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…
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48 results for “Wintgen inequalities”
The study improves Wintgen inequalities for submanifolds in specific geometric spaces.
problem Improving Wintgen inequalities for submanifolds in various geometric spaces.
method Analyzing submanifolds in conformally flat manifolds and deriving inequalities for different types of spaces.
result Derived inequalities for submanifolds in various geometric spaces, including Riemannian manifolds of quasi-constant curvature and warped products.
Main interest of the present paper is to obtain the generalized Wintgen inequality for Legendrian submanifolds in almost Kenmotsu manifolds.
The paper classifies submanifolds in space forms that meet curvature conditions.
problem Characterizing submanifolds in space forms that meet specific curvature conditions.
method Local parametric classification based on the DDVV inequality.
result Local parametric classification of Wintgen ideal submanifolds.
A submanifold in space forms satisfies the well-known DDVV inequality due to De Smet, Dillen, Verstraelen and Vrancken. The submanifold attaining equality in the DDVV inequality at every point is called Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are studied in this paper using…
On Wintgen ideal surfacesmath.DG
Wintgen proved in [P. Wintgen, Sur l'inégalité de Chen-Willmore, C. R. Acad. Sci. Paris, 288 (1979), 993--995] that the Gauss curvature and the normal curvature of a surface in the Euclidean 4-space satisfy where is the squared mean curvature. A surface in $\E4$ is called …
Wintgen ideal submanifolds in space forms are those ones attaining equality at every point in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the normal scalar curvature. This property is conformal invariant; hence we study them in the framework of Moebius geometry, and restrict…
Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. Using the framework of Moebius geometry, we show that in the codimension two case, the mean curvature sphere of t…
Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. They are Moebius invariant objects. The mean curvature sphere defines a conformal Gauss map into a Grassmann mani…
A submanifold in a real space form attaining equality in the DDVV inequality at every point is called a Wintgen ideal submanifold. They are invariant objects under the Moebius transformations. In this paper, we classify those Wintgen ideal submanifolds of dimension m>3 which are Moebius homogeneous. There are three cla…
The paper derives optimal inequalities for bi-slant submanifolds in metallic Riemannian space forms.
problem Understanding geometric properties of bi-slant submanifolds in metallic Riemannian product space forms.
method Deriving generalized Wintgen inequality, optimal inequalities involving δ-invariants, Ricci curvature, shape operator invariants, and generalized normalized δ-Casorati curvatures.
result Established optimal inequalities for bi-slant submanifolds in metallic Riemannian product space forms.
Study curvatures of submanifolds in space forms with topological obstructions.
problem Investigate intrinsic curvatures and their implications for submanifolds in space forms.
method Derive inequalities involving mean curvature and normal scalar curvature, derive topological obstructions.
result Prove existence of compact 3-dimensional minimal Wintgen ideal submanifolds in even-dimensional spheres.
Wintgen ideal surfaces in E^4 form an important family of surfaces, namely surfaces with circular ellipse of curvature. Obviously, Wintgen ideal surfaces satisfy the pointwise equality K+K_N=H^2. In the present study we consider the Wintgen ideal surfaces in n-dimensional Euclidean space E^4. We have shown that Wintgen…
Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvaturemath.DG
Every biharmonic Wintgen ideal submanifold in a Riemannian manifold of constant sectional curvature is either minimal or has constant mean curvature.
problem Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
method Show that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of nonpositive constant sectional curvature is minimal and that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of positive constant sectional curvature has constant mean curvature.
result Partial affirmative answers to Chen's conjecture, generalized Chen's conjecture in hyperbolic spaces, and Balmuş-Montaldo-Oniciuc conjecture in spheres within the class of Wintgen ideal submanifolds.
Let be a space-like surface immersed in a 4-dimensional pseudo-Riemannian space form with constant sectional curvature and index two. In the first part of this article, we prove that the Gauss curvature , the normal curvature , and mean curvature vector of satisfy the general inequali…
Study Wintgen ideal submanifolds in curved spaces with specific curvature conditions.
problem Characterize Wintgen ideal submanifolds in curved spaces under certain curvature constraints.
method Analyze submanifolds in real space forms R^{n+m}(k) with specific curvature conditions.
result Identify conditions under which submanifolds satisfy given pseudo-symmetry type curvature conditions.
