In this paper, we study the Lagrangian F-stability of closed Lagrangian self-shrinkers immersed in complex Euclidean space. We show that any closed Lagrangian self-shrinker with first Betti number greater than one is Lagrangian F-unstable. In particular, any two-dimensional embedded closed Lagrangian self-shrinker is L…
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48 results for “Lagrangian self-shrinker”
The Clifford torus is uniquely identified as a Lagrangian self-shrinker in complex space.
problem Characterizing the Clifford torus as a Lagrangian self-shrinker in complex space.
method Analyzing the Clifford torus in with specific curvature conditions.
result The Clifford torus is the unique compact orientable Lagrangian self-shrinker in with .
The study classifies complete Lagrangian self-shrinkers in 4D space.
problem Classifying complete Lagrangian self-shrinkers in 4D space.
method Complete classification of 2D complete Lagrangian self-shrinkers with constant squared norm of the second fundamental form.
result A complete classification for 2-dimensional complete Lagrangian self-shrinkers in with constant squared norm of the second fundamental form.
The paper identifies special Lagrangian shapes in 4D space.
problem Characterizing Lagrangian self-shrinkers in 4D space.
method Proving shapes are products of Abresch-Langer curves.
result The Clifford torus is the unique embedded closed Lagrangian self-shrinker.
We provide several rigidity results for the Clifford torus in the class of compact self-shrinkers for Lagrangian mean curvature flow.
In this paper, we study the Lagrangian F-stability and Hamiltonian F-stability of Lagrangian self-shrinkers. We prove a characterization theorem for the Hamiltonian F-stability of -dimensional complete Lagrangian self-shrinkers without boundary, with polynomial volume growth and with the second fundamental form sati…
The paper studies shrinking Kähler-Ricci solitons and their self-shrinkers.
problem Estimating the diameter of Lagrangian self-shrinkers in gradient shrinking Kähler-Ricci solitons.
method Lower bound estimate and non-existence proof for self-shrinkers.
result Established a lower bound for the diameter of Lagrangian self-shrinkers.
Paper proves rigidity of certain 2D Lagrangian shapes in 4D space.
problem Proving rigidity of specific Lagrangian shapes in 4D space.
method Used a rigidity theorem for 2D complete Lagrangian self-shrinkers.
result Rigidity of 2D complete Lagrangian self-shrinkers with constant squared norm of mean curvature vector.
In this paper, we generalize Colding and Minicozzi's work \cite{CM} on the stability of self-shrinkers in the hypersurface case to higher co-dimensional cases. The first and second variation formulae of the -functional are derived and an equivalent condition to the stability in general codimension is found. Moreover…
On the Lagrangian angle and the Kähler angle of immersed surfaces in the complex plane $\bbc^2$math.DG
The paper extends Lagrangian and Kähler angles to immersed surfaces in complex plane and proves pinching results.
problem Extending Lagrangian and Kähler angles to immersed surfaces in complex plane.
method Extending Lagrangian angle, Maslov form, and Maslov class to immersed surfaces in ; proving pinching results for Kähler angle.
result Two pinching results for the Kähler angle imply rigidity theorems of self-shrinkers with Kähler angle.
The study classifies and proves rigidity of Legendrian self-shrinkers in 3D and 5D.
problem Classifying and proving rigidity of Legendrian self-shrinkers.
method Classification and rigidity theorem proof.
result Compact Legendrian self-shrinkers in are rigid and must be a specific type of minimal generalized Legendrian Clifford torus.
Let be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics converges smoothly to a Riemannian metric . We show that a subsequence of converges smoothly to …
In [LW], we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak form…
The paper constructs examples of Lagrangian flows using moment maps.
problem Constructing Lagrangian mean curvature flows in Calabi-Yau manifolds.
method Using moment maps for abelian Lie group actions.
result Examples of Lagrangian self-shrinkers and translating solitons.
