2011, Progress in mathematics, ISBN 9783034801447, Volume 290, xii, 166

This introduction to the subject of mean curvature flow of hypersurfaces has a special emphasis on the analysis of singularities and provides a detailed...

Global differential geometry | Curvature | Flows (Differentiable dynamical systems) | Mathematics | Differential equations, Parabolic

Global differential geometry | Curvature | Flows (Differentiable dynamical systems) | Mathematics | Differential equations, Parabolic

Book

Geometriae Dedicata, ISSN 0046-5755, 8/2019, Volume 201, Issue 1, pp. 281 - 315

Let M be a globally hyperbolic maximal compact 3-dimensional spacetime locally modelled on Minkowski, anti-de Sitter or de Sitter space. It is well known that...

53C42 | 53C50 | Mathematics | Topology | Conical singularities | Lorentzian geometry | Constant curvature surfaces | Convex and Discrete Geometry | Teichmuller theory | Algebraic Geometry | Hyperbolic Geometry | Projective Geometry | Differential Geometry | SPACETIMES | MATHEMATICS | POINT PARTICLES | SURFACES

53C42 | 53C50 | Mathematics | Topology | Conical singularities | Lorentzian geometry | Constant curvature surfaces | Convex and Discrete Geometry | Teichmuller theory | Algebraic Geometry | Hyperbolic Geometry | Projective Geometry | Differential Geometry | SPACETIMES | MATHEMATICS | POINT PARTICLES | SURFACES

Journal Article

Journal of Geometric Analysis, ISSN 1050-6926, 10/2017, Volume 27, Issue 4, pp. 3441 - 3473

We obtain area growth estimates for constant mean curvature graphs in E(kappa, tau)-spaces with k <= 0, by finding sharp upper bounds for the volume of...

Area estimates | Heisenberg group | Homogeneous 3-manifolds | Minimal surfaces | Constant mean curvature | Height estimates | MINIMAL GRAPHS | MINKOWSKI SPACE | X R | MATHEMATICS | HYPERSURFACES | UNBOUNDED-DOMAINS | MANIFOLDS | SURFACE EQUATION | Mathematics - Differential Geometry

Area estimates | Heisenberg group | Homogeneous 3-manifolds | Minimal surfaces | Constant mean curvature | Height estimates | MINIMAL GRAPHS | MINKOWSKI SPACE | X R | MATHEMATICS | HYPERSURFACES | UNBOUNDED-DOMAINS | MANIFOLDS | SURFACE EQUATION | Mathematics - Differential Geometry

Journal Article

Advances in Mathematics, ISSN 0001-8708, 09/2018, Volume 335, pp. 809 - 837

Given a closed flat 3-torus N, for each H>0 and each non-negative integer g, we obtain area estimates for closed surfaces with genus g and constant mean...

Curvature estimates | Area estimates | H-lamination | Minimal surface | Injectivity radius | Constant mean curvature | EXISTENCE | MINIMAL-SURFACES | H-SURFACES | GENUS | UNIQUENESS | EMBEDDED SURFACES | MATHEMATICS

Curvature estimates | Area estimates | H-lamination | Minimal surface | Injectivity radius | Constant mean curvature | EXISTENCE | MINIMAL-SURFACES | H-SURFACES | GENUS | UNIQUENESS | EMBEDDED SURFACES | MATHEMATICS

Journal Article

Journal de mathématiques pures et appliquées, ISSN 0021-7824, 02/2018, Volume 110, pp. 32 - 70

We study hypersurfaces of RN with constant nonlocal (or fractional) mean curvature. This is the equation associated with critical points of the fractional...

