The study classifies area-maximizing hypersurfaces with singularities and exterior domains.
problem Classifying area-maximizing hypersurfaces with singularities and exterior domains.
method Complete classification for entire area maximizing hypersurfaces with isolated singularities. Construction of an example. Partial result on asymptotic behavior for exterior domains. Solvability of exterior Dirichlet problems.
result Complete classification and partial results on asymptotic behavior for area maximizing hypersurfaces.
Study of maximal hypersurfaces in spacetimes, proving uniqueness results.
problem Uniqueness of compact maximal hypersurfaces in stably causal spacetimes.
method Analysis of a distinguished function on maximal hypersurfaces under first order conditions of the spacetime.
result Several uniqueness results on compact maximal hypersurfaces in stably causal spacetimes.
Maximal hypersurfaces in spacetimes with translational symmetry are characterized and their properties studied.
problem Characterizing maximal hypersurfaces in spacetimes with translational symmetry.
method Analyzing quotient spacetimes and using properties of maximal hypersurfaces.
result Complete noncompact maximal hypersurfaces in quotient spacetimes are either cylinders or conformal to the Euclidean plane.
Study on maximal hypersurfaces in pp-wave spacetimes, proving non-existence and total geodesic properties.
problem Existence and properties of maximal hypersurfaces in Lorentz manifolds.
method Analysis of constant mean curvature spacelike hypersurfaces in pp-wave spacetimes.
result Non-existence of compact spacelike hypersurfaces with non-zero constant mean curvature and total geodesic property of maximal hypersurfaces.
New complete hypersurfaces found in affine geometry.
problem Finding new complete hypersurfaces in affine geometry.
method Classifying special Calabi hypersurfaces and solving equations.
result Found a class of new Euclidean and Calabi complete affine hypersurfaces.
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 maximal hypersurfaces in open spacetimes using a maximum principle.
problem Characterize maximal hypersurfaces in open spacetimes.
method Use a generalized maximum principle to analyze hypersurfaces in spatially open Generalized Robertson-Walker spacetimes.
result Provide new uniqueness and non-existence results for complete maximal hypersurfaces in open Robertson-Walker spacetimes.
Proves properties of maximal hypersurfaces in specific spacetimes.
problem Maximal hypersurfaces in asymptotically AdS spacetimes.
method Uniqueness, existence, and regularity results via mathematical proofs.
result Proves uniqueness, existence, and regularity of maximal hypersurfaces.
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.
Paper revisits rigidity of hypersurfaces in Euclidean space.
problem Rigidity of hypersurfaces in Euclidean space.
method Energy method and maximal principle.
result New proof of rigidity of hypersurfaces.
Study stable maximal hypersurfaces in various spacetimes.
problem Characterize stability of maximal hypersurfaces in Lorentzian spacetimes.
method Analyze hypersurfaces in spacetimes with curvature assumptions, proving stability conditions and rigidity results.
result Characterize stability in spaces with constant sectional curvature and sufficient conditions for stability in general spacetimes.
Paper finds unique maximal hypersurfaces in open spacetimes.
problem Finding unique maximal hypersurfaces in open spacetimes.
method Natural geometric and physical assumptions applied to spatially open Robertson-Walker spacetimes with flat fiber.
result New uniqueness and non-existence results for complete maximal hypersurfaces.
Maximally hyperbolic solutions contain future neighborhoods of intersecting hypersurfaces.
problem Maximally globally hyperbolic solutions of higher-dimensional vacuum Einstein equations.
method Analyzing intersections of characteristic hypersurfaces.
result Contains a future neighborhood of intersecting hypersurfaces.
Optimizes the first p-Laplacian eigenvalue on hypersurfaces.
problem Maximizing the first eigenvalue of the p-Laplacian on hypersurfaces. method Equivariant maximization of the first eigenvalue for p-Laplacian on hypersurfaces. result The maximizing surface is either Simons' cone or a C1 hypersurface. We introduce in this paper the concept of tropical mirror hypersurfaces and we prove a complex tropical localization Theorem which is a version of Kapranov's Theorem \cite{K-00} in tropical geometry. We give a geometric and a topological equivalence between coamoebas of complex algebraic hypersurfaces defined by a maxi…
New findings on hypersurfaces in Euclidean space that are both maximal and minimal.
problem Characterizing hypersurfaces in Euclidean space that are both maximal and minimal.
method Analyzing the level curves of the hypersurfaces and showing they are minimal hypersurfaces in the lower-dimensional Euclidean space.
result The level curves of these hypersurfaces are minimal hypersurfaces in the lower-dimensional Euclidean space.
