Newly confirmed area-minimizing properties of Lawson-Osserman cones.
problem Verifying the area-minimizing property of Lawson-Osserman cones.
method Analyzing cones of type (n, p, 2) constructed in [XYZ].
result All Lawson-Osserman cones of type (n, p, 2) are area-minimizing.
The study proves stability inequalities for specific area-minimizing Lawson cones.
problem Stability of area-minimizing Lawson cones.
method Proof of stability inequalities for Lawson cones with specific parameters.
result Extends results to all area-minimizing Lawson cones.
Characterizes eigenfunctions of Lawson-Osserman cone and proves its integrability.
problem Eigenfunctions and integrability of Lawson-Osserman cone.
method Characterization of eigenfunctions and proof of integrability.
result Lawson-Osserman cone is integrable, generating all Jacobi fields via rotations and translations.
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.
Study shows instability of specific cone solutions in high-dimensional spaces.
problem Unstable solutions of minimal graphs in high codimension.
method Min-max technique applied to Euclidean spaces.
result First examples of non-smooth unstable minimal graphs.
We prove the existence of the analog of Lawson's minimal cones for a notion of nonlocal minimal surface introduced by Caffarelli, Roquejoffre and Savin, and establish their stability/instability in low dimensions. In particular we find that there are nonlocal stable minimal cones in dimension 7, in contrast with the ca…
New solutions found for a complex equation, diverging from a cone.
problem Finding new solutions to a complex equation with specific symmetry properties.
method Careful study of a Jacobi-Toda system on an invariant manifold, asymptotic to a cone.
result Constructs new families of two-ended solutions diverging logarithmically from a cone.
The paper constructs solutions to the Allen-Cahn equation using special minimal hypersurfaces.
problem Constructing solutions to the Allen-Cahn equation with specific properties.
method Using special minimal hypersurfaces asymptotic to a Lawson cone.
result Constructs solutions to the Allen-Cahn equation with infinite Morse index.
Smooths out complex shapes into simpler forms.
problem Transforming complex shapes into simpler, smooth forms.
method Perturbing minimizing hypercones and viscosity mean convex cones into smooth, properly embedded hypersurfaces.
result Properly embedded smooth minimizing hypersurfaces and self-expanders are achieved.
Our goal is to generalize the Choe-Hoppe helicoid and Clifford cones in Euclidean space. By sweeping out L indpendent Clifford cones in R2N+2 via the multi-screw motion, we construct minimal submanifolds in RL(2N+2)+1. Also, we sweep out the L-rays Clifford cone (introduced in Sectio…
Global minimizers found in high dimensions for a specific equation.
problem Existence of global minimizers for the Allen-Cahn equation in high dimensions.
method Proof using strictly area-minimizing Lawson's cones and Jerison-Monneau's program.
result Existence of global minimizers whose nodal sets are asymptotic to specific cones.
To study the Lawson-Osserman's counterexample to the Bernstein problem for minimal submanifolds of higher codimension, a new geometric concept, submanifolds in Euclidean space with constant Jordan angles(CJA), is introduced. By exploring the second fundamental form of submanifolds with CJA, we can characterize the Laws…
Develops a more general scheme for constructing non-parametric minimal cones.
problem Minimal surface equations and their solutions.
method Using harmonic Riemannian submersions and homothetic minimal immersions.
result Uncover a constellation of uncountably many non-parametric minimal cones.
The paper lifts Lagrangian immersions to cones in complex space.
problem Creating Lagrangian cones from immersions in complex projective space.
method Developing a method to lift immersions to cones, producing examples and analyzing projections.
result Examples of Lagrangian cones and special cones are produced, with projections showing few transverse double points.
Researchers prove uniqueness of certain cylindrical tangent cones for special Lagrangians.
problem Proving uniqueness of cylindrical tangent cones for special Lagrangians.
method Analyzing exact special Lagrangian submanifolds with multiplicity one and cylindrical tangent cones.
result The cylindrical tangent cones are unique under specific conditions.