Constant nonlocal mean curvature | Delaunay periodic cylinders | MATHEMATICS | MATHEMATICS, APPLIED | MINIMAL-SURFACES | REGULARITY | FLOWS | 35J Partial differential equations of elliptic type | 35B Qualitative properties of solutions | Operadors, teoria dels | 35 Partial differential equations | Classificació AMS | Operator theory | 47 Operator theory | 47G Integral, integro-differential, and pseudodifferential operators | Equacions diferencials el·líptiques | Matemàtiques i estadística | Differential equations, Partial | Àrees temàtiques de la UPC

Constant nonlocal mean curvature | Delaunay periodic cylinders | MATHEMATICS | MATHEMATICS, APPLIED | MINIMAL-SURFACES | REGULARITY | FLOWS | 35J Partial differential equations of elliptic type | 35B Qualitative properties of solutions | Operadors, teoria dels | 35 Partial differential equations | Classificació AMS | Operator theory | 47 Operator theory | 47G Integral, integro-differential, and pseudodifferential operators | Equacions diferencials el·líptiques | Matemàtiques i estadística | Differential equations, Partial | Àrees temàtiques de la UPC

Journal Article

ACM Transactions on Graphics (TOG), ISSN 0730-0301, 08/2012, Volume 31, Issue 4, pp. 1 - 11

We present a new method for modeling discrete constant mean curvature (CMC) surfaces, which arise frequently in nature and are highly demanded in architecture...

constant mean curvature | mesh quality | centroidal Voronoi tessellation | surface modeling | Surface modeling | Centroidal Voronoi tessellation | Constant mean curvature | Mesh quality | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MINIMAL-SURFACES | PARTIAL-DIFFERENTIAL-EQUATIONS | TENSION MEMBRANE STRUCTURES | COMPUTATION | DIAGRAM | CENTROIDAL VORONOI TESSELLATIONS | Asymptotic properties | Finite element method | Equivalence | Mathematical analysis | Evolution | Mathematical models | Curvature | Free boundaries

constant mean curvature | mesh quality | centroidal Voronoi tessellation | surface modeling | Surface modeling | Centroidal Voronoi tessellation | Constant mean curvature | Mesh quality | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MINIMAL-SURFACES | PARTIAL-DIFFERENTIAL-EQUATIONS | TENSION MEMBRANE STRUCTURES | COMPUTATION | DIAGRAM | CENTROIDAL VORONOI TESSELLATIONS | Asymptotic properties | Finite element method | Equivalence | Mathematical analysis | Evolution | Mathematical models | Curvature | Free boundaries

Journal Article

Tohoku Mathematical Journal, ISSN 0040-8735, 09/2018, Volume 70, Issue 3, pp. 475 - 485

We explicitly determine tori that have a parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane.

Parallel mean curvature vector | Constant mean curvature surfaces in complex space forms | MATHEMATICS | constant mean curvature surfaces in complex space forms | MINIMAL-SURFACES | TOTALLY-REAL SUBMANIFOLDS | TERMS | CLASSIFICATION | X R | VECTOR

Parallel mean curvature vector | Constant mean curvature surfaces in complex space forms | MATHEMATICS | constant mean curvature surfaces in complex space forms | MINIMAL-SURFACES | TOTALLY-REAL SUBMANIFOLDS | TERMS | CLASSIFICATION | X R | VECTOR

Journal Article

General Relativity and Gravitation, ISSN 0001-7701, 9/2019, Volume 51, Issue 9, pp. 1 - 14

A sequence of constant-mean-curvature (CMC) slices in the Swiss-Cheese (SC) Universe is investigated. We focus on the CMC slices which smoothly connect to the...

Black hole | Constant-mean-curvature | Theoretical, Mathematical and Computational Physics | Swiss-cheese universe | Quantum Physics | Differential Geometry | Classical and Quantum Gravitation, Relativity Theory | Time slice | Physics | Astronomy, Astrophysics and Cosmology | SPACE | PHYSICS, MULTIDISCIPLINARY | ASTRONOMY & ASTROPHYSICS | EXPANSION | PHYSICS, PARTICLES & FIELDS | Cheese | Physics - General Relativity and Quantum Cosmology

Black hole | Constant-mean-curvature | Theoretical, Mathematical and Computational Physics | Swiss-cheese universe | Quantum Physics | Differential Geometry | Classical and Quantum Gravitation, Relativity Theory | Time slice | Physics | Astronomy, Astrophysics and Cosmology | SPACE | PHYSICS, MULTIDISCIPLINARY | ASTRONOMY & ASTROPHYSICS | EXPANSION | PHYSICS, PARTICLES & FIELDS | Cheese | Physics - General Relativity and Quantum Cosmology

Journal Article

Classical and Quantum Gravity, ISSN 0264-9381, 02/2015, Volume 32, Issue 3, pp. 35018 - 35034

For each spacetime of a family of static spacetimes, we prove the existence of entire spherically symmetric spacelike graphs with prescribed mean curvature...

