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Departamento de Geometría y Topología 
Facultad de Ciencias
Universidad de Granada 
E-18071 Granada. Spain

34 958 243281

 

 

Antonio Ros homepage Department of Geometry Granada University 

Links

a course on the isoperimetric problem

a course on minimal surfaces

research group on minimal surfaces and geometric analysis

geometry seminar

workshops, courses,...   

docencia

 

 

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Recent papers

 

Geometry and topology of some overdetermined elliptic problems, Pieralberto Sicbaldi and Antonio Ros el articulo ha salido: J. Diff. Eq., 255  (2013), 951-977, pdf.

Superficies mínimas, A. Ros, aparecerá en Florentino García Santos: In Memoriam, Universidad de Granada, pdf.

 

The local removable singularity theorem for minimal laminations, William H. Meeks III, Joaquín Pérez & Antonio Ros, pdf.

In this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a Riemannian three-manifold.  Then we show that a complete embedded minimal surface in R3 with quadratic decay of curvature has finite total curvature

 

Stable constant mean curvature surfaces are area minimizing in small L^1 neighborhoods, Frank Morgan and Antonio Ros, Interfaces and Free Boundaries 12  (2010), 151-155,  pdf.

We prove that a strictly stable constant-mean-curvature surface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L^1 neighborhood.

 

Properly embedded surfaces with constant mean curvature, Antonio Ros and Harold Rosenberg, American Journal of Mathematics, 132 (2010), 1429-1443,  pdf.

We study some global properties of surfaces with nonzero constant mean curvature in the Euclidean 3-space. We show that given two surfaces of this type, one of them cannot stay at the convex side of the other. If the surface is symmetric and lies in a narrow slab we prove that shape of the surface is similar to the one of Lawson double periodic examples. 

 

Limit leaves of a cmc lamination are stable, William H. Meeks III, Joaquín Pérez and Antonio Ros,  Journal of  Differential Geometry 84  (2010), 179-189, pdf                     

 

Stability of minimal and constant mean curvature surfaces with free boundary, Matemática Contemporânea 35 (2009), 221-240, pdf.

We prove that stable balance minimal surfaces with free boundary in a centrally symmetric mean-convex region of R^3 are topological disks. For surfaces with constant mean curvature and free boundary, we prove that volume-preserving stability implies that the surface has either genus zero with at most four boundary components or genus one with 1 or 2 curves at its boundary. This paper is dedicated to Manfredo do Carmo on his 80th Birthday

 

Properly embedded minimal planar domains, William H. Meeks III, Joaquín Pérez and Antonio Ros, pdf

The only properly embedded minimal surfaces in the euclidean 3-space are the plane, the helicoid, the catenoid and the Riemann minimal examples

 

Stable constant mean curvature surfaces, William H. Meeks III, Joaquín Pérez and Antonio Ros, in Handbook of Geometric Analysis nº 1 (2008) 301-380. Editors: Lizhen Ji, Peter Li, Richard Schoen, Leon Simon. International Press. pdf

 

Stable periodic constant mean curvature surfaces and mesoscopic phase separation. Interfaces and Free Boundaries Volume 9, Issue 3 (2007), 355–365. pdf

We give a comprehensive description of the stable solutions of the periodic isoperimetric problem in the case of lattice symmetry. This result is intended to elucidate the geometry of certain sophisticated interfaces appearing in mesoscale phase separation phenomena.

 

Properly Embedded Minimal Surfaces with Finite Topology, Proceedings of the International Congress of Mathematics (Madrid 2006), Volume II, EMS Pub., Zürich 2006, 907-926. pdf

We present a synthesis of the situation as it now stands about the various moduli spaces of properly embedded minimal surfaces of finite topology in flat 3-manifolds. This family includes the case of minimal surfaces with finite total curvature in R3 as well as singly, doubly and triply periodic minimal surfaces.

