Classifies minimal submanifolds in complex hyperbolic spaces.
problem Identifying minimal submanifolds in complex hyperbolic spaces.
method Classification based on extrinsic homogeneity.
result Classification of minimal extrinsically homogeneous submanifolds.
Uniqueness and stability of minimal submanifolds proved.
problem Uniqueness and stability of minimal submanifolds.
method Proved a strong stability condition on minimal submanifolds.
result Existence and convergence of mean curvature flow for minimal submanifolds.
Minimal Lagrangian submanifolds deform to J-minimal ones under small perturbations.
problem Understanding how minimal Lagrangian submanifolds behave under small perturbations in Kaehler-Einstein manifolds.
method Analyzing the deformation of minimal Lagrangian submanifolds under Kaehler-Einstein perturbations.
result Deformed submanifolds remain J-minimal under certain conditions.
Study shows area-minimizing submanifolds are mostly rough, not smooth.
problem Understanding the smoothness of area-minimizing submanifolds.
method Proved non-smoothness by contradiction and established Hausdorff dimension bounds.
result Area-minimizing submanifolds are not generically smooth, resolving a conjecture.
Study shows area-minimizing submanifolds are mostly smooth except for specific types.
problem Understanding when area-minimizing submanifolds are smooth in mod 2 homology.
method Proved area-minimizing submanifolds are not generically smooth except for geodesics, minimal surfaces, and minimal hypersurfaces.
result Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces.
Study shows area-minimizing submanifolds are mostly smooth except for geodesics, minimal surfaces, and hypersurfaces.
problem Understanding when area-minimizing submanifolds are smooth in mod 2 homology.
method Proving the mod 2 area-minimizing submanifolds are smooth in specific cases and establishing lower bounds on singular sets.
result Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces.
Hasse principle applied to area-minimizing submanifolds across different homology types.
problem Understanding the behavior of area-minimizing submanifolds in various homology contexts.
method Extending the Hasse principle from number theory to geometric variational problems.
result Recovering information about area-minimizing submanifolds in integral homology from those in real and mod n homology. Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total (2k)-th Gauss-Bonnet curvature function, called (2k)-minimal su…
The study finds conditions for area-minimizing cones over submanifolds.
problem Conditions for area-minimizing cones over submanifolds.
method General configuration results for area-minimizing cones.
result Cone over the minimal product of submanifolds and spheres are area-minimizing.
Uniqueness of minimal submanifolds in specific Riemannian manifolds.
problem Understanding uniqueness of minimal submanifolds in various Riemannian manifolds.
method Analyzing compact minimal submanifolds in specific classes of Riemannian manifolds.
result Uniqueness results for compact minimal submanifolds in large classes of Riemannian manifolds.
Study finds formulas for minimal submanifolds using Möbius transformations.
problem Understanding minimal submanifolds in Euclidean space.
method Monotonicity formulas for minimal submanifolds involving Möbius transformations.
result Proved formulas for minimal submanifolds under Möbius transformations.
Constructs minimal submanifolds in symmetric spaces using eigenfunctions.
problem Finding minimal submanifolds in symmetric spaces.
method Employing recent results from S. Gudmundsson and T.J. Munn, constructing submanifolds using eigenfunctions.
result Constructs minimal submanifolds of classical compact Riemannian symmetric spaces.
Paper studies second variation for L-minimal submanifolds in pseudo-Sasakian manifolds.
problem Analyzing stability of L-minimal submanifolds in pseudo-Sasakian manifolds.
method Provides a second variation formula and applies it to Lorentzian-Sasakian manifolds.
result Relates L-stability of Legendrians in a Sasakian manifold to their stability in an associated Lorentzian-Sasakian structure.
Develops methods for computing conformal invariants of submanifolds.
problem Computing conformal invariants of submanifolds.
method Direct construction of extrinsic ambient space, global invariants of conformally compact minimal submanifolds, introduction of conformal submanifold scalars.
result Derives an explicit Gauss--Bonnet--Chern-type formula and proves a rigidity result.
Paper proves existence of area-minimizing submanifolds on almost any manifold with fractal singular sets.
problem Existence of area-minimizing submanifolds with fractal singular sets.
method Constructing and proving existence on almost any smooth manifold.
result Existence of area-minimizing submanifolds with fractal singular sets on almost any smooth manifold.
Minimal submanifolds in spheres can be produced via Clifford type minimal products, and their Morse indices and nullities are calculated.
problem Understanding the properties of minimal submanifolds in spheres via Clifford products.
method Analyzing the first eigenfunctions and Morse indices of minimal products of minimal submanifolds.
result The Morse index and nullity of the minimal product are calculated and shown for specific cases.
