
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.
In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-…
Gradient flow expands curves to round shapes.
problem Expanding curves to round shapes.
method Steepest descent L2-gradient flow of entropy.
result Flow converges to a round expanding circle for various initial curves.
Expander graphs have been intensively studied in the last four decades. In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every d there are unbounded degree simplicial co…
We construct expanding endomorphisms on smooth manifolds that are homeomorphic to tori yet have exotic underlying PL-structures.
Paper reviews and proves volume growth estimates for different types of gradient Ricci solitons.
problem Estimating volume growth for gradient Ricci solitons.
method Survey and prove new volume growth estimates.
result New volume growth estimates for expanding gradient Ricci solitons.