We give a local parametric description of all holomorphic hypersurfaces in complex Euclidean and projective spaces with constant index of relative nullity, together with applications. This is a complex analogue to the parametrization for real hypersurfaces in Euclidean space known as the Gauss parametrization.
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48 results for “parametric hypersurfaces”
Paper develops an algorithm for finding lines of curvature on 4D hypersurfaces.
problem Finding lines of curvature on parametric hypersurfaces in Euclidean 4-space.
method Develops an algorithm using the extended Darboux frame.
result Obtains curvatures of lines of curvature on hypersurfaces.
We obtain an explicit parametrization of stationary discs glued to some Levi non-degenerate hypersurfaces. These discs form a family which is invariant under the action of biholomorphisms. We use this parametrization to construct a local circular representation of these hypersurfaces. As a corollary, we get the uniquen…
Small time heat expansion of the laplacian on an analytic hypersurface with an isolated singularitymath.AP
Proves heat expansion for Laplacian on a singularity.
problem Analytic hypersurface with isolated singularity and Laplacian heat expansion.
method Local parametrization, Newton scheme, quasihomogeneous tangent cone, local models with irregular singularities.
result Existence of small time heat expansion for Laplace operator.
In this paper, we study the problem of finding a hypersurface family from a given spatial geodesic curve in R4. We obtain the parametric representation for a hypersurface family whose members have the same curve as a given geodesic curve. Using the Frenet frame of the given geodesic curve, we present the hypersurface a…
A parametric manifold can be viewed as the manifold of orbits of a (regular) foliation of a manifold by means of a family of curves. If the foliation is hypersurface orthogonal, the parametric manifold is equivalent to the 1-parameter family of hypersurfaces orthogonal to the curves, each of which inherits a metric and…
A new Tzitzeica hypersurfacemath.DG
A new hypersurface of Tzitzeica type is obtained in all three forms: parametric, implicit and explicit. Its two-dimensional version, although well-known from a theoretical point of view, is plotted with Matlab.
Study classifies special Hessian rank 2 hypersurfaces in 4D space.
problem Classifying hypersurfaces with constant Hessian rank 2.
method Power series method of equivalence, Lie's classification spirit.
result 34 inequivalent terminal branches, each with a nonempty moduli space.
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.
Study invariant hypersurfaces with linear mean curvature in Euclidean space.
problem Understanding hypersurfaces with linear mean curvature.
method Explicit parametrizations and classification of rotationally invariant hypersurfaces.
result Obtained explicit parametrizations of constant curvature hypersurfaces.
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 must be zero.
The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.
problem Characterizing canal hypersurfaces formed by pseudo null, partially null, and null curves.
method Obtained parametric expressions and geometric invariants of canal hypersurfaces.
result Characterizations of tubular hypersurfaces in .
A parametric manifold is a manifold on which all tensor fields depend on an additional parameter, such as time, together with a parametric structure, namely a given (parametric) 1-form field. Such a manifold admits natural generalizations of Lie differentiation, exterior differentiation, and covariant differentiation, …
Study of hypersurfaces in static spacetimes with curvature bounds.
problem Characterizing hypersurfaces in static spacetimes with curvature constraints.
method Mean curvature and gradient estimates for spacelike hypersurfaces.
result Complete CMC hypersurfaces in static spacetimes are maximal under certain curvature conditions.
We consider the inverse curvature flows in the anti-de Sitter-Schwarzschild manifold with star-shaped initial hypersurface, driven by the 1-homogeneous curvature function. We show that the solutions exist for all time and the principle curvatures of the hypersurface converges to 1 exponentially fast.
Complete classification of homogeneous real hypersurfaces in complex 3-space.
problem Classifying locally homogeneous real hypersurfaces in .
method Classification of abstract 5-dimensional real Lie algebras and their representations by algebras of holomorphic vector fields in complex 3-space.
result 47 types of homogeneous hypersurfaces, including 1- or 2-parametric families and single hypersurfaces/families.
