Paper derives Chen's inequality for a specific type of submanifold in a generalized space form.
problem Deriving Chen's inequality for C-totally real submanifolds in a generalized (κ,μ)-space form. method Using intrinsic and extrinsic invariants, the paper derives Chen's inequality involving scalar curvature, sectional curvatures, and squared mean curvature.
result Inequalities between squared mean curvature and Ricci curvature and between squared mean curvature and k-Ricci curvature are obtained. Study finds specific complex projective plane hypersurfaces hitting equality in curvature inequality.
problem Determining hypersurfaces in complex projective planes hitting equality in curvature inequality.
method Analyzing non-Hopf hypersurfaces with constant mean curvature in the complex projective plane.
result Identifies specific hypersurfaces achieving equality in a curvature inequality.
Extends Chen's work to Bochner Kahler manifolds.
problem Relationship between Ricci curvature and mean curvature vectors.
method Generalizes Chen's relationship to Bochner Kahler manifolds.
result Establishes new relationship for Bochner Kahler manifolds.
For submanifolds tangent to the structure vector field in cosymplectic space forms, we establish a basic inequality between the main intrinsic invariants of the submanifold, namely its sectional curvature and scalar curvature on one side; and its main extrinsic invariant, namely squared mean curvature on the other side…
Proves a pinching theorem for self-shrinkers of mean curvature flow.
problem Pinch on the squared norm of the second fundamental form of self-shrinkers.
method Proves a theorem for n−dimensional closed self-shrinkers. result Closed self-shrinkers must be the standard sphere if pinched.
Study of equifocal hypersurfaces in symmetric spaces and backward mean curvature flow.
problem Understanding the behavior of equifocal hypersurfaces under backward mean curvature flow.
method Derive formulas for mean curvature and shape operator, analyze long-time existence and evolution of flow.
result Generalize results for isoparametric hypersurfaces in the sphere to symmetric spaces of compact type.
The paper studies how submanifolds of a sphere evolve over time.
problem Evolution of pinched submanifolds in the sphere.
method High codimension mean curvature flow with pinching conditions.
result Convergence to a round point or totally geodesic sphere under pinching conditions.
The paper extends Chen's inequalities to Bochner Kaehler manifolds.
problem Generalizing Chen's inequalities to a new class of manifolds.
method Extending Chen's inequalities to Bochner Kaehler manifolds and CR-warped product submanifolds.
result Established an inequality for scalar curvature in CR-warped product submanifolds of Bochner Kaehler manifolds.
We consider a complete biharmonic hypersurface with nowhere zero mean curvature vector field φ:(Mm,g)→(Sm+1,h) in a sphere. If the squared norm of the second fundamental form B is bounded from above by m, and ∫MH−pdvg<∞, for some 0<p<∞, then the mean curvature is constant.
We study perturbations of the Allen-Cahn equation and prove the convergence to forced mean curvature flow in the sharp interface limit. We allow for perturbations that are square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and ap…
In this paper we prove that an embedded and simply connected constant mean curvature surface with curvature large at a point contains a multi-valued graph around that point on the scale of ∣A∣2, where ∣A∣2 is the norm squared of the second fundamental form. This generalizes Colding and Minicozzi's result for mini…
Study on Hawking and Bartnik masses for specific surfaces.
problem Analyzing the positivity and bounds of Hawking and Bartnik masses for constant mean curvature surfaces.
method Intrinsic conditions and estimates for the masses of constant mean curvature surfaces.
result Positivity and estimates of Hawking and Bartnik masses for surfaces with nonnegative scalar curvature.
The paper proves conditions for stable constant mean curvature surfaces in specific manifolds.
problem Conditions for stable constant mean curvature surfaces in warped product manifolds.
method Analyzes de Sitter-Schwarzschild and Reissner-Nordstrom manifolds, then generalizes to a broader class of three-dimensional warped product manifolds.
result Stable, compact surfaces in specific manifolds are embedded topological spheres.
