Paper solves Dirichlet problem for p-convex hypersurfaces with curvature constraints.
problem Solving the Dirichlet problem for p-convex hypersurfaces with prescribed curvature. method Proved existence of a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition, obtained an interior curvature estimate.
result Existence of a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition.
Paper estimates curvature of convex hypersurfaces with prescribed curvature.
problem Estimating curvature of p-convex hypersurfaces with prescribed curvature. method Establishes curvature estimates for p-convex hypersurfaces in Rn+1 with p≥2n. result Proves existence of star-shaped hypersurface of prescribed curvature and interior C2 estimates. Three results in p-convex geometry are established. First is the analogue of the Levi problem in several complex variables, namely: local p-convexity implies global p-convexity. The second asserts that the support of a minimal p-dimensional current is contained in the p-hull of the boundary union with the "core" of the…
In this paper, we establish various L2-estimates for the exterior differential operator on p-convex Riemannian manifolds in the sense of Harvey and Lawson. As geometric applications, we prove vanishing and finiteness results for the de Rham cohomology groups.
In view of A. Andreotti and H. Grauert's vanishing theorem for q-complete domains in C^n, (Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193--259,) we re-prove a vanishing result by J.-P. Sha, (p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437--447,)…
Smoothly bounded domains have special functions that are plurisubharmonic.
problem Finding smooth functions that are plurisubharmonic on bounded domains.
method Proving existence of smooth defining functions that are p-plurisubharmonic. result Smooth domains with smooth p-convex boundaries admit smooth defining functions that are p-plurisubharmonic. Recently the authors have explored new concepts of plurisubharmonicity and pseudoconvexity, with much of the attendant analysis, in the context of calibrated manifolds. Here a much broader extension is made. This development covers a wide variety of geometric situations, including, for example, Lagrangian plurisubhamon…
Paper constructs L2 estimates for flat vector bundles and generalizes Prékopa's theorem.
problem Constructing L2 estimates for flat vector bundles. method Using Hörmander's L2-estimate for the operator d on a flat vector bundle over a p-convex Riemannian manifold. result Generalizes Prékopa's theorem in convex analysis.
We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices, with bdy(F) contined in the set {f=0}. We establish the existence and uniquenes…
The paper classifies Codazzi hypersurfaces and characterizes minimal hypersurfaces in Nil^4.
problem Classifying hypersurfaces in Nil^4.
method Using Lie group structure and Codazzi conditions.
result Characterization and classification of minimal hypersurfaces in Nil^4.
Classification of hypersurfaces in homogeneous spaces with specific properties.
problem Classifying hypersurfaces in Riemannian homogeneous spaces with additional assumptions.
method Analyzing hypersurfaces under various conditions in homogeneous spaces CP3. result All extrinsically homogeneous hypersurfaces are classified in all homogeneous CP3 spaces. Study on biconservative hypersurfaces with constant scalar curvature in space forms.
problem Characterize biconservative hypersurfaces with constant scalar curvature in space forms.
method Analyzing biconservative hypersurfaces in space forms Nn+1(c), proving properties and finding specific examples. result Proves that biconservative hypersurfaces with constant scalar curvature in N4(c) have constant mean curvature, and in N5(c), they are either rotational or constant mean curvature. The paper classifies various types of hypersurfaces in a product space.
problem Classifying hypersurfaces in a specific product space.
method Analyzing hypersurfaces with constant curvatures, product angle functions, and additional conditions.
result Different types of hypersurfaces are classified based on their properties.
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. The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in Rn for any n≥3. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immer…
Classifies hypersurfaces with constant isotropic curvature in space forms.
problem Identifying hypersurfaces with constant isotropic curvature in space forms.
method Analyzing complete orientable hypersurfaces and their properties.
result Hypersurfaces have constant mean curvature only if they are isoparametric, and are minimal under specific conditions.