Schwarzschild spacetime | POSITIVE RADIAL SOLUTIONS | quasilinear elliptic equation | PHYSICS, MULTIDISCIPLINARY | MINKOWSKI SPACE | Dirichlet boundary condition | singular phi-Laplacian | Reissner-Nordstrom spacetime | entire graph | HYPERSURFACES | ASTRONOMY & ASTROPHYSICS | DIRICHLET PROBLEM | prescribed mean curvature function | OPERATORS | EQUATION | PHYSICS, PARTICLES & FIELDS | Euclidean geometry | Asymptotic properties | Mathematical analysis | Constants | Graphs | Curvature | Quantum gravity | Symmetry

Schwarzschild spacetime | POSITIVE RADIAL SOLUTIONS | quasilinear elliptic equation | PHYSICS, MULTIDISCIPLINARY | MINKOWSKI SPACE | Dirichlet boundary condition | singular phi-Laplacian | Reissner-Nordstrom spacetime | entire graph | HYPERSURFACES | ASTRONOMY & ASTROPHYSICS | DIRICHLET PROBLEM | prescribed mean curvature function | OPERATORS | EQUATION | PHYSICS, PARTICLES & FIELDS | Euclidean geometry | Asymptotic properties | Mathematical analysis | Constants | Graphs | Curvature | Quantum gravity | Symmetry

Journal Article

Journal of Statistical Physics, ISSN 0022-4715, 7/2018, Volume 172, Issue 2, pp. 458 - 476

We consider the dynamics of small closed submanifolds (‘bubbles’) under the volume preserving mean curvature flow. We construct a map from ($$\text {n}+1$$ n+1...

Volume preserving mean curvature flow | Submanifold | Constant mean curvature surface | Physical Chemistry | Theoretical, Mathematical and Computational Physics | Effective dynamic | Quantum Physics | Adiabatic dynamic | Physics | Mean curvature flow | Statistical Physics and Dynamical Systems | SPHERES | STABILITY | LIMIT | PHYSICS, MATHEMATICAL | MOTION | CONVERGENCE | MAP

Volume preserving mean curvature flow | Submanifold | Constant mean curvature surface | Physical Chemistry | Theoretical, Mathematical and Computational Physics | Effective dynamic | Quantum Physics | Adiabatic dynamic | Physics | Mean curvature flow | Statistical Physics and Dynamical Systems | SPHERES | STABILITY | LIMIT | PHYSICS, MATHEMATICAL | MOTION | CONVERGENCE | MAP

Journal Article

Archiv der Mathematik, ISSN 0003-889X, 05/2018, Volume 110, Issue 5, pp. 523 - 531

We provide a classification of Einstein submanifolds in space forms with flat normal bundle and parallel mean curvature. This extends a previous result due to...

Einstein submanifolds | Parallel mean curvature | Flat normal bundle | Principal normals | MATHEMATICS | SPACES | CONSTANT CURVATURE | ISOMETRIC IMMERSIONS

Einstein submanifolds | Parallel mean curvature | Flat normal bundle | Principal normals | MATHEMATICS | SPACES | CONSTANT CURVATURE | ISOMETRIC IMMERSIONS

Journal Article

Journal of Geometry and Physics, ISSN 0393-0440, 10/2019, Volume 144, pp. 121 - 125

We show that ruled real hypersurfaces with constant mean curvature in the complex projective and hyperbolic spaces must be minimal. This provides their...

Minimal hypersurface | Complex hyperbolic space | Constant mean curvature | Complex space form | Complex projective space | Ruled hypersurface | MATHEMATICS | SUBMANIFOLDS | PHYSICS, MATHEMATICAL | REAL HYPERSURFACES

Minimal hypersurface | Complex hyperbolic space | Constant mean curvature | Complex space form | Complex projective space | Ruled hypersurface | MATHEMATICS | SUBMANIFOLDS | PHYSICS, MATHEMATICAL | REAL HYPERSURFACES

Journal Article

Annals of Global Analysis and Geometry, ISSN 0232-704X, 7/2019, Volume 56, Issue 1, pp. 57 - 86

In this paper, we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter–Schwarzschild and...