Constructs special Lagrangians with stable singularities.
problem Creating compact special Lagrangians with stable singularities.
method Constructs families of compact almost Calabi-Yau manifolds and special Lagrangians.
result Models stable T^2-cones as compact special Lagrangians.
The study examines stability and classification of special minimal hypersurfaces in high dimensions.
problem Stability and classification of special minimal hypersurfaces in high dimensions.
method Analysis of stability and nondegeneracy properties using Jacobi fields and Morse index.
result In high dimensions, these hypersurfaces are strictly stable and have a full classification of bounded Jacobi fields.
In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved the existence of self-expanding hypersurfaces with prescribed tangent cones at infinity. If the cone is C3,α-regular and mean convex (but not area-minimizing…
Proof shows cones minimize certain geometric functionals.
problem Minimizing cones over spheres in geometric functionals.
method Proof by foliation analysis of cone leaves.
result Cone minimizes functionals for SkimesSl. Extends Smale's principle to produce minimal graphs with singularities.
problem Creating minimal graphs with isolated singularities in higher dimensions.
method Extends Smale's singular bridge principle to arbitrary codimension and applies it to specific minimal cones.
result Produces a minimal graph in 7D with any number of isolated singularities.
This study addresses transitions in conically singular associative submanifolds and their desingularizations.
problem Counting closed associative submanifolds of G2-manifolds and understanding transitions arising from degenerations. method Analysis of moduli spaces, transversality results, and desingularization techniques for conically singular associative submanifolds.
result For generic co-closed G2-structures, there are no CS associative submanifolds with stability-index greater than 0 or 1. Recent progress on minimal surface system and cones in Euclidean spaces.
problem Exploring the Dirichlet problem for minimal surfaces and cones.
method Systematic developments and new families of minimizing cones.
result New families of minimizing cones of different types.
It is shown that coassociative cones in R^7 that are r-oriented and ruled by 2-planes are equivalent to CR-holomorphic curves in the oriented Grassmanian of 2-planes in R^7. The geometry of these CR-holomorphic curves is studied and related to holomorphic curves in S^6. This leads to an equivalence between associative …
Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.
problem Proves existence and uniqueness of minimal capillary cones with bi-orthogonal symmetry.
method Solves a nonlinear free boundary equation parametrized by the contact angle and uses monotonicity properties.
result Demonstrates that minimizing capillary hypersurfaces can have singularities in codimension 7.
The nonlocal s-fractional minimal surface equation for Σ=∂E where E is an open set in RN is given by HΣs(p):=∫RN∣x−p∣N+sχE(x)−χEc(x)dx = 0for all p∈Σ. Here 0<s<1, χ designates characteristic function, and the integral is understood i…
Minimal surfaces in spheres constructed from symmetry reductions of ODEs.
problem Construct minimal surfaces in spheres with symmetry.
method Doubling links of free-boundary minimal cones in R^(p+q+3) with bi-orthogonal symmetry.
result Existence of minimal embeddings of S^p × S^q × S^1 in S^(p+q+2).
Ancient solutions in Lagrangian flow are classified based on their blow-down.
problem Understanding ancient solutions in Lagrangian mean curvature flow.
method Structural and classification results for ancient solutions, focusing on the almost calibrated case.
result Classification of Type II blow-ups in terms of their blow-down.
Study symplectically aspherical Kähler manifolds with unique properties.
problem Existence and properties of symplectically aspherical Kähler manifolds.
method Detailed study and analysis of geometric and topological features.
result Existence of symplectically aspherical Kähler manifolds with large fundamental groups.
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 R5 are rigid and must be a specific type of minimal generalized Legendrian Clifford torus. Construct locally minimizing (1,2)-clusters with prescribed asymptotic geometry.
problem Minimizing clusters with prescribed asymptotic geometry.
method Develop a refined construction using the Hardt-Simon foliation.
result Produce a countably infinite family of distinct locally minimizing clusters asymptotic to a singular area-minimizing hypercone.
Constructs new coassociative fibrations for G2 manifolds.
problem Tackles the construction of new coassociative fibrations for G2 manifolds.
method Constructs fibrations by coassociative 4-folds, relates to hypersymplectic geometry and Donaldson's work.
result Shows natural generalizations of known coassociative fibrations.