 

Liouville type properties for embedded minimal surfaces, with William H. Meeks III and Joaquín Pérez. Communications in Analysis and Geometry 14  (2006), 703-723. pdf

We prove that a positive harmonic function on a periodic minimal surface must be constant.

 

One-sided complete stable minimal surfaces, J. Diff. Geom, 74 (2006), 69-92.  pdf  

We prove that there are not complete one-sided stable minimal surfaces in the euclidean 3-space. We classify least area surfaces in the quotient of  R^3 by one or two linearly independent translations and we give sharp upper bounds of the genus of compact two-sided index one minimal surfaces in non-negatively curved ambient spaces.

 

The Geometry of minimal surfaces of finite genus I: curvature estimates and quasiperiodicity, with William H. Meeks III and Joaquín Pérez, J. Diff. Geom. 66 (2004), 1-45. pdf .

We prove curvature estimates for a sequence of properly embedded minimal surfaces of finite genus and two limit ends, in terms of the horizontal part of their normalized fluxes.

 

The Geometry of minimal surfaces of finite genus II: nonexistente of one limit end examples, with William H. Meeks III and Joaquín Pérez, Invent. Math. 158 (2004). 323 - 341pdf.

 

Isoperimetric inequalities in crystallography, J. Amer. Math. Soc. 17 (2004), 373-388.

We prove sharp isoperimetric inequalities for regions which are invariant under a given cubic space group, more...

 

The periodic isoperimetric problem, with L. Hauswirth, J. Perez and P. Romon, Transactions of the AMS  356  (2004),  no. 5, 2025--2047

Given a discrete group G of isometries in the 3-space, we study among G-invariant regions with prescribed volume fraction, those whose boundary has least area, more...

 

The Gauss map of minimal surfaces, Differential Geometry, Valencia 2001, Proceedings of the conference in honour of Antonio M. Naveira, O. Gil-Medrano and V. Miquel ed., World Scientific, (2002)  235-252, pdf.

We study in a unified way several properties of the Gauss map of a minimal surface in the Euclidean 3-space. In particular we consider stable minimal surfaces and minimal surfaces whose Gauss map image omits  certain points of the sphere. The paper contains also an insight into the classical little and great Picard theorems in one complex variable.

 

The isoperimetric problem, Lecture series at the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, summer 2001,  Mathematical Sciences Research Institute, Berkeley, California, more... 

 

Some updates on isoperimetric problems, with M. Ritore, Mathematical  Intelligencer, 24 (2002) 9-14, pdf.

 

Proof of the double bubble conjecture, with M. Hutchings, F. Morgan and M. Ritoré, Ann. of  Math. 155 (2002), 459-489.

The standard double bubble minimizes area among all compact surfaces in the Euclidean 3-space which enclose and separate two regions of prescribed volumes, pdf

 

 

 

Minimal surfaces

 

Minimal immersions of surfaces by the first eigenfunctions and conformal area, with S. Montiel, Invent. Math. 83 (1986) 153-166.

 

Complete minimal surfaces with index one and stable constant mean curvature surfaces, with F. J. López, Comment. Math. Helvet. 64 (1989) 34-43.

 

On embedded complete minimal surfaces of genus zero, with F. J. López, J. Diff. Geom. 33 (1991) 293-300.

 

Schrodinger operators associated with a holomorphic map, with S. Montiel,  Proceedings conference on global anal. and global diff. geom. At Berlin, 1990. Lecture notes in Math, 1481, Springer Verlag (1991) 147-174, pdf.

 

The Gauss map of minimal surfaces, Differential Geometry, Valencia 2001, Proceedings of the conference in honour of Antonio M. Naveira, O. Gil-Medrano and V. Miquel ed., World Scientific, (2002)  235-252, pdf.

 

Some uniqueness and nonexistence theorems for Embedded minimal surfaces, with J. Pérez, Math. Ann. 295 (1993) 513-525, pdf.

 

Compactness of spaces of properly embedded minimal surfaces with finite total curvature, Indiana Univ. Math. J. 44 (1995) 139-152.