Special Lagrangians and Lagrangian self-similar solutions in cones over toric Sasaki manifoldsmath.DG
We construct some examples of special Lagrangian submanifolds and Lagrangian self-similar solutions in almost Calabi-Yau cones over toric Sasaki manifolds. For example, for any integer g>0, we can construct a real 6 dimensional Calabi-Yau cone M_g and a 3 dimensional special Lagrangian submanifold L^1_g in M_g which is…
By the integral method we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in with the indefinite metric is flat. This result improves the previous ones in \cite{HW} and \cite{CCY} by removing the additional assumption in their results. I…
We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regular…
Paper proves rigidity for special submanifolds in complex plane.
problem Proving rigidity for a specific type of submanifolds in complex plane.
method Introduced -submanifolds and proved rigidity theorem for Lagrangian ones.
result Rigidity theorem for Lagrangian -submanifolds in .
Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow
problem Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow
method Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow
result Identifying the lowest eigenfunction with the scaling mode of the special Lagrangian desingularization
The article recovers the Smale conjecture on a Sasakian 3-sphere using Legendrian mean curvature flow.
problem Recovering the Smale conjecture on a Sasakian 3-sphere.
method Using Legendrian mean curvature flow to deform area-preserving contactomorphisms to isometries.
result Obtained the minimal Legendrian graph in S² × S³.
Classifies regularity for Lagrangian mean curvature type equations.
problem Classifying regularity for Lagrangian mean curvature type equations.
method Generalized constant rank theorem for Legendre transform, constructed convex solutions, and showed regularity conditions.
result Optimal regularity conditions for Lagrangian mean curvature type equations.
Researchers set entropy limits for specific types of self-shrinkers.
problem Understanding entropy limits for self-shrinkers with symmetries.
method Derived explicit entropy bounds for two specific classes of self-shrinkers using isoparametric foliations and symmetry analysis.
result Entropy bounds generalized to new classes of self-shrinkers, extending previous findings.
In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower diameter bound for compact manifolds with shrinking Ricci solitons. Such a boun…
Study bounds on self-shrinkers with bounded HA for applications.
problem Understanding bounds on self-shrinkers with bounded HA.
method Integral and pointwise bounds on the second fundamental form of self-shrinkers.
result Gap and compactness results for self-shrinkers.
Survey of self-shrinkers with symmetry and new existence problems.
problem Existence and uniqueness of closed self-shrinkers with specific symmetries.
method Review of known constructions and introduction of new problems.
result Proposed new existence problems for self-shrinkers with bi-rotational symmetry.
Study on 2D self-shrinkers, proving their local behavior and boundedness.
problem Understanding the geometry and behavior of 2D self-shrinkers.
method Local graphical theorem, asymptotic behavior analysis, boundedness proof.
result Uniform boundedness of second fundamental form for noncompact self-shrinkers.
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.
A rigidity theorem for smooth Legendrian self-shrinkers is proven.
problem Understanding the structure of Legendrian self-shrinkers.
method Estimating weighted volume to prove optimal volume growth.
result Rigidity theorem for entire smooth Legendrian self-shrinkers.
The paper studies scalar curvature of self-shrinkers and proves curvature bounds.
problem Proving curvature bounds for self-shrinkers.
method Analyzing scalar curvature of self-shrinkers in Euclidean space.
result Proves that the scalar curvature of self-shrinkers is bounded by .
Generalizes halfspace theorems to higher dimensions for self-shrinkers.
problem Limitations of halfspace theorems in higher dimensions for self-shrinkers.
method Extends codimension 1 results to arbitrary codimension.
result Establishes new halfspace theorems for self-shrinkers in arbitrary codimension.
Compact self shrinkers in 3D are topologically standard.
problem Understanding the topological structure of self shrinkers in 3D.
method Demonstrated through ambient isotopy and genus analysis.
result Compact self shrinkers in 3D are topologically standard.
We investigate Mean Curvature Flow self-shrinking hypersurfaces with polynomial growth. It is known that such self shrinkers are unstable. We focus mostly on self-shrinkers of the form . We use a connection between the stability operator and the quantum harmonic oscillator Ham…
New theorem shows noncompact self shrinkers are unknotted.
problem Understanding the structure of noncompact self shrinkers.
method Used mean curvature flow to extend theorem to noncompact cases.
result Noncompact self shrinkers without knotted components.