Geometry | 53C42 | 53A10 | Stability | Mathematical Physics | Constant mean curvature | Analysis | Global Analysis and Analysis on Manifolds | Warped product manifolds | Mathematics | Differential Geometry | 49Q10 | MATHEMATICS | Lower bounds | Euclidean geometry | Submerging | Euclidean space | Fields (mathematics) | Surface stability | Dimensional stability | Manifolds (mathematics) | Curvature | Mathematics - Differential Geometry

Geometry | 53C42 | 53A10 | Stability | Mathematical Physics | Constant mean curvature | Analysis | Global Analysis and Analysis on Manifolds | Warped product manifolds | Mathematics | Differential Geometry | 49Q10 | MATHEMATICS | Lower bounds | Euclidean geometry | Submerging | Euclidean space | Fields (mathematics) | Surface stability | Dimensional stability | Manifolds (mathematics) | Curvature | Mathematics - Differential Geometry

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 09/2014, Volume 106, pp. 57 - 69

In this paper we provide several uniqueness and non-existence results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian...

Calabi–Bernstein problem | Parabolicity | Lorentzian warped products | Constant mean curvature spacelike hypersurfaces | Uniformly elliptic EDP’s | Calabi-Bernstein problem | Uniformly elliptic EDP's | MATHEMATICS, APPLIED | MINKOWSKI SPACE | MATHEMATICS | MAXIMAL SURFACES | HARMONIC-FUNCTIONS | ROBERTSON-WALKER SPACETIMES | RIEMANNIAN-MANIFOLDS | GEOMETRY | Nonlinearity | Constants | Mathematical analysis | Curvature | Uniqueness

Calabi–Bernstein problem | Parabolicity | Lorentzian warped products | Constant mean curvature spacelike hypersurfaces | Uniformly elliptic EDP’s | Calabi-Bernstein problem | Uniformly elliptic EDP's | MATHEMATICS, APPLIED | MINKOWSKI SPACE | MATHEMATICS | MAXIMAL SURFACES | HARMONIC-FUNCTIONS | ROBERTSON-WALKER SPACETIMES | RIEMANNIAN-MANIFOLDS | GEOMETRY | Nonlinearity | Constants | Mathematical analysis | Curvature | Uniqueness

Journal Article

The Journal of Geometric Analysis, ISSN 1050-6926, 12/2019, Volume 29, Issue 4, pp. 3293 - 3307

In this paper, we provide under certain geometric and physical assumptions new uniqueness and non-existence results for complete spacelike hypersurfaces of...

53C42 | 53C50 | Mathematics | Abstract Harmonic Analysis | Spacelike hypersurface | Fourier Analysis | Constant mean curvature | Convex and Discrete Geometry | Global Analysis and Analysis on Manifolds | Differential Geometry | Dynamical Systems and Ergodic Theory | 53C80 | Generalized Robertson–Walker spacetime

53C42 | 53C50 | Mathematics | Abstract Harmonic Analysis | Spacelike hypersurface | Fourier Analysis | Constant mean curvature | Convex and Discrete Geometry | Global Analysis and Analysis on Manifolds | Differential Geometry | Dynamical Systems and Ergodic Theory | 53C80 | Generalized Robertson–Walker spacetime

Journal Article

Bulletin of the Brazilian Mathematical Society, ISSN 1678-7544, 06/2017, Volume 48, Issue 2, pp. 187 - 198

In this paper, we provided conditions for an entire constant mean curvature Killing graph lying inside a possible unbounded region to be necessarily a slice.

Entire graph | Constant mean curvature | Killing graph | Unbounded region | MATHEMATICS

Entire graph | Constant mean curvature | Killing graph | Unbounded region | MATHEMATICS

Journal Article

Duke Mathematical Journal, ISSN 0012-7094, 2015, Volume 164, Issue 14, pp. 2643 - 2722

For all N >= 9, we find smooth entire epigraphs in R-N, namely, smooth domains of the form Omega := {x is an element of R-N broken vertical bar X-N broken...

