Paper proves new rigidity results for biconservative hypersurfaces.
problem Rigidity of non-negatively curved compact biconservative hypersurfaces.
method Alternative proofs and new estimates of the squared norm of the shape operator.
result New rigidity results for biconservative hypersurfaces in space forms.
Local rigidity proved for convex hypersurfaces in spaces of constant curvature.
problem Proving rigidity of convex hypersurfaces in spaces of constant curvature.
method Analyzing isometric convex hypersurfaces in spaces of constant curvature of dimension n≥4. result Two convex isometric hypersurfaces are congruent locally around their corresponding under strict convexity isometries.
We provide an explicit description of all rigid hypersurfaces that are equivalent to a Heisenberg sphere. These hypersurfaces are determined by 4 real parameters. The defining equations of the rigid spheres can also be viewed as the complete solution of a non-linear PDE that expresses the vanishing Cartan curvature con…
The paper proves rigidity results for capillary hypersurfaces in hyperbolic space.
problem Understanding the rigidity of capillary hypersurfaces in hyperbolic space.
method Proving a Heintze-Karcher type inequality and applying it to Alexandrov type theorems.
result Rigidity results for capillary hypersurfaces, including totally umbilical and totally geodesic cases.
Paper proves rigidity for certain spacelike hypersurfaces in de Sitter space.
problem Proving rigidity for hypersurfaces with curvature restrictions.
method Analogue to Guan and Shen's theorem for Riemannian space forms.
result Rigidity theorem for locally isometric hypersurfaces in de Sitter space.
In the present paper, we revisit the rigidity of hypersurfaces in Euclidean space. We highlight Darboux equation and give new proof of rigidity of hypersurfaces by energy method and maximal principle.
New rigidity results for specific hypersurfaces in spacetimes.
problem Characterizing maximal hypersurfaces in Generalized Robertson-Walker spacetimes.
method Applying rigidity results under geometric assumptions and the Null Energy Condition.
result New parametric uniqueness and nonexistence results for maximal hypersurfaces.
Paper proves rigidity of convex hypersurfaces in various spaces.
problem Proving the uniqueness of convex hypersurfaces in multidimensional spaces.
method Generalizing Senkin's theorem to higher dimensions and constant curvature spaces.
result Rigidity of convex hypersurfaces in En+1, n≥3. Study on biharmonic hypersurfaces in spheres and space forms, proving rigidity under scalar curvature condition.
problem Characterizing biharmonic hypersurfaces in space forms.
method Proved a rigidity result and established an integral formula for biharmonic hypersurfaces.
result Rigidity result under a scalar curvature condition for biharmonic hypersurfaces in space forms.
Double-well transitions are stiffer than minimal surfaces.
problem Rigidity of double-well phase transitions compared to minimal hypersurfaces.
method Comparison of rigidity properties between double-well phase transitions and minimal hypersurfaces.
result Double-well phase transitions exhibit more rigidity than minimal hypersurfaces.
Since n-dimensional λ-hypersurfaces in the Euclidean space Rn+1 are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of complete λ-hypersurfaces. We give a gap theorem of complete λ-hypersurfaces with po…
Paper shows rigidity of rotationally symmetric minimal hypersurfaces in 5D spaces.
problem Understanding rigidity of rotationally symmetric minimal hypersurfaces in 5D spaces.
method Pointwise hypersurface invariant analysis for minimal hypersurfaces in spaces of constant curvature.
result Rotationally symmetric minimal hypersurfaces in 5D spaces are rigid.
Study proves rigidity of certain hypersurfaces in 5- and 6-manifolds.
problem Proving rigidity of stable minimal hypersurfaces in 5- and 6-manifolds.
method Nonnegative 3-intermediate Ricci curvature combined with uniformly positive k-triRic curvature.
result No complete noncompact stable minimal hypersurface in a closed 5-dimensional manifold with positive sectional curvature.
The paper proves rigidity for hypersurfaces with constant shifted curvature functions in warped product manifolds.
problem Characterizing and proving rigidity for hypersurfaces with constant shifted curvature functions.
method Using integral inequalities and Minkowski-type formulas, the paper derives rigidity theorems in sub-static warped product manifolds.
result The paper provides new characterizations and rigidity results for hypersurfaces with constant shifted curvature functions in warped product manifolds.
The paper explores rigidity of hypersurfaces with constant curvature in Euclidean spaces.
problem Rigidity of hypersurfaces with constant mean and scalar curvature.
method Characterizations and rigidity results under various conditions of Gaussian-Kronecker and r-th mean curvatures. result Rigidity theorems for hypersurfaces in dimensions 4, 5, and 6, and general dimensions under pinching conditions.