Minimal surfaces match symmetries and topology exactly.
problem Matching symmetries and topology in minimal surfaces.
method Proved congruence of surfaces with specific symmetries and topology.
result Lawson surfaces are uniquely determined by their symmetries and topology.
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.
Extremal spectral properties of Lawson tau-surfaces are investigated. The Lawson tau-surfaces form a two-parametric family of tori or Klein bottles minimally immersed in the standard unitary three-dimensional sphere. A Lawson tau-surface carries an extremal metric for some eigenvalue of the Laplace-Beltrami operator. U…
A natural map from Lawson homology to Deligne cohomology groups for smooth complex projective varieties is constructed by using the Harvey-Lawson spark complexes. We also compare this to Abel-Jacobi type constructions by others.
The Lawson surface ξ_{g,1} has index 2g+3 and nullity 6.
problem Characterizing the geometric properties of Lawson surfaces.
method Analyzing the linearized stability of the Lawson surface.
result The Lawson surface ξ_{g,1} has no exceptional Jacobi fields.
New proof of high genus Lawson surfaces with area estimates.
problem Existence and area estimates of high genus Lawson surfaces.
method Deforming DPW potential to prove existence and calculating area.
result Estimates on area of Lawson surfaces in terms of genus.
Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces τr,m minimally immersed in spheres to a three-parametric family Ta,b,c of tori and Klein bottles minimally immersed in spheres. It was remarked that this family inclu…
This is the fourth in a series of papers math.DG/0008021, math.DG/0008155, math.DG/0010036 constructing explicit examples of special Lagrangian submanifolds (SL m-folds) in C^m. A submanifold of C^m is ruled if it is fibred by a family of real straight lines in C^m. This paper studies ruled special Lagrangian 3-folds i…
Strengthened surgery theorem for positive scalar curvature metrics.
problem Metrics of positive scalar curvature on manifolds.
method Complete account of Chernysh's strengthening of the Gromov-Lawson surgery theorem.
result Complete account of the strengthened theorem.
Extremal spectral properties of the Lawson tori are studied. A Lawson torus carries an extremal metric for some eigenvalue of the Laplace-Beltrami operator. The main result of this paper is that the number of this eigenvalue is expressed in terms of fundamental tones of auxiliary periodic Sturm-Liouville problems.
Study on Lawson surfaces' first Laplace eigenvalue using symmetry and algebraic methods.
problem Yau's conjecture on first eigenvalue of minimal hypersurfaces in the sphere.
method Symmetry-based approach exploiting discrete reflection symmetries and algebraic structure of reflection groups.
result Equality λ1(ξ_{m,k})=2 for Lawson surfaces with m and k even.
Topology classifies bipolar surfaces; they are not embedded.
problem Classifying bipolar surfaces in the 5-sphere.
method Topological classification and area bounds calculation.
result Bipolar surfaces are not embedded.
Minimal and CMC surfaces in S3 can be treated via their associated family of flat $\SL(2,\C)$-connections. In this the paper we parametrize the moduli space of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of Lawson's geometric construc…
Study Cayley fibrations on Bryant-Salamon Spin(7) manifolds.
problem Investigate Cayley fibrations on specific Spin(7) manifolds. method Analyze invariant Cayley fibrations for each SU(2) action. result Explicitly describe the fibres of Cayley fibrations.
Proves planar Lipschitz critical points of area functional are smooth.
problem Lawson-Osserman conjecture about smoothness of critical points.
method Outer variations to prove smoothness of critical points.
result Proves conjecture for planar case.
We show that any embedded minimal torus in S^3 is congruent to the Clifford torus. This answers a question posed by H.B. Lawson, Jr., in 1970.
Study crystallographic groups for positive scalar curvature conditions.
problem Examining positive and negative results for Gromov-Lawson-Rosenberg Conjecture.
method Analyzing split extensions of free abelian by cyclic groups.
result Produce infinite counterexamples for the Gromov-Lawson-Rosenberg Conjecture.