Torically maximal curves (known also as simple Harnack curves) are real algebraic curves in the projective plane such that their logarithmic Gauß map is totally real. In this paper we show that hyperplanes in projective spaces are the only torically maximal hypersurfaces of higher dimensions.
New non-quadratic hypersurfaces found for higher dimensions.
problem Bernstein problem for affine maximal type hypersurfaces in higher dimensions.
method Constructing non-quadratic hypersurfaces for N>=3, θ\in(1/2,(N-1)/N).
result Found non-quadratic affine maximal type hypersurfaces for N>=3.
Classifies homogeneous CR hypersurfaces in low dimensions with maximal symmetry.
problem Classifying CR hypersurfaces with maximal symmetry in low dimensions.
method Introduced modified CR symbols to organize local invariants, classified hypersurfaces through modified symbols, and used Lie group structures.
result Found nine model structures among locally homogeneous 2-nondegenerate hypersurfaces in C4. On real hypersurfaces in complex space forms many results are proven. In this paper we generalize some results concerning extrinsic geometry of real hypersurfaces, to CR submanifolds of maximal CR dimension in complex space forms.
The paper studies volumes of conformally flat manifolds in light-cone geometry.
problem Volume maximization of conformally flat manifolds in light-cone geometry.
method Computes variational formulas for the volume of hypersurfaces in light-cone.
result Hypersurfaces of conformally flat manifolds maximize volume in certain null hypersurfaces.
Infinite G-invariant minimal hypersurfaces found in Riemannian manifolds.
problem Finding minimal hypersurfaces in manifolds with group actions.
method New algorithm using multi-stage maximal cuttings.
result Each G-homology class admits infinitely many distinct realizations by embedded minimal G-hypersurfaces. New examples of real hypersurfaces found in complex hyperbolic quadrics.
problem Existence of specific types of real hypersurfaces in complex hyperbolic quadrics.
method Construction of a one-parameter family of homogeneous Hopf hypersurfaces.
result First known examples of real hypersurfaces with integrable maximal complex subbundle in irreducible Kahler manifolds.
We construct maximal hypersurfaces with a Neumann boundary condition in Minkowski space via mean curvature flow. In doing this we give general conditions for long time existence of the flow with boundary conditions with assumptions on the curvature of a the Lorentz boundary manifold.
Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.
problem Evolution of hypersurfaces in spacetime.
method Weak solutions for hypersurfaces evolving along inverse spacetime mean curvature in asymptotically flat maximal initial data sets.
result Weak solution detects both future- and past-trapped apparent horizons.
The paper proves the existence of boundary minimal hypersurfaces in compact manifolds with boundary.
problem Existence of boundary minimal hypersurfaces in compact manifolds with boundary.
method Min-max theory applied to local maximizers of width in conformal classes.
result Existence of a sequence of properly embedded equidistributed boundary minimal hypersurfaces.
New maximal surfaces solve Bernstein problems.
problem Bernstein problems in centroaffine geometry.
method Calabi affine maximal surfaces and orthonormal frame fields.
result Complete centroaffine extremal hypersurfaces solve all Bernstein problems.
This paper studies ruled real hypersurfaces in indefinite complex projective space.
problem Characterizing and classifying ruled real hypersurfaces in indefinite complex projective space.
method Introduced and studied ruled real hypersurfaces with maximal holomorphic distribution integrable and leaves totally geodesic holomorphic hyperplanes. Detailed shape operator computation and method of construction by gluing totally geodesic hyperplanes along a curve.
result Classification of all minimal ruled real hypersurfaces in terms of three main families of curves.
The paper solves splitting problems for specific spacetimes.
problem Splitting globally hyperbolic spacetimes with certain conditions.
method Geometric maximum principle and elementary proof for the second case.
result Proves splitting theorems under specific conditions.
New non-quadratic Euclidean complete affine maximal type hypersurfaces found for N≥2, θ∈(0,(N-1)/N].
problem Bernstein problem for affine maximal type equation.
method Constructing explicit examples of hypersurfaces.
result Found new non-quadratic Euclidean complete affine maximal type hypersurfaces for N≥2, θ∈(0,(N-1)/N].
Study finds maximal symmetry groups for CR structures with specific properties.
problem Determining the maximal dimension of symmetry groups for CR structures.
method Proved the sharp upper bound for the dimension of symmetry groups for homogeneous, 2-nondegenerate CR manifolds.
result The maximal dimension is n2+7 for n≥3. We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in AdSn+1, any subset E of the boundary at infinity which is the boun…
Study inverse mean curvature flow on non-compact hypersurfaces, proving long-term existence and characterizing maximal time.
problem Evolution of non-compact convex hypersurfaces in Rn+1 by inverse mean curvature. method Establish long-term existence via pointwise mean curvature estimate and viscosity solutions for strict convexity.
result Characterization of maximal time of existence in terms of tangent cone at infinity.
