Researchers describe a specific type of submanifolds in Euclidean space.
problem Understanding inhomogeneous almost symmetric submanifolds.
method Completely describing submanifolds as unions of parallel symmetric submanifolds.
result Described inhomogeneous properly embedded almost symmetric submanifolds as unions of symmetric submanifolds.
The study of Euclidean submanifolds with incompressible canonical vector fields.
problem Characterizing Euclidean submanifolds with incompressible canonical vector fields.
method Analyzing the canonical vector field properties and conditions for incompressibility.
result Necessary and sufficient conditions for the canonical vector field of a Euclidean submanifold to be incompressible.
The study characterizes Euclidean submanifolds with a conformal canonical vector field.
problem Characterizing Euclidean submanifolds with a specific vector field property.
method Investigating properties of the canonical vector field and its conformal nature.
result Characterization of Euclidean submanifolds with conformal canonical vector fields.
Direct proof for all conformally flat isoparametric submanifolds in Euclidean space.
problem Classifying conformally flat isoparametric submanifolds in Euclidean space.
method Direct proof approach.
result Complete classification of conformally flat isoparametric submanifolds of Euclidean space.
Local and global classifications of Einstein submanifolds in Euclidean space.
problem Classifying Einstein submanifolds in Euclidean space.
method Local and global parametric classifications with emphasis on intrinsic assumptions.
result Local and global classifications of Einstein submanifolds of codimension two.
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.
Sharp inequality for submanifolds in Euclidean space.
problem Logarithmic Sobolev inequality for submanifolds.
method Proved a sharp inequality with mean curvature term.
result Submanifolds in Euclidean space have a logarithmic Sobolev inequality.
Study on harmonic Gauss maps for submanifolds in Euclidean space and sphere.
problem Existence and non-existence of unit normal sections with harmonic associated Gauss maps.
method Proof of existence and non-existence results for submanifolds in Euclidean space and sphere.
result Obtained applications to CMC hypersurfaces of the sphere and isoparametric submanifolds.
Study on Yamabe and quasi-Yamabe solitons on Euclidean submanifolds.
problem Characterizing solitons on Euclidean submanifolds.
method Investigation of solitons with tangential components of position vector fields.
result Classification of Yamabe and quasi-Yamabe solitons on Euclidean hypersurfaces.
The paper classifies and studies conformal variations of submanifolds.
problem Classifying and understanding conformal variations of submanifolds.
method Develops a Fundamental theorem and a rigidity theorem for Euclidean submanifolds.
result Fundamental theorem and rigidity theorem for Euclidean submanifolds.
Survey on complex submanifolds in Euclidean spaces.
problem Existence of complete bounded complex submanifolds.
method Historical overview and recent progress.
result Discussion of open questions in the field.
Study on submanifolds of Euclidean space, classifying their symmetry types.
problem Classifying symmetry types of submanifolds in Euclidean space.
method Analyzing properties of full irreducible almost symmetric submanifolds and their cohomogeneity.
result Classification of almost symmetric submanifolds into specific types.
The paper studies essential spectra of submanifolds in Euclidean spaces.
problem Investigating the essential spectrum of submanifolds under geometric conditions.
method Analyzing submanifolds in Euclidean spaces with various geometric constraints.
result The essential spectrum of a complete non-compact submanifold is [0,+∞) if the second fundamental form satisfies certain Lp norms. Study variational submanifolds in Euclidean spaces from dynamical systems.
problem Formulate and solve conditions for variationality of induced systems on submanifolds.
method Employ variational sequence theory on sheaves of differential forms to analyze local and global variationality.
result Solve the problem of existence of variational submanifolds for second-order systems.
Proves Chen's conjecture on biharmonic submanifolds in Euclidean space and space forms.
problem Chen's conjecture on biharmonic submanifolds in Euclidean space.
method Derived a fundamental identity involving the mean curvature vector field and used it to prove the conjecture.
result Proved Chen's conjecture on biharmonic submanifolds in a Euclidean space and space forms.
We introduce a class of potential submanifolds in pseudo-Euclidean spaces (each N-dimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space) and prove that each N-dimensional Frobenius manifold can be locally represented as an N-dimensional potential submanif…
Paper disproves Kähler submanifolds in symmetrized polydisc.
problem Existence of Kähler submanifolds in symmetrized polydisc.
method Proof by contradiction using canonical metrics.
result Non-existence of Kähler submanifolds in symmetrized polydisc.
The paper studies infinitesimal variations of submanifolds in Euclidean space.
problem Understanding infinitesimal variations of submanifolds of arbitrary dimension and codimension.
method Proving a system of three equations as integrability conditions for the differential equation determining infinitesimal variations.
result Established a Fundamental theorem for infinitesimal variations of submanifolds.
Study on minimal submanifolds with finite curvature in Euclidean space.
problem Finite diffeomorphism types of complete immersed minimal submanifolds with finite total curvature.
method Adapted method from Chodosh, Ketover, and Maximo for hypersurfaces to submanifolds of arbitrary codimension.
result Proved finite diffeomorphism types for complete immersed minimal submanifolds with finite total curvature.
The paper studies submanifolds of Euclidean space from Kaehler manifolds.
problem Understanding submanifolds of Euclidean space from Kaehler manifolds.
method Analyzing conformal immersions and isometric embeddings.
result Local constructions of submanifolds from Kaehler 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.
Sharp upper bound found for Steklov spectrum on revolution submanifolds.
problem Finding bounds for Steklov spectrum on specific submanifolds.
method Analyzing submanifolds of revolution in Euclidean space.
result Sharp upper bound established for Steklov spectrum.
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.