Superconformal surfaces in Euclidean space are the ones for which the ellipse of curvature at any point is a nondegenerate circle. They can be characterized as the surfaces for which a well-known pointwise inequality relating the intrinsic Gauss curvature with the extrinsic normal and mean curvatures, due to Wintgen (\…
The paper studies geometric properties of soliton surfaces using an extended Darboux frame field.
problem Geometric analysis of soliton surfaces associated with the Betchov-Da Rios equation.
method Derivative formulas of an extended Darboux frame field, geometric invariants, curvature calculations.
result Construction of curvature ellipse and Wintgen ideal soliton surfaces.
The paper examines soliton surfaces using a parallel transport frame field in 4D space.
problem Geometric properties of soliton surfaces associated with the Betchov-Da Rios equation.
method Parallel transport frame field approach in four-dimensional Euclidean space.
result Characterization of soliton surfaces as flat, minimal, semi-umbilic, or Wintgen ideal.
The isoperimetric inequalitymath.DG
The isoperimetric inequality and related inequalities are explored.
problem Proving the isoperimetric inequality and related inequalities.
method Discussing classical and recent proofs.
result Various proofs of the isoperimetric inequality and Sobolev inequality.
New proof of Willmore inequality using geometric divergence inequality.
problem Proving the Willmore inequality for bounded domains.
method Using a parametric geometric inequality derived from a divergence form geometric differential inequality.
result New proofs of quantitative Willmore-type and weighted Minkowski inequalities.
Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.
problem Deriving new inequalities on manifolds and convex hypersurfaces.
method Using Fourier theory and geometric implications of Poincare-type inequalities.
result Sharp Minkowski-type inequalities, including stability and Alexandrov-Fenchel inequalities.
The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.
problem Proving -Sobolev and Nash inequalities on Finsler metric measure manifolds.
method Global -Poincaré inequality, -Sobolev inequality, Nash inequality derivation.
result Established global optimal -Sobolev inequality with a sharp constant.
New inequality on sphere generalizes circle inequality.
problem Generalizing circle inequality to sphere.
method Develops a new inequality on the sphere that incorporates mass center deviation.
result Improves Aubin's inequality and Onofri's inequality.
Paper proves anisotropic Minkowski inequality and related inequalities.
problem Proving anisotropic Minkowski inequality and related inequalities.
method Utilizes a nonlinear potential theoretic approach.
result Sharp anisotropic Minkowski inequality and related inequalities proved.
Explains geometric inequalities for minimal hypersurfaces.
problem Geometric inequalities for minimal hypersurfaces.
method Expository discussion of known inequalities.
result Discussion of classical inequalities for minimal hypersurfaces.
The paper finds new inequalities for convex polygons.
problem Finding precise inequalities for convex polygons.
method Analytic isoperimetric inequalities based on Schur convex functions, followed by Bonnesen-style and inverse Bonnesen-style inequalities.
result Sharp discrete isoperimetric inequalities for planar convex polygons.
The study improves Bochner inequality on Finsler manifolds to derive important inequalities.
problem Improving Bochner inequality on Finsler manifolds to derive new inequalities.
method Using improved Bochner inequality and its integrated form, the study derives a sharp Poincaré-Lichnerowicz inequality, a new proof for logarithmic Sobolev inequality, and an estimate of geodesic ball volumes.
result Derivation of new inequalities and estimates on Finsler manifolds.
The paper proves various inequalities on gradient shrinking Ricci solitons.
problem Understanding geometric inequalities on gradient shrinking Ricci solitons.
method Proving multiple inequalities equivalent on complete gradient shrinking Ricci solitons.
result Various inequalities (Sobolev, logarithmic Sobolev, Schrödinger, etc.) are equivalent on gradient shrinking Ricci solitons.
Sharp inequality found on three-balls for fourth order Sobolev traces.
problem Fourth order Sobolev trace inequality on three-balls.
method Established through equivalence to a third order Sobolev inequality on two-spheres.
result Sharp fourth order Sobolev trace inequality on three-balls.
Sharp inequalities for star bodies in 2D space.
problem Understanding star bodies in 2D space.
method Sharp inequalities for star bodies in .
result New inequalities and proofs for star bodies.
The paper develops inequalities for log-concave functions and related surface areas.
problem Understanding log-concave functions and their inequalities.
method Establishing new inequalities through f-divergences and functional affine surface areas.
result New inequalities on functional affine surface area and bounds for Kullback-Leibler divergence.
Study on functional inequalities on simple edge spaces.
problem Whether classical functional inequalities hold in simple edge spaces.
method Analyzing Sobolev and Poincaré inequalities, proving optimality of Sobolev constant.
result Optimality result concerning the B-constant of the Sobolev inequality.