MATHEMATICS | SYMMETRY | PHASE-TRANSITIONS | HYPERSURFACES | GIORGI | ELLIPTIC-EQUATIONS | INDEX | DOMAINS | EMBEDDED MINIMAL-SURFACES | RIEMANNIAN-MANIFOLDS | CONJECTURE

MATHEMATICS | SYMMETRY | PHASE-TRANSITIONS | HYPERSURFACES | GIORGI | ELLIPTIC-EQUATIONS | INDEX | DOMAINS | EMBEDDED MINIMAL-SURFACES | RIEMANNIAN-MANIFOLDS | CONJECTURE

Journal Article

Nonlinear Differential Equations and Applications NoDEA, ISSN 1021-9722, 10/2015, Volume 22, Issue 5, pp. 1047 - 1066

The Hardy–Sobolev trace inequality can be obtained via harmonic extensions on the half-space of the Stein and Weiss weighted Hardy–Littlewood–Sobolev...

Weighted trace Sobolev inequality | 35J60 | Analysis | Mathematics | 35B40 | Hardy–Sobolev inequality | Mean curvature | MATHEMATICS, APPLIED | EXPONENTS | SYMMETRY | CRITICAL GROWTH | CONSTANT | Hardy-Sobolev inequality | ELLIPTIC-EQUATIONS | BOUNDARY SINGULARITIES | RIEMANNIAN-MANIFOLDS | Equality

Weighted trace Sobolev inequality | 35J60 | Analysis | Mathematics | 35B40 | Hardy–Sobolev inequality | Mean curvature | MATHEMATICS, APPLIED | EXPONENTS | SYMMETRY | CRITICAL GROWTH | CONSTANT | Hardy-Sobolev inequality | ELLIPTIC-EQUATIONS | BOUNDARY SINGULARITIES | RIEMANNIAN-MANIFOLDS | Equality

Journal Article

Journal of Geometry and Physics, ISSN 0393-0440, 11/2018, Volume 133, pp. 91 - 101

A 3-dimensional Riemannian manifold is called Killing submersion if it admits a Riemannian submersion over a surface such that its fibers are the trajectories...

Constant mean curvature | Killing submersions | Biharmonic surfaces | MATHEMATICS | PHYSICS, MATHEMATICAL | RIEMANNIAN-MANIFOLDS | Mathematics - Differential Geometry

Constant mean curvature | Killing submersions | Biharmonic surfaces | MATHEMATICS | PHYSICS, MATHEMATICAL | RIEMANNIAN-MANIFOLDS | Mathematics - Differential Geometry

Journal Article

Letters in Mathematical Physics, ISSN 0377-9017, 7/2010, Volume 93, Issue 1, pp. 85 - 105

Several uniqueness and non-existence results on complete constant mean curvature spacelike surfaces lying between two slices in certain three-dimensional...

Geometry | 53C42 | 53C50 | 35J60 | Theoretical, Mathematical and Computational Physics | spacelike surfaces | constant mean curvature | Calabi–Bernstein problem | Group Theory and Generalizations | generalized Robertson–Walker spacetimes | Statistical Physics, Dynamical Systems and Complexity | Physics | Calabi-Bernstein problem | Spacelike surfaces | Constant mean curvature | Generalized Robertson-Walker spacetimes | generalized Robertson-Walker spacetimes | HYPERSURFACES | MINKOWSKI SPACE | PHYSICS, MATHEMATICAL | RIEMANNIAN MANIFOLDS | UNIQUENESS

Geometry | 53C42 | 53C50 | 35J60 | Theoretical, Mathematical and Computational Physics | spacelike surfaces | constant mean curvature | Calabi–Bernstein problem | Group Theory and Generalizations | generalized Robertson–Walker spacetimes | Statistical Physics, Dynamical Systems and Complexity | Physics | Calabi-Bernstein problem | Spacelike surfaces | Constant mean curvature | Generalized Robertson-Walker spacetimes | generalized Robertson-Walker spacetimes | HYPERSURFACES | MINKOWSKI SPACE | PHYSICS, MATHEMATICAL | RIEMANNIAN MANIFOLDS | UNIQUENESS

Journal Article

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