The paper classifies product minimal Lagrangian submanifolds in complex space forms.
problem Understanding minimal Lagrangian submanifolds in complex space forms.
method Examining submanifolds as Riemannian products with constant sectional curvature.
result Complete classification of product minimal Lagrangian submanifolds.
The paper studies stability and instability of minimal submanifolds in complex Einstein spaces.
problem Stability and instability of minimal submanifolds in complex Einstein spaces.
method Computation of index and nullity, investigation of stability, and algorithm for higher eigenvalues.
result Criterion for instability of minimal submanifolds in some cases.
New minimal submanifolds in spheres share properties of the Clifford torus.
problem Finding new minimal surfaces in spheres.
method Analyzing properties of the Clifford torus and extending to other minimal submanifolds.
result More minimal submanifolds in spheres have helicoidal properties.
Paper constructs new minimal submanifolds in spheres by spinning given ones.
problem Creating new minimal submanifolds in spheres from given ones.
method Spin given minimal submanifolds by a curve γ in a balanced way. result Generates spiral minimal products forming a two-dimensional family.
Minimal totally real submanifolds in complex space forms have special umbilical properties.
problem Characterizing minimal totally real submanifolds in complex space forms.
method Analyzing the position of umbilical normal vectors in the normal bundle.
result Pseudo-umbilical totally real submanifolds with flat normal connection in non-flat complex space forms are minimal.
Characterizes harmonic morphisms preserving minimal submanifolds and finds novel area-minimising hypercones.
problem Understanding harmonic morphisms and their relationship to minimal submanifolds.
method Characterization of harmonic morphisms as weakly horizontally conformal maps preserving minimal submanifold equations, derivation of reduction properties for other co-dimensions, application to find novel area-minimising hypercones.
result Novel family of degree 4 area-minimising hypercones in R^m, m≥32.
Minimal submanifolds are stable in certain conformal spheres.
problem Stability of minimal submanifolds in conformal spheres.
method Analyzing n-dimensional Riemannian spheres with specific curvature conditions. result Closed stable minimal submanifolds are not found in δ-pinched conformal spheres. Extends submanifold minimality to multi-index u-minimality using frame bundle analysis.
problem Generalizing submanifold minimality to multi-index settings.
method Analysis of frame bundle and associated vector bundles; variation of σ_u-symmetric function.
result Alternative definition of u-minimality and examples of u-minimal submanifolds.
The paper introduces austere and arid submanifolds in Hilbert spaces.
problem Classifying minimal orbits in hyperpolar PF actions on Hilbert spaces.
method Introducing austere and arid submanifolds into PF submanifolds in Hilbert spaces.
result Examples of infinite dimensional austere and arid PF submanifolds in Hilbert spaces.
Hamiltonian minimality (H-minimality) for Lagrangian submanifolds is a symplectic analogue of Riemannian minimality. A Lagrangian submanifold is called H-minimal if the variations of its volume along all Hamiltonian vector fields are zero. This notion was introduced in the work of Y.-G. Oh in connection with the celebr…
The study of stable and index compact minimal submanifolds in Berger spheres.
problem Stability and index of compact minimal submanifolds in Berger spheres.
method Analyzing stability and index properties of compact minimal submanifolds in Berger spheres.
result Stable compact minimal submanifolds exist in Berger spheres for specific values of τ, and their classification is provided.
The paper studies minimal submanifolds with specific curvature properties in Euclidean space.
problem Minimal submanifolds with (n−2)-umbilical properties in Euclidean space. method Established a correspondence and developed a Weierstrass type method for local parametrization.
result Minimal, generic, (n−2)-umbilic submanifolds are (n−2)-rotational and have a parametric description. New formulas limit minimal submanifolds' area in curved spaces.
problem Bounding minimal submanifolds' area in curved spaces.
method Developed new monotonicity formulae involving energy-like integrals over non-geodesic sets.
result Imply sharp area bounds for minimal submanifolds through a prescribed point.
Sharp isoperimetric inequality for minimal submanifolds in 2D and higher Euclidean space.
problem Finding the minimum surface area for a given volume in submanifolds.
method Proving a Sobolev inequality and using it to derive a sharp isoperimetric inequality.
result Sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2.
Constructs area-minimizing submanifolds with fractal singularities.
problem Area-minimizing submanifolds with fractal singular sets.
method Integral currents, mod v currents, stable stationary varifolds.
result Sharp dimensionwise solution to Almgren's conjecture.