Existence proof of noncompact self-shrinkers with arbitrary genus.
problem Existence of noncompact self-shrinkers with arbitrary genus.
method Employing min-max techniques to rigorously prove existence.
result Confirmation of one asymptotically conical end for large genus self-shrinkers.
Study self shrinkers with medium entropy in 4D space.
problem Analyzing self shrinkers with entropy bounds.
method Smooth asymptotically conical self shrinkers in R^4.
result Entropy bounded above by Λ_1.
Paper proves finite Morse index for certain self-shrinkers.
problem Finite Morse index of self-shrinkers.
method Sufficient condition for finite Morse index of complete properly self-shrinkers.
result Proves finite Morse index for self-shrinkers with finite asymptotically conical or cylindrical ends.
Using a maximum principle for self-shrinkers of the mean curvature flow, we give new proofs of a rigidity theorem for rotationally symmetric compact self-shrinkers and a result about the asymptotic behavior of self-shrinkers. This comparison argument also implies a linear bound for the second fundamental form of self-s…
Self-shrinkers in 3D have simple ends.
problem Understanding the structure of self-shrinkers.
method Analyzing asymptotic behavior of noncompact self-shrinkers.
result Each end of a self-shrinker is asymptotic to a cone or cylinder.
New self-shrinkers found in higher dimensions.
problem Existence of specific types of self-shrinkers in higher-dimensional spaces.
method Construction of closed embedded self-shrinkers with specific topological types.
result Existence of new closed self-shrinkers in .
New topological property of self-shrinkers with small entropy.
problem Understanding the topological structure of self-shrinkers with small entropy.
method Analyzing asymptotically conical self-shrinkers with entropy ≤ cylinder entropy.
result The link of the asymptotic cone separates the unit sphere into two diffeomorphic components.
New self-shrinkers of Platonic solids found.
problem Finding new embedded self-shrinkers of specific genus.
method Variational methods, numerical discovery by D. Chopp.
result Constructed self-shrinkers resembling doublings of Platonic solids.
New theorems on compactness and finiteness for specific types of self-shrinkers.
problem Characterizing rotationally symmetric self-shrinkers with constraints.
method Compactness and finiteness theorems for self-shrinkers with specific symmetries and constraints.
result Existence of entropy minimizing self-shrinkers diffeomorphic to for each .
We study geometric properties of complete non-compact bounded self-shrinkers and obtain natural restrictions that force these hypersurfaces to be compact. Furthermore, we observe that, to a certain extent, complete self-shrinkers intersect transversally a hyperplane through the origin. When such an intersection is comp…
Study classifies 3D self-shrinkers with constant second form norm.
problem Classifying self-shrinkers with specific geometric properties.
method Analyzes 3D self-shrinkers in Euclidean space with constant second form norm.
result Classifies complete self-shrinkers with constant norm of the second fundamental form.
Study of 2D self-shrinkers in 3D space without polynomial volume growth.
problem Classify 2D complete self-shrinkers in .
method Used generalized maximum principle for -operator to classify self-shrinkers.
result Complete classification of 2D complete self-shrinkers with constant squared norm of the second fundamental form in .
The study proves properties of self-shrinkers with bounded curvature.
problem Characterizing self-shrinkers with bounded curvature.
method Analyzing properties of self-shrinkers in with bounded second fundamental form.
result Proves that if the squared norm of the second fundamental form is bounded, it must be constant.
The paper proves gap results for self-shrinkers in -mean curvature flow.
problem Understanding the gap in properties of self-shrinkers in -mean curvature flow.
method Proving gap results using a modified second fundamental form and a differential operator.
result Proper self-shrinkers are parabolic for a certain second-order differential operator.
Study proves spacelike self-shrinkers are hyperplanes under certain conditions.
problem Classifying spacelike self-shrinkers in pseudo-Euclidean space.
method Applied maximum principles to show rigidity.
result Spacelike self-shrinkers are rigid and must be hyperplanes.