Proves inequalities on curved spaces with positive curvature.
problem Proving inequalities on manifolds with nonnegative Ricci curvature.
method Analyzes manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Proves Heisenberg-Pauli-Weyl, Hardy-Sobolev, and Caffarelli-Kohn-Nirenberg inequalities.
The paper derives inequalities on Finsler manifolds, influenced by their curvatures.
problem Deriving inequalities on Finsler manifolds.
method Local and global geometric inequalities on Riemannian and Finsler manifolds.
result Generalized Caffarelli-Kohn-Nirenberg and Hardy type inequalities on Finsler manifolds.
Paper refines Talagrand inequality on Euclidean spaces.
problem Improving Talagrand inequality for Euclidean spaces.
method Symmetrization and alternative proof methods.
result Several refined functional inequalities derived.
New inequalities for convex curves with multiple geometric factors.
problem Establishing inequalities for convex curves with multiple geometric factors.
method Parametric isoperimetric-type inequalities for closed convex curves with parameter conditions and equality conditions.
result Derived new inequalities and improved versions of existing inequalities.
Alternative proofs for various inequalities on Riemannian manifolds.
problem Various functional inequalities on Riemannian manifolds.
method Generic functional inequality, Riccati pairs, solving Riccati-type ODE.
result Alternative proofs for multiple inequalities, including Hardy-type and Caccioppoli inequalities.
The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.
problem Functional and geometric inequalities on Finsler measure spaces.
method Local uniform Poincaré and Sobolev inequalities, mean value inequality, Harnack inequalities, and gradient estimates.
result Global gradient estimates for positive harmonic functions on Finsler measure spaces.
The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.
problem Proving a Harnack inequality for positive solutions to heat equations on Finsler metric measure manifolds.
method Volume comparison theorem, weighted Poincaré inequality, local uniform Sobolev inequality, mean value inequalities.
result Derives a Harnack inequality for positive solutions to heat equations.
Extends Riemannian geometry inequalities with sharper estimates.
problem Deriving new inequalities on Riemannian manifolds.
method Investigates advanced Hardy and Rellich-type inequalities on complete noncompact manifolds with weight functions.
result Provides sharper estimates conforming to the geometry and structure of the manifold.
The paper explores how information geometry impacts classical CR inequalities.
problem Deriving and generalizing CR inequalities using information geometry.
method Examining Eguchi's theory and applying Amari-Nagoaka's theory to KL-divergence, and then extending to other divergences.
result Generalized CR inequalities derived from various divergences.
The paper proves inequalities for hypersurfaces in weighted manifolds.
problem Willmore-type inequalities for closed hypersurfaces in weighted manifolds.
method Analyzes weighted manifolds with nonnegative Bakry-Émery Ricci curvature, proving sharp inequalities and characterizing equality cases.
result Derives sharp Willmore-type and Willmore-like inequalities in steady and shrinking gradient Ricci solitons.
Extends Gromov's optimal systolic inequality to manifolds with specific cohomology properties.
problem Finding optimal systolic inequalities for manifolds with complex cohomology structures.
method Extends Gromov's inequality to manifolds with fundamental cohomology classes as cup products of 2-dimensional classes.
result Provides an optimal systolic inequality for a new class of manifolds.
Logarithmic Sobolev, Poincaré and Beckner Inequalities on Hyperbolic Spaces and Riemannian Manifoldsmath.AP
Study inequalities on hyperbolic spaces and Riemannian manifolds using symmetrization and heat semigroup.
problem Investigate functional and geometric inequalities on hyperbolic spaces and Riemannian manifolds.
method Employ symmetrization and semigroup approach based on sharp estimates for heat semigroup.
result Developed robust inequalities and methods relying on geometric and isoperimetric properties.
Fenchel-Willmore and Sobolev-type inequalities for submanifolds in non-negatively curved manifoldsmath.DG
Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.
problem Developing inequalities for submanifolds in curved spaces.
method Connecting Fenchel-Willmore and logarithmic Sobolev inequalities for mean-convex submanifolds.
result Established extensions of Fenchel-Willmore inequality and derived new Sobolev-type inequalities.
Sharp inequalities on curved spaces with bounded curvature.
problem Establishing inequalities on curved spaces with curvature constraints.
method Using Sobolev and Moser-Trudinger inequalities on noncompact Riemannian manifolds with Ricci curvature bounded below.
result Best constants for inequalities on curved spaces with curvature constraints.