Rigidity results for hypersurfaces in warped spacetimes.
problem Characterizing hypersurfaces in specific spacetime geometries.
method Analyzing mean curvature and conformal vectors in warped spacetimes.
result Compact hypersurfaces in certain spacetimes are slices.
Study proves rigidity of minimal hypersurfaces in specific manifolds.
problem Proving rigidity of complete free boundary minimal hypersurfaces.
method Warped θ-bubble method, generalizing capillary surfaces. result No complete two-sided stable free boundary immersions in unit ball of R4. Paper proves rigidity results for anisotropic capillary hypersurfaces.
problem Rigidity of anisotropic capillary hypersurfaces.
method New Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces.
result Uniqueness of solution to anisotropic Orlicz-Christoffel-Minkowski problem.
A minimal hypersurface in a sphere is uniquely determined.
problem Characterizing closed minimal hypersurfaces in spheres.
method Proving strong rigidity of closed minimal hypersurfaces.
result Closed minimal hypersurfaces in spheres are uniquely determined.
New method proves rigidity of minimal hypersurfaces in curved 4-manifolds.
problem Proving rigidity of minimal hypersurfaces in curved 4-manifolds.
method Combining nonnegative 2-intermediate Ricci curvature and strict positivity of scalar curvature, extending Chodosh-Li-Stryker method.
result Rigidity of two-sided free boundary stable minimal hypersurfaces in 4-manifolds with bounded geometry and weakly convex boundary.
The paper explores rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.
problem Rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.
method Characterizations and rigidity investigations for hypersurfaces with constant weighted shifted mean curvatures or ratios.
result Rigidity results for hypersurfaces with constant linear combinations of weighted shifted mean curvatures and radially symmetric shifted mean curvatures.
The study of Laguerre isotropic hypersurfaces with rigidity and isoparametric properties.
problem Characterizing and understanding Laguerre isotropic hypersurfaces.
method Analyzing hypersurfaces with zero Laguerre form and constant eigenvalues of the Laguerre tensor.
result For L-isotropic hypersurfaces, if they are also L-isoparametric, the constant λ must be zero. We prove that every smooth rigid spherical hypersurface in C2 is in fact real-analytic. As an application of this result, it follows that the classification of real-analytic rigid spherical hypersurfaces in C2 found by V. Ezhov and G. Schmalz applies in the smooth case.
The paper bounds eigenvalues of the Jacobi operator and derives rigidity results for CMC hypersurfaces.
problem Bounding the first eigenvalue of the Jacobi operator for CMC hypersurfaces.
method Geometric upper bounds for eigenvalues and rigidity results.
result New rigidity results for the area and length of CMC hypersurfaces.
Survey on extending rigidity theorems to Riemannian manifolds.
problem Extending classical rigidity theorems to Riemannian manifolds.
method Review and extension of existing rigidity theorems.
result Rigidity results for convex hypersurfaces of homogeneous 3-manifolds.
Study finds rigidity of biconservative hypersurfaces in space forms without curvature assumptions.
problem Investigating biconservative hypersurfaces in space forms without scalar curvature assumptions.
method Introduced a novel divergence-free tensor to derive results without curvature assumptions.
result Rigidity results for biconservative hypersurfaces in space forms without scalar curvature assumptions.
The article proves a Poincaré inequality for hypersurfaces and applies it to rigidity results.
problem Proving rigidity results for hypersurfaces under curvature constraints.
method Using a divergence formula for symmetric endomorphisms, the article deduces a Poincaré type inequality and applies it to higher-order mean curvature of hypersurfaces.
result The article proves several rigidity results for complete r-minimal hypersurfaces.
We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is diffeomorphic to a sphere or to a quotient of a sphere by a group action. We also…
New insights on solutions to Allen-Cahn equation with degenerate minimal hypersurfaces.
problem Existence and rigidity of solutions to the Allen-Cahn equation.
method Analysis of degenerate minimal hypersurfaces as limit interfaces.
result New observations and examples of solutions to the Allen-Cahn equation.
Proves rigidity of stable minimal hypersurfaces in low dimensions.
problem Rigidity of stable minimal hypersurfaces in low dimensions.
method Conformal method inspired by Fischer-Colbrie.
result No stable minimal hypersurfaces in positively curved closed Riemannian manifolds when dimension is 5 or less.
The study constructs balanced and rigid curves on specific types of hypersurfaces and complete intersections.
problem Constructing balanced and rigid curves on Calabi-Yau and general-type complete intersections.
method Balanced and rigid curves are constructed using specific hypersurfaces and complete intersections.
result Rigid curves of various genera and balanced rational curves of high degrees are constructed.