Given a sequence of minimal surfaces properly embedded in the  euclidean 3-space which have  genus one and r finite total curvature  ends, r > 4,   there exists a subsequence which converge (with multiplicity one) to a properly embedded minimal surface with the same topology.

 

A two-piece property for compact minimal surfaces in a three-sphere, Indiana Univ. Math. J. 44 (1995) 841-849.
If  M is a compact minimal surface embedded in the unit 3-sphere,  then any 2-equator divides M just in two connected pieces.

 

Embedded minimal surfaces : forces, topology and symmetries, Calculus of variations and PDE, 4 (1996) 469-496.

 

The space of properly embedded minimal surfaces with finite total curvaturewith J. Pérez, Indiana Univ. Math. J. 45 (1996) 177-204, pdf.

 

Uniqueness of the Riemann minimal examples, with W. Meeks III and J. Pérez,  Invent. Math. 131 (1998) 107-132. pdf

 

The space of complete minimal surfaces with finite total curvature as lagrangian submanifold, with J. Pérez, Trans. A.M.S. 351 (1999) 3935-3952, pdf.

 

A Plateau problem at infinity for properly immersed minimal surfaces with finite total curvature, with C. Cosín, Indiana Univ. Math. J. 50 (2001), 847-878.

We construct properly immersed minimal surfaces in Euclidean 3-space with prescribed asymptotic behaviour. more...

 

Properly embedded minimal surfaces with finite total curvature, with J. Pérez, Lecture series at  the CIME course The Global Theory of Minimal Surfaces in Flat spaces at Martina-Franca, Italy, summer 1999, Lecture Notes in Math., Springer, edited. by G. P. Pirola, 1775 (2002), 15-66. more...

 

Properly Embedded Minimal Annuli Bounded by a Convex Curve, with J. Pérez, Journal de l'Institut Mathématique de Jussieu, 1 (2002) 293-305, pdf.
We prove that, given a convex Jordan curve C in the plane z = 0, the space of properly embedded  minimal annuli in the upper halfspace with boundary C is diffeomorphic to the interval [0,1[.
We also study the corresponding Plateau Problem at infinity.  

 

The Geometry of minimal surfaces of finite genus I: curvature estimates and quasiperiodicity,  with William H. Meeks III and Joaquín Pérez,  J. Diff. Geom. 66 (2004), 1-45. pdf .

 

The Geometry of minimal surfaces of finite genus II: nonexistente of one limit end examples,  with William H. Meeks III and Joaquín Pérez,  Invent. Math. 158  (2004). 323 - 341pdf.

 

One-sided complete stable minimal surfaces,    J. Diff. Geom, 74 (2006), 69-92.  pdf  

We prove that there are not complete one-sided stable minimal surfaces in the euclidean 3-space. We classify least area surfaces in the quotient of  R^3 by one or two linearly independent translations and we give sharp upper bounds of the genus of compact two-sided index one minimal surfaces in non-negatively curved ambient spaces.

 

 

 

Isoperimetric problems

 

Complete minimal surfaces with index one and stable constant mean curvature surfaces, with F. J. López, Comment. Math. Helvet. 64 (1989) 34-43.

 

Schrodinger operators associated with a holomorphic map, with S. Montiel, Proceedings conference on global anal. and global diff. geom. At Berlin, 1990. Lecture notes in Math, 1481, Springer Verlag (1991) 147-174, pdf.

 

Stable constant mean curvature tori and the isoperimetric problem in three-space forms, with M. Ritoré, . Comment. Math. Helvet. 67 (1992) 293-305, pdf.

 

Stability for hypersurfaces of constant mean curvature with free boundary, with E. Vergasta, . Geometriae dedicata 56 (1995) 19-33. pdf.

 

The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds, with M. Ritoré ,Trans. A.M.S. 348 (1996) 391-410, pdf.