We consider strictly convex hypersurfaces which are evolving by the non-parametric logarithmic Gauss curvature flow subject to a Neumann boundary condition. Solutions are shown to converge smoothly to hypersurfaces moving by translation. In particular, for bounded domains we prove that convex functions with prescribed …
The study shows properties of stable anisotropic minimal hypersurfaces in 4D space.
problem Characterizing stable anisotropic minimal hypersurfaces in .
method Analyzing the intrinsic cubic volume growth and interior volume upper bounds for stable anisotropic minimal hypersurfaces.
result Explicit estimates of constants for stable anisotropic minimal hypersurfaces in .
Classifies hypersurfaces preserving Gauss map with two conditions.
problem Classify nonflat hypersurfaces preserving the Gauss map.
method Analyzes conditions for hypersurfaces to preserve the Gauss map.
result Hypersurfaces must be Kaehler.
In this paper we give a complete local parametric classification of the hypersurfaces with dimension at least three of a space form that carry a totally geodesic foliation of codimension one. A classification under the assumption that the leaves of the foliation are complete was given in \cite{drt} for Euclidean hypers…
The study characterizes canal hypersurfaces formed by non-null curves with parallel frame in Minkowski space-time.
problem Characterizing canal hypersurfaces in Minkowski space-time.
method Obtained canal hypersurfaces by pseudo hyperspheres or pseudo hyperbolic hyperspheres with parallel timelike normal vector field.
result Gaussian curvature, mean curvature, and principal curvatures of canal hypersurfaces were derived.
Unified framework for photon and massive particle hypersurfaces in stationary spacetimes.
problem Understanding photon and massive particle hypersurfaces in stationary spacetimes.
method Unified framework using Killing-invariant timelike hypersurfaces and associated Finsler structures.
result Conditions for a hypersurface to be a photon or massive particle hypersurface are established.
In this paper we provide several uniqueness and non-existence results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian warped products under appropriate geometric assumptions. As a consequence of this parametric study, we obtain very general uniqueness and non-existence results for a…
We study Jang's equation on a one-parameter family of asymptotically flat, spherically symmetric Cauchy hypersurfaces in the maximally extended Schwarzschild spacetime. The hypersurfaces contain apparent horizons and are parametrized by their proximity to the singularity at . We show that on those hypersurfaces …
In this paper, we investigate a holonomy invariant elliptic anisotropic surface energy for hypersurfaces in a complete Riemannian manifold, where "holonomy invariant" means that the elliptic parametric Lagrangian (i.e., a Finsler metric) of the Riemannian manifold used to define the anisotropic surface energy is consta…
Synthetic framework for null hypersurfaces in non-smooth spacetimes.
problem Analyzing null hypersurfaces in non-smooth spacetimes.
method Develops synthetic null hypersurfaces using optimal transport and Lorentzian geometry.
result Synthetic null energy condition stabilizes under convergence and applies to low-regularity spacetimes.
We present a novel and comprehensive approach to the study of the parametric Plateau problem for locally strictly convex (LSC) hypersurfaces of prescribed curvature for general convex curvature functions inside general Riemannian manifolds. We prove existence of solutions to the Plateau problem with outer barrier for L…
In this paper we present the algorithms for calculating the differential geometric properties {t,n,b1,b2,b3,k1,k2,k3,k4} along-with geodesic curvature and geodesic torsion of the transversal intersection curve of four hypersurfaces (given by parametric representation) in Euclidean space R^5. In transversal intersection…
A Levi nondegenerate real analytic hypersurface M of C^2 represented in local coordinates (z, w) in C^2 by a complex defining equation of the form w = Theta (z, \bar z, \bar w) which satisfies an appropriate reality condition, is spherical if and only if its complex graphing function Theta satisfies an explicitly writt…
The paper characterizes gaps in minimal foliations on tori using energy criteria.
problem Characterizing gaps in minimal foliations on tori.
method Introduced an energy to study min-max theory and applied it to Almgren-Pitts min-max theory.
result For a generic metric, if a lamination contains a gap, there exists a non-area-minimizing minimal hypersurface inside the gap.
Minimal foliation of symmetric space by Damek-Ricci spaces found.
problem Finding minimal foliation in symmetric spaces.
method Analyzing hypersurfaces in $SL(3,\C)/SU(3)$ and parametrizing them via angles.
result Existence of a minimal leaf isometric to Damek-Ricci space.