We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area m…
New global section found for geodesic flows on convex hypersurfaces.
problem Finding global sections for geodesic flows on convex hypersurfaces.
method Constructing a global hypersurface of section with an isometric involution.
result Generalized Birkhoff annulus to higher dimensions.
New tensors capture intrinsic embedding data of conformal hypersurfaces.
problem Classifying hypersurface invariants in conformal manifolds.
method Constructing curvatures and conformal fundamental forms.
result Finite family of tensors captures extrinsic embedding data.
The study characterizes canal hypersurfaces in Euclidean spaces and their curvature properties.
problem Characterizing canal hypersurfaces in Euclidean spaces.
method Analyzing canal hypersurfaces in Euclidean n-space, focusing on E4, computing curvature properties, and proving specific cases.
result Flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones, and minimal canal hypersurfaces are only generalized catenoids.
Survey on Dupin hypersurfaces and isoparametric hypersurfaces in spheres.
problem Understanding the relationship between compact proper Dupin hypersurfaces and isoparametric hypersurfaces in spheres.
method Survey of existing results and related developments.
result Progress on Cecil and Ryan's conjecture on compact proper Dupin hypersurfaces.
In this paper we introduce radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric. The study of these hypersurfaces is motivated by the fact that for indefinite almost Hermitian manifolds this class of lightlike hypersurfaces does not exist. We also establish that radical transversal…
Study on Dirac operators on lightlike hypersurfaces in 4D Lorentzian manifolds.
problem Investigating Dirac operators on hypersurfaces with degenerate metrics.
method Spinorial Gauss formula, investigation of Dirac operator, relation with Riemannian curvatures.
result Established relation between Dirac operators and curvatures of the manifold and hypersurface.
In this paper we show that a Dupin hypersurface with constant Möbius curvatures is Möbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat L…
New compact mean convex hypersurfaces found for positive λ.
problem Finding compact embedded hypersurfaces for positive λ.
method Constructing compact mean convex hypersurfaces diffeomorphic to spheres.
result No compact convex embedded λ-hypersurfaces except a round sphere for λ > 0.
In this paper, we study generalized constant ratio (GCR) hypersurfaces in Euclidean spaces. We mainly focus on the hypersurfaces in E4. First, we deal with δ(2)-ideal GCR hypersurfaces. Then, we study on hypersurfaces with constant (first) mean curvature. Finally, we obtain the complete classification of G…
Study on minimal hypersurfaces in a special normed space.
problem Characterizing minimal hypersurfaces in a specific normed space.
method Investigate translation and separable minimal hypersurfaces.
result New insights into the properties of minimal hypersurfaces.
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.
Classifies and describes hypersurfaces in Siklos spacetimes.
problem Characterizing hypersurfaces in Siklos spacetimes.
method Classification and description of totally geodesic and parallel hypersurfaces.
result A large class of minimal hypersurfaces is described.
Proves convexity of certain hypersurfaces with negative λ.
problem Understanding convexity of hypersurfaces with specific λ values.
method Analyzes mean convex hypersurfaces and proves convexity for λ ≤ 0.
result Closed n-dimensional mean convex λ-hypersurfaces are convex if λ≤0. Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.
problem Exploring hypersurfaces with constant Gauss-Kronecker curvature.
method Solving ODE for generating curves and analyzing geometric properties.
result Discovery of non-compact rotational hypersurfaces with negative Gauss-Kronecker curvature and finite volume.
Minimal hypersurfaces are the only H-tensional in 4D space forms.
problem Classifying H-tensional hypersurfaces in 4D space forms. method Investigation of H-tensional hypersurfaces in 4-dimensional space forms of constant sectional curvature. result Minimal hypersurfaces are the only H-tensional hypersurfaces in 4D space forms. Study shows constant curvature convex hypersurfaces on hyperboloids are parts of hyperboloids.
problem Characterizing convex hypersurfaces with constant curvature on hyperboloids.
method Analyzing hypersurfaces with constant higher order mean curvature and constant boundary angle.
result Hypersurfaces with constant curvature on hyperboloids are parts of hyperboloids.