New method creates minimal submanifolds using complex-valued eigenfunctions.
problem Creating minimal submanifolds in compact Riemannian manifolds.
method Employing complex-valued eigenfunctions.
result Manufactured minimal submanifolds in compact Riemannian manifolds.
New minimal submanifolds found in Lie group duals.
problem Finding minimal submanifolds in Lie group duals.
method Constructing families of complete minimal submanifolds of codimension two.
result New families of complete minimal submanifolds of codimension two in Lie group duals constructed.
Study on minimal submanifolds with nullity in hyperbolic space.
problem Characterize complete minimal submanifolds with nullity in hyperbolic space.
method Investigate properties of submanifolds with index of relative nullity at least one in hyperbolic space, considering scalar curvature bounds.
result If the scalar curvature is bounded from below, submanifolds are either totally geodesic or generalized cones over complete minimal surfaces.
Totally geodesic submanifolds in hyperbolic space up to codimension two.
problem Characterizing minimal homogeneous submanifolds in hyperbolic spaces.
method Analyzing properties of minimal submanifolds in hyperbolic spaces up to codimension two.
result Minimal homogeneous submanifolds of hyperbolic space up to codimension two are totally geodesic.
No global solutions found for time-like minimal submanifolds in Minkowski space.
problem Existence of global-in-time axisymmetric solutions to time-like minimal submanifolds in Minkowski space.
method Analysis of limiting geometry as maximal time of existence is approached.
result No global solutions found for time-like minimal submanifolds in Minkowski space.
The study constructs minimal submanifolds in complex and quaternionic projective spaces.
problem Finding minimal submanifolds in complex and quaternionic projective spaces.
method Using complex-valued harmonic morphisms.
result Complete minimal submanifolds of odd-dimensional complex projective spaces and their dual hyperbolic spaces are constructed.
The paper finds many infinite-dimensional weakly reflective PF submanifolds in Hilbert spaces.
problem Minimal submanifolds in Hilbert spaces with reflective properties.
method Introduced weakly reflective PF submanifolds into Hilbert spaces and showed their existence.
result Existence of infinite-dimensional weakly reflective PF submanifolds in Hilbert spaces.
Barrier methods classify minimal submanifolds in hyperkaehler spaces.
problem Classifying compact minimal submanifolds in hyperkaehler spaces.
method Barrier argument and strong stability condition analysis.
result Results towards a classification of compact minimal submanifolds.
We show that a totally geodesic submanifold of a symmetric space satisfying certain conditions admits an extension to a minimal submanifold of dimension one higher, and we apply this result to construct new examples of complete embedded minimal submanifolds in simply connected noncompact globally symmetric spaces.
New stability result extends to enhanced Brakke flows for minimal submanifolds.
problem Stability of minimal submanifolds under perturbations.
method Extending recent dynamical stability result to enhanced Brakke flows.
result Local uniqueness statement for strongly stable minimal submanifolds.
Study on ball widths and minimal submanifolds in space forms.
problem Understanding widths of balls and minimal submanifolds.
method Analyzing the area of equatorial balls and related bounds for minimal submanifolds.
result Lower bounds for the area of free boundary minimal submanifolds.
Ricci curvature links volume convexity and minimal submanifolds.
problem Volume functional convexity and minimal submanifolds in Kaehler geometry.
method Toric Kaehler geometry and quasi-homogeneous manifolds.
result Sign of Ricci curvature correlates with volume functional convexity.
Formula derived for renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
problem Calculating the renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
method Decomposition of extrinsic Q-curvature and application to renormalized area. result Renormalized area formula expressed as a linear combination of Euler characteristic and scalar conformal submanifold invariant.
Study complex properties of minimal Lagrangian submanifolds in Kaehler spaces.
problem Complex properties of minimal Lagrangian submanifolds in Kaehler spaces.
method Mix of holomorphic curve techniques and convexity results.
result Minimal Lagrangians do not admit fillings by holomorphic discs in negative curvature case.
Study f-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials.
problem Variational properties of f-minimal Lagrangian submanifolds. method Derive second variation formula and study stability of solitons.
result Stability of expanding and translating solitons for LMCF.
Study harmonic mappings and submanifolds using Bochner technique.
problem Classical theorems in harmonic mappings and submanifolds.
method Generalized Bochner technique.
result New insights into classical theorems.
Study of higher-dimensional Willmore energies via minimal submanifold asymptotics.
problem Understanding conformally invariant generalizations of Willmore energy.
method Derives and studies a new energy functional for submanifolds, connects it to minimal submanifold asymptotics in Poincare-Einstein spaces.
result Explicitly identifies the energy for four-dimensional submanifolds and studies its variational properties.