Study flat surfaces along curves in higher-dimensional spaces.
problem Existence and uniqueness of flat surfaces along curves.
method Explicit parametric construction using flat surfaces as ruled surfaces with stable tangent planes.
result Explicit construction of flat approximations of hypersurfaces along curves.
The goal of this paper is to classify parametrically parabolic submanifolds in any codimension. First, we describe the ones that are ruled and show that they are the only parabolic submanifolds that admit an isometric immersion as a hypersurface. Then, we classify the nonruled ones by two different means. In fact, we p…
The paper studies conformal-biharmonic hypersurfaces in spheres and product spaces.
problem Characterizing conformal-biharmonic hypersurfaces in spheres and product spaces.
method Analyzing critical points of the conformal-bienergy functional and studying properties of hypersurfaces in product spaces.
result Characterization of conformal-biharmonic hypersurfaces in spheres and product spaces.
We prove that there exist solutions for a non-parametric capillary problem in a wide class of Riemannian manifolds endowed with a Killing vector field. In other terms, we prove the existence of Killing graphs with prescribed mean curvature and prescribed contact angle along its boundary. These results may be useful for…
Study on isometric submanifolds with preserved Gauss map metrics.
problem Investigating isometric immersions with preserved Gauss map metrics.
method Local analysis of isometric immersions and deformations in Euclidean space.
result Local characterization of non-minimal and non-reducible isometric immersions.
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.
The paper studies curvature flows in Euclidean and hyperbolic spaces, proving smooth convergence to spheres.
problem Analyzing curvature flows in Euclidean and hyperbolic spaces.
method Introduced a class of expanding flows with specific speed functions and proved their longtime existence and smooth convergence.
result The flows converge smoothly to spheres in Euclidean and hyperbolic spaces under certain conditions.
Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.
problem Proves existence and uniqueness of minimal capillary cones with bi-orthogonal symmetry.
method Solves a nonlinear free boundary equation parametrized by the contact angle and uses monotonicity properties.
result Demonstrates that minimizing capillary hypersurfaces can have singularities in codimension 7.
Study pp-waves with lightlike parallel spinors in vacuum spacetimes.
problem Characterize pp-waves with lightlike parallel spinors in vacuum spacetimes.
method Parametrize pp-wave spacetimes, show correspondence with Riemannian metrics, prove parallel spinor condition.
result A pp-wave spacetime with a lightlike parallel spinor corresponds to a Ricci-flat metric with a parallel spinor.
All parabolic geometries, i.e. Cartan geometries with homogeneous model a real generalized flag manifold, admit highly interesting classes of distinguished curves. The geodesics of a projective class of connections on a manifold, conformal circles on conformal Riemannian manifolds, and Chern--Moser chains on CR--manifo…
The paper studies minimal submanifolds with specific curvature properties in Euclidean space.
problem Minimal submanifolds with -umbilical properties in Euclidean space.
method Established a correspondence and developed a Weierstrass type method for local parametrization.
result Minimal, generic, -umbilic submanifolds are -rotational and have a parametric description.
In this paper we construct the space of smooth 4-manifolds and find the homotopy model for the connected components of the complement to the discriminant. The discriminant of this space is a singular hypersurface and its generic points correspond to manifolds with isolated Morse singularities. These spaces can be consi…
In this paper we study Clifford and harmonic analysis on some conformal flat spin manifolds. In particular we treat manifolds that can be parametrized by where is a simply connected subdomain of either or and is a Kleinian group acting discontinuously on . Examples of such manifolds t…
The paper classifies ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.
problem Classifying ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.
method Maximum principle applications, classification based on causal character of rulings.
result Planes are the only cylindrical stationary surfaces. For non-cylindrical surfaces, classification depends on the causal character of the rulings.
An almost contact metric structure is parametrized by a section of an associated homogeneous fibre bundle, and conditions for this to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field, and the almost complex structure in the contact subbundle. Several examples are giv…
We continue the investigation of the correspondence between systems of conservation laws and congruences of lines in projective space. Relationship between "additional" conservation laws and hypersurfaces conjugate to a congruence is established. This construction allows us to introduce, in a purely geometric way, the …
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.