The paper calculates S-curvature for specific Finsler metrics.
problem Calculating curvature properties of homogeneous Finsler spaces with (α,β)-metrics. method Proved existence of invariant vector fields and derived explicit formula for S-curvature. result Explicit formula for S-curvature of (α,β)-metrics is established. Proves curvature bounds for submanifolds in negatively curved spaces.
problem Bounding the total squared mean curvature of submanifolds in negatively curved spaces.
method Analyzes the eigenvalue and curvature of submanifolds homotopic to a point in a negatively curved manifold.
result Establishes a new inequality relating the first eigenvalue and the total squared mean curvature of submanifolds.
In this note we investigate the behaviour at finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M^m_t\hookrightarrow (N^{m+n}, h). We show that they are characterized by the blow-up of a trace A = H \cdot II of the square of the second fundamental form.
The Wintgen inequality (1979) is a sharp geometric inequality for surfaces in the 4-dimensional Euclidean space involving the Gauss curvature (intrinsic invariant) and the normal curvature and squared mean curvature (extrinsic invariants), respectively. In the present paper we obtain a Wintgen inequality for statistica…
Classically, isothermic surfaces are characterized as those surfaces which are "divisible into infinitesimal squares by their curvature lines". This characterization is the direct analogue to the definition of discrete isothermic nets. In order to understand the relations between the discrete and the smooth theory bett…
Study on surface configurations with curvature and elasticity.
problem Equilibrium configurations of surfaces with curvature and elasticity.
method Investigates the Euler-Helfrich functional, focusing on axially symmetric surfaces and their variational problems.
result Critical surfaces for the Euler-Helfrich functional, if axially symmetric, satisfy a simpler second order variational problem.
New gap theorem for hypersurfaces with constant mean curvature in space forms.
problem Finding a gap for hypersurfaces with constant mean curvature and scalar curvature.
method Analyzing the relationship between the squared length of the second fundamental form and the mean curvature.
result Proving a new gap theorem for hypersurfaces with constant mean curvature and constant scalar curvature.
We derive, for the square operator of Yau, an analogue of the Omori-Yau maximum principle for the Laplacian. We then apply it to obtain nonexistence results concerning complete spacelike hypersurfaces with constant higher order mean curvature in the Steady State space.
We show that for an isometric immersion of a complete Riemannian manifold into a Riemannian manifold with non-positive curvature, the norm of the mean curvature vector field is square integrable, then it is minimal. This is a partial affirmative answer of the B. Y. Chen's conjecture.
Formula found for surfaces in Sol_3, leading to gap results.
problem Finding constant mean curvature surfaces in Sol_3.
method Computed Laplacian of squared norm of second fundamental form, used Simons type formula.
result Gap results for compact constant mean curvature surfaces.
Study curvature inequalities for real hypersurfaces in complex space forms.
problem Understanding curvature properties of real hypersurfaces in complex space forms.
method Established an inequality relating Ricci curvature, mean curvature, and normal curvature; classified hypersurfaces achieving equality.
result Classified real hypersurfaces in two-dimensional non-flat complex space forms achieving equality in curvature inequality.
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 R4 with constant squared norm of the second fundamental form. Paper proves non-positivity of eigenvalue for Schrödinger operator on hypersurfaces.
problem Eigenvalue non-positivity of Schrödinger operator on hypersurfaces.
method Penalizes (r+1)-mean curvature, establishes non-positivity. result Characterizes the sphere when eigenvalue is null.
We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequa…
In this note we prove that every two-dimensional entire Willmore graph in R3 with square integrable mean curvature is a plane.
Study CR-statistical submanifolds in holomorphic statistical spaces.
problem Characterize CR-statistical submanifolds and their properties.
method Optimization technique to relate Ricci curvature and mean curvature.
result Established relationship between Ricci curvature and mean curvature.
The paper studies the blow-up of conformal mean curvature flow in higher codimension.
problem Analyzing the blow-up behavior of conformal mean curvature flow.
method Introduced and studied conformal mean curvature flow, derived blow-up theorem and evolution formulas.
result Maximum of the square norm of the second fundamental form tends to infinity in finite time.