The Riemannian symmetric space SU_{2,m}/S(U_2U_m) is both Hermitian symmetric and quaternionic Kahler symmetric. Let M be a hypersurface in SU_{2,m}/S(U_2U_m) and denote by TM its tangent bundle. The complex structure of SU_{2,m}/S(U_2U_m) determines a maximal complex subbundle C of TM, and the quaternionic structure o…
In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.
The paper studies inverse mean curvature flow on hypersurfaces in space forms.
problem Analyzing the inverse mean curvature flow on hypersurfaces in space forms.
method Investigates the existence and properties of the flow for isoparametric hypersurfaces.
result Characterizes the flow and solutions explicitly under certain conditions.
Study of spacelike hypersurfaces in twisted product spacetimes with specific conditions.
problem Characterizing and understanding spacelike hypersurfaces in twisted product spacetimes.
method Analysis of conditions ensuring global hyperbolicity, use of conformal geometry results.
result Non-existence results for constant mean curvature hypersurfaces under certain conditions.
Study exterior maximal surfaces with precise asymptotic behavior.
problem Maximal surfaces in exterior domains with prescribed asymptotic behavior.
method Analyze the maximal surface equation and prove the exterior Dirichlet problem's uniqueness.
result Uniquely solvable exterior Dirichlet problem with given asymptotic behavior.
Paper studies Einstein vacuum equations with low regularity data.
problem Einstein vacuum equations with low regularity initial data.
method Combines Klainerman-Szeftel-Rodnianski curvature theorem, Czimek's extension procedure, and global elliptic estimates.
result Time of existence controlled by low regularity bounds on curvature in L2. The study of curvature in spacelike submanifolds in pseudo-hyperbolic spaces.
problem Curvature properties of spacelike submanifolds in pseudo-hyperbolic spaces.
method Analyzing scalar and Ricci curvatures, proving bounds and characterizing submanifolds.
result Nonpositive scalar curvature for complete maximal spacelike submanifolds in pseudo-hyperbolic spaces.
Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
The study finds hypersurfaces with constant scalar curvature in Minkowski space.
problem Finding hypersurfaces with constant scalar curvature in Minkowski space.
method Analyzes regular domains in Minkowski space and constructs hypersurfaces with constant scalar curvature.
result Hypersurfaces with constant scalar curvature exist and provide foliations of domains.
Study on flows in hyperbolic and de Sitter spaces, linking shrinking and expanding surfaces.
problem Understanding flows in hyperbolic and de Sitter spaces.
method Analyzing strictly convex hypersurfaces in hyperbolic and de Sitter spaces, relating contracting and expanding flows.
result Rescaled contracting and expanding hypersurfaces converge to specific geometric shapes.
In this paper we prove that the H^k (k is odd and larger than 2) mean curvature flow of a closed convex hypersurface can be extended over the maximal time provided that the total L^p integral of the mean curvature is finite for some p
Maximizes eigenvalue of Jacobi operator on spheres.
problem Finding the maximum eigenvalue of Jacobi operator on spheres.
method Local generalization of Willmore functional for hypersurfaces.
result First eigenvalue of Jacobi operator is a local maximum in Euclidean spheres.
The paper studies marginally trapped submanifolds in Lorentzian manifolds under null energy condition.
problem Understanding marginally trapped submanifolds in Lorentzian manifolds.
method Analyzes properties of marginally trapped submanifolds in a Lorentzian manifold satisfying the null energy condition.
result Marginally trapped submanifolds have locally volume-maximizing properties in certain null hypersurfaces.
Rotationally symmetric hypersurfaces converge to cylinders under area-preserving flow.
problem Convergence of rotationally symmetric hypersurfaces to cylinders under area-preserving mean curvature flow.
method Geometric properties and maximal principle used for gradient and curvature estimates, leading to long-time existence and convergence.
result Rotationally symmetric hypersurfaces converge to cylinders under area-preserving mean curvature flow.
It has been proved that there are no real hypersurfaces satisfying RA = 0 in non-flat complex space forms. In this paper we prove that the same is true in the case of CR submanifolds of maximal CR dimension, that is there are no CR submanifolds of maximal CR dimension satisfying RA = 0 in non-flat complex space forms.