In this short note, we study the asymptotic property of Huisken's functional for mean curvature flow on the minimal submanifolds of Euclidean space. We prove that the limit of Huisken's functional equals to the extrinsic asymptotic volume ratio on the minimal submanifold of Euclidean space.
Study examines homology of contact CR-submanifolds in complex Euclidean space.
problem Investigating the homology of contact CR-submanifolds.
method Analyzes stable integral currents and homology groups of contact CR-submanifolds in Cm. result Nonexistence of stable integral currents and vanishing theorems for homology groups.
The study characterizes compact submanifolds with pinched Ricci curvature in Euclidean and spherical space forms.
problem Characterizing compact submanifolds with specific Ricci curvature bounds.
method Proving rigidity results for submanifolds with Ricci curvature bounded below by a function of mean curvature.
result Submanifolds are either isometric to the Einstein Clifford torus or have vanishing homology groups up to a certain degree.
Classifies rank-one submanifolds in Euclidean space.
problem Classifying submanifolds with singularities.
method Associate degree to ruled submanifolds and analyze singularities.
result An open and dense subset of rank-one submanifolds is the union of cylindrical, conical, and tangent regions.
The study finds larger gaps in mean curvature for biharmonic submanifolds in spheres.
problem Understanding gaps in mean curvature for biharmonic submanifolds.
method Analyzing proper biharmonic submanifolds with parallel mean curvature vector field in Euclidean spheres.
result Determining larger gaps in mean curvature for specific biharmonic submanifolds.
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. Paper defines minimal hypersurfaces in Euclidean and Riemannian spaces.
problem Characterizing minimal hypersurfaces in different spaces.
method Analyzes conditions for hypersurfaces to be minimal or stable.
result Minimal and stable hypersurfaces are hyperplanes in Euclidean spaces and totally geodesic submanifolds in Riemannian manifolds.
Develops PCA for symmetric spaces using geodesic submanifolds.
problem PCA for data on Riemannian symmetric spaces.
method Totally geodesic submanifolds for approximation.
result Illustrated on n-spheres, Grassmannians, n-tori and polyspheres.
The authors obtained the classification theorem for Lagrangian H-umbilical submanifolds in quaternion Euclidean spaces using the idea of quaternion extensors.
The paper classifies almost Yamabe solitons on hypersurfaces and submanifolds in Euclidean spaces.
problem Classifying almost Yamabe solitons on various geometric structures.
method Analyzing hypersurfaces and submanifolds with position and concurrent vector fields.
result Complete classification of almost Yamabe solitons on hypersurfaces and submanifolds.
We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat…
The study examines complete conformally flat submanifolds with nullity in Euclidean space.
problem Investigating properties of conformally flat submanifolds with nullity.
method Analyzing the index of relative nullity and scalar curvature to deduce manifold properties.
result Conditions for the manifold to be flat and the immersion to be a cylinder over a submanifold.
The paper proves a Wulff inequality for minimal submanifolds with boundary in Euclidean space.
problem Proving a Wulff inequality for minimal submanifolds with boundary.
method Associating a nonnegative anisotropic weight to the boundary of minimal submanifolds and proving the inequality.
result The Wulff inequality constant is independent of the weights and depends only on m and n. The study examines principal directions and curvatures of Lagrangian submanifolds.
problem Understanding the geometry of Lagrangian submanifolds.
method Recalling and analyzing the extrinsic principal tangential and normal directions, and their corresponding curvatures for Lagrangian submanifolds in complex Euclidean spaces.
result Established natural relationships between distinguished tangential and normal directions and their curvatures for Lagrangian submanifolds.
Study of filtering and smoothing in submanifolds of Euclidean space.
problem Filtering and smoothing in continuous-discrete time on submanifolds.
method Formal expressions and projection approach for prediction and smoothing.
result Agreement with classical results for prediction, differences for smoothing.
In this note we prove a Weierstrass representation formula for pluriminimal submanifolds of euclidean spaces. We use this formula to produce new families of examples of pluriminimal submanifolds. We also prove that any affine algebraic manifold can be pluriminimally embedded into some euclidean space in a non holomorph…
We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.
Using the idea of special Legendre curves, the authors obtained the explicit description of flat Lagrangian H-umbilical submanifolds in quaternion Euclidean spaces.
A space curve in a Euclidean 3-space E3 is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf 110} (2003), no. 2, 147-152]. In this present article, we introduce and study the n…
We describe several families of Lagrangian submanifolds in the complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of th…
Proves inequalities for tensor fields on submanifolds using ABP method.
problem Proving Michael-Simon inequalities for tensor fields.
method Alexandrov-Bakelman-Pucci (ABP) method
result Proved Michael-Simon inequalities for tensor fields.
Sharp Lp-logarithmic-Sobolev inequalities on submanifolds with applications to hypercontractivity.
problem Developing inequalities on submanifolds of Euclidean space.
method Optimal mass transport theory on submanifolds, sharpness analysis.
result Sharp inequalities and equality conditions for submanifolds.
In this paper we develop the vectorial Ribaucour transformation for Euclidean submanifolds. We prove a general decomposition theorem showing that under {appropriate} conditions the composition of two or more vectorial Ribaucour transformations is again a vectorial Ribaucour transformation. An immediate consequence of t…
The purpose of this work is to close the local deformation problem of rank two Euclidean submanifolds in codimension two by describing their moduli space of deformations. In the process, we provide an explicit simple representation of these submanifolds, a result of independent interest by its applications. We also det…
Upper and lower bounds for Steklov eigenvalues of submanifolds with fixed boundary.
problem Finding bounds for Steklov eigenvalues of submanifolds with prescribed boundary.
method General upper bound and sharp lower bounds for hypersurfaces of revolution.
result Each eigenvalue is uniquely minimized by the ball for hypersurfaces of revolution with connected boundary.