Paper characterizes a special hypersurface in 5D sphere.
problem Characterizing minimal hypersurfaces in S5. method Analyzes hypersurfaces satisfying a specific curvature condition.
result Closed minimal hypersurfaces in S5 satisfying a certain curvature condition are either totally geodesic or congruent to the Cartan minimal hypersurface. The problem of determining the {\it Bonnet hypersurfaces in} Rn+1, for n>1, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/o…
The paper classifies isoparametric hypersurfaces in conic Finsler spaces.
problem Identifying new isoparametric hypersurfaces in conic Finsler spaces.
method Introduced isoparametric functions and hypersurfaces in conic Finsler spaces, classified them in specific spaces.
result Found additional isoparametric hypersurfaces in conic Minkowski spaces, such as helicoids.
Paper introduces new hypersurface types on statistical manifolds.
problem Understanding new hypersurface types on statistical manifolds.
method Introducing and analyzing new classes of hypersurfaces.
result Obtained properties and relations on specific hypersurface types.
Study on properties of tangential hypersurfaces in product-like manifolds.
problem Investigating properties of tangential hypersurfaces in product-like manifolds.
method Analyzing basic properties and computing curvature tensor relations.
result Computed relations involving the Riemannian curvature tensor of tangential hypersurfaces.
In this paper, geometric characterizations of conformally flat and radially flat hypersurfaces in Sn×R and Hn×R are given by means of their extrinsic geometry. Under suitable conditions on the shape operator, we classify conformally flat hypersurfaces in terms of …
Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.
problem Characterizing and constructing hypersurfaces with specific curvature properties in Finsler geometry.
method Analyzing isoparametric and constant-curvature hypersurfaces on Randers manifolds.
result Construction of a conformally flat Randers manifold with nonisoparametric hyperplanes of constant curvatures.
Similar to the definition of Dupin hypersurface in Riemannian space forms, we define the spacelike Dupin hypersurface in Lorentzian space forms. As conformal invariant objects, spacelike Dupin hypersurfaces are studied in this paper using the framework of conformal geometry. Further we classify the spacelike Dupin hype…
Classifies isoparametric hypersurfaces in 3D manifolds.
problem Identifying isoparametric hypersurfaces in specific 3D manifolds.
method By proving constant angle and principal curvatures.
result Established classification of hypersurfaces.
Study on hypersurfaces in pseudo-Euclidean space with constant curvature or rotational properties.
problem Characterizing hypersurfaces in pseudo-Euclidean space.
method Defined and studied warped product hypersurfaces with constant sectional curvature or rotational properties.
result Hypersurfaces in pseudo-Euclidean space either have constant curvature or are contained in rotational hypersurfaces.
Revisits and applies a formula for hypersurface Euler characteristics.
problem Understanding isoparametric hypersurfaces in space forms.
method Re-proves Allendoerfer-Weil's formula and applies it to isoparametric hypersurfaces.
result New insights into isoparametric hypersurfaces.
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 ruled real hypersurfaces in nonflat complex space forms with constant norm shape operators.
problem Proving nonexistence and characterizing ruled hypersurfaces in nonflat complex space forms.
method Analyzing shape operators with constant squared norm and using biharmonic properties.
result Biharmonic ruled real hypersurfaces are minimal, leading to their classification.
There are three types of hypersurfaces in a pseudoconformal space C^n_1 of Lorentzian signature: spacelike, timelike, and lightlike. These three types of hypersurfaces are considered in parallel. Spacelike hypersurfaces are endowed with a proper conformal structure, and timelike hypersurfaces are endowed with a conform…
Study examines tangential real hypersurfaces on Hermite-like manifolds.
problem Characterizing real hypersurfaces on Hermite-like manifolds.
method Introduced tangential real hypersurfaces and derived main identities.
result Discussed contact metric structures in K-contact and cosymplectic cases.