Paper proves rigidity of certain minimal hypersurfaces in a 5D sphere.
problem Characterizing closed minimal hypersurfaces in a 5D sphere.
method Analyzes mean curvature and principal curvatures to prove rigidity.
result Closed minimal hypersurfaces with constant 3-mean curvature and distinct principal curvatures are isoparametric.
Proves rigidity in product spaces using index theory.
problem Scalar curvature rigidity in product spaces.
method Fredholm family index theorem.
result Recover corresponding results of Clifford-linear index theory.
New findings on stable minimal hypersurfaces in curved 4-manifolds.
problem Nonexistence of complete stable minimal hypersurfaces in positively curved 4-manifolds.
method Combination of non-negative sectional curvature and strict positivity of scalar curvature.
result Rigidity of complete stable minimal hypersurfaces in 4-manifolds with positive curvature.
Paper characterizes umbilical hypersurfaces using a generalized overdetermined problem.
problem Characterizing umbilical hypersurfaces in space forms.
method Using a Serrin-type partially overdetermined problem with inhomogeneous Robin boundary condition.
result Any contact angle θ ∈ (0, π) can be achieved, generalizing previous results.
Study on rigidity of translating hypersurfaces not in graphical direction.
problem Rigidity of translating hypersurfaces not in graphical direction.
method Proved rigidity results for complete graphical translating hypersurfaces under specific conditions.
result Entire graphical translating surfaces are flat under certain conditions.
Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.
problem Characterize biharmonic hypersurfaces in spheres.
method Prove CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres.
result Supports the conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.
Study on compact biconservative hypersurfaces in de Sitter space.
problem Understanding compact biconservative hypersurfaces in de Sitter space.
method Investigation of hypersurfaces with constant scalar curvature under geometric constraints.
result Extension of rigidity properties of biconservative hypersurfaces.
Compact method proves Brown-York mass positivity and connects to major conjectures.
problem Proving positivity of Brown-York's mass and its connections to conjectures.
method Compact approach to proving mass positivity and exploring connections.
result Proved the positivity of Brown-York's mass and its relation to conjectures.
Totally geodesic hypersurfaces in hyperbolic manifolds are rigid under certain conditions.
problem Conditions under which totally geodesic hypersurfaces in hyperbolic manifolds are rigid.
method Study of homotopy equivalence and sectional curvature properties.
result Conditions for rigidity of totally geodesic hypersurfaces in hyperbolic manifolds.
The paper studies hypersurfaces in 5D space forms with topological and rigidity results.
problem Characterizing and bounding hypersurfaces in 5D space forms.
method Analyzing the Weyl tensor, deriving topological bounds, and using integral inequalities.
result Sharp topological bounds on the Weyl functional for closed, minimal hypersurfaces.
In this paper, we study complete space-like λ-hypersurfaces in the Lorentzian space R1n+1. As the result, we prove some rigidity theorems for these hypersurfaces including the complete space-like self-shrinkers in $\bbr^{n+1}_1$.
Study on real hypersurfaces in products of complex space forms, proving rigidity and nonexistence results.
problem Existence and properties of totally umbilical real hypersurfaces in complex space forms.
method Analyzing shape operators and local product structures in products of complex space forms.
result Nonexistence and rigidity results for totally umbilical real hypersurfaces in products of complex space forms.
Proves rigidity of stable free boundary hypersurfaces in 5-manifolds.
problem Stability and rigidity of free boundary hypersurfaces in 5-manifolds.
method Combining k-tri-Ricci curvature and 3-intermediate Ricci curvature. result Improves rigidity result to 5-dimensions and extends to free boundary case.
Proves Green function rigidity for specific operators and obtains new ADM mass formula.
problem Proving Green function rigidity for specific operators and obtaining new ADM mass formula.
method Positive mass theorem and positive energy theorem for Paneitz operator.
result Obtained new formula for the ADM mass of asymptotically flat hypersurfaces.
The paper generalizes a rigidity theorem for hypersurfaces with constant weighted mean curvature.
problem Classifying hypersurfaces with constant weighted mean curvature.
method Using polynomial volume growth and specific curvature conditions, the authors prove rigidity theorems.
result Hypersurfaces with constant weighted mean curvature must be either a hyperplane or a generalized cylinder under certain conditions.
Sharp focal radius estimate for hypersurfaces in manifolds with positive curvature.
problem Estimating the focal radius of hypersurfaces in manifolds with positive curvature.
method Proved a sharp Clifford-threshold focal-radius estimate and rigidity under specific curvature conditions.
result Any closed two-sided immersion satisfies a focal radius estimate of π/4, with equality case rigid.
The notion of ideal immersions was introduced by the author in 1990s. Roughly speaking, an ideal immersion of a Riemannian manifold into a real space form is a nice isometric immersion which produces the least possible amount of tension from the ambient space at each point. In this paper, we classify all ideal hypersur…