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 higher dimensions, we classify hypersurfaces with constant mean and scalar curvatures.
problem Classifying hypersurfaces with specific curvature conditions in Euclidean spaces.
method Using principal curvature theorem and a formula for the Laplacian of the squared norm of the second fundamental form.
result Characterization of hypersurfaces with constant mean and scalar curvatures in R5. Paper establishes inequality for submanifolds in real space forms with semi-symmetric non-metric connection.
problem Deriving a sharp lower bound for Ricci curvature of submanifolds.
method Using semi-symmetric non-metric connection, derive a lower bound for Ricci curvature in terms of mean curvature vector and second fundamental form.
result Established Hineva inequality for submanifolds with semi-symmetric non-metric connection.
In this paper we introduce slant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We obtain some results on slant Riemannian submersions of a cosymplectic manifolds. We also give examples and inequalities between the scalar curvature and squared mean curvature of fibres of such slant submer…
Study on surfaces minimizing elastic energy with boundary constraints.
problem Finding stable configurations of surfaces with elastic boundaries and surface energy.
method Investigation of critical surfaces with mean curvature and spontaneous curvature, coupled to boundary elastic energy.
result Characterization and minimization of surface energy for specific topological shapes.
We establish some inequalities of Chen's type between certain intrinsic invariants (involving sectional, Ricci and scalar curvatures) and the squared mean curvature of submanifolds tangent to the structure vector fields of a generalized S-space-form and we discuss the equality cases of them. We apply the obtained resul…
In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in Euclidean three-space is at least 2π^2. We prove this conjecture using the min-max theory of minimal surfaces.
The paper examines spectral properties and rigidity of self-expanding solutions in mean curvature flows.
problem Spectral properties and rigidity of self-expanding solutions in mean curvature flows.
method Analysis of the spectrum of the drifted Laplacian and weighted stability operator.
result The Euclidian subspace through the origin is the unique self-expander where the bottom of the spectrum of the drifted Laplacian is achieved.
Paper examines conditions for singular square metrics to have constant curvature.
problem Conditions for constant curvature in singular square metrics.
method Analyzes Finsler metrics, introduces singular square metrics, provides necessary and sufficient conditions.
result Necessary and sufficient conditions for constant Ricci or flag curvature in singular square metrics.
Estimates mean curvature in warped product spaces, generalizing known results.
problem Estimating mean curvature in warped product spaces.
method Using a graph submanifold and parallel mean curvature, the mean curvature is estimated using a Heinz type inequality.
result A Heinz type estimation of the mean curvature in compact domains of warped product spaces.
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.
The paper proves conditions for a 4D minimal surface to be isoparametric.
problem Conditions for a 4D minimal surface to be isoparametric.
method Analyzes the properties of a closed immersed minimal hypersurface in S5 with specific curvature conditions. result If conditions on curvature are met, the surface is isoparametric.
For constant mean curvature surfaces of class C2 immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classi…
The paper classifies certain types of Lagrangian translators and self-expanders in complex 2-space.
problem Classifying Lagrangian translators and self-expanders in complex 2-space.
method Using a new Omori-Yau type maximum principle proved by Chen and Qiu.
result Several classification results of 2-dimensional complete Lagrangian translators and self-expanders.
We consider the reduced Allen-Cahn action functional, which appears as the sharp interface limit of the Allen-Cahn action functional and can be understood as a formal action functional for a stochastically perturbed mean curvature flow. For suitable evolutions of generalized hypersurfaces this functional consists of th…
Proves inequalities for convex hypersurfaces in Euclidean space.
problem Geometric inequalities for hypersurfaces in Euclidean space.
method Proves two weighted geometric inequalities involving area, volume, and mean curvature.
result Examples of convex surfaces with ratios less than the round sphere.
In this paper we establish a general inequality involving the Laplacian of the warping functions and the squared mean curvature of any doubly warped product isometrically immersed in a Riemannian manifold. Moreover, we obtain some geometric inequalities for C-totally real doubly warped product submanifolds of generaliz…
The paper studies how submanifolds in Gaussian space behave under mean curvature flow, showing they typically blow up.
problem Behavior of submanifolds in Gaussian space under mean curvature flow.
method Analysis of mean curvature flow in the standard Gaussian metric space.
result Submanifolds in Gaussian space with non-zero square norm of position vector blow up under mean curvature flow.