 

On stability of capillary surfaces in a ball, with R. Souam, Pacific Math. J. 178 (1997) 345-361.

 

Compact minimal hypersurfaces with index one in the real projective space, with M. do Carmo and M. Ritoré, Comment. Math.Helvet. 75 (2000) 247-254, pdf.

The only (two-sided) compact minimal hypersurfaces in the real projective space whose Jacobi operator has index one are the minimal hyperquadrics.

 

Proof of the double bubble conjecture, with M. Hutchings, F. Morgan and M. Ritoré,  Electron. Res. Announc. Amer. Math. Soc. 6 (2000) 45-49.

 

The isoperimetric and Willmore problems, Global Differential Geometry: the mathematical legacy of Alfred Gray,  M. Fernandez  &  J. A: Wolf ed., Contemporay Mathematics  288 (2001) 149-161, pdf.

Lecture given by the author at the International Congress on Differential Geometry, in memory of Alfred Gray, September, 2000, Bilbao (Spain). We review some of the methods used to study the isoperimetric problem in 3-dimensional Riemannian manifolds. We also give a new result about the topology of the isoperimetric regions in the positive curvature case and we prove the Willmore conjecture for tori in Euclidean 3-space  which are symmetric with respect to a point. 

 

Proof of the double bubble conjecture, with M. Hutchings, F. Morgan and M. Ritoré,  Ann. of  Math. 155 (2002), no. 2, 459-489.
The standard double bubble minimizes area among all compact surfaces in the Euclidean 3-space which enclose and separate two regions of prescribed volumes
pdf

 

The isoperimetric problem, Lecture series at the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, summer 2001,  Mathematical Sciences Research Institute, Berkeley, California, more... 

 

Some updates on isoperimetric problems, with M. Ritore, Mathematical  Intelligencer (to appear) pdf.

 

The periodic isoperimetric problem. with L. Hauswirth, J. Perez and P. Romon, Transactions of the AMS  356  (2004),  no. 5, 2025--2047, pdf.

Given a discrete group G of isometries in the 3-space, we study among G-invariant regions with prescribed volume fraction, those whose boundary has least area, more...

 

Isoperimetric inequalities in crystallography, J. Amer. Math. Soc. 17 (2004), 373-388.

We prove sharp isoperimetric inequalities for regions which are invariant  under a given cubic space group. more...

 

 

 

Constant mean curvature

 

Compact hypersurfaces with constant scalar curvature and a congruence theorem, J. Diff. Geom. 27 (1988) 215-220.

 

Compact hypersurfaces with constant higher order mean curvatures, Revista Mat. Iberoamer. 3 (1987) 447-453.

 

The Alexandrov theorem for higher order mean curvatures, with S. Montiel, Proceedings conference in honour of Manfredo do Carmo, Pitman survey in pure and. appl. math . 52 (1991) 280-296.

 

Constant mean curvature surfaces in a halfspace of R3 with boundary in the boundary of the halfspace, with H. Rosenberg, J. Diff. Geom. 44 (1996) 807-817, pdf.

 

Lagrangian submanifolds of C^n with conformal Maslov form and the Withney sphere, with F. Urbano, J. Math. Soc. Japan (1998) 203-226, pdf.

 

Properly embedded surfaces with constant mean curvature, with H. Rosenberg, pdf.
We study some global properties of surfaces with nonzero constant mean curvature in the Euclidean 3-space. We show that  given two surfaces of this type, one of them cannot stay at the convex side of the other. If the surface is  symmetric and lies  in a narrow slab we prove that shape of the surface is similar to the one of Lawson double periodic examples. 

 

 

 

Bending energy and Willmore conjecture

 

Minimal immersions of surfaces by the first eigenfunctions and conformal area, with S. Montiel, Invent. Math. 83 (1986) 153-166.

 

The Willmore conjecture in the real projective space, Math. Research Letters 6 (1999) 487-494, pdf.

Any antipodal invariant torus in the unit three sphere has total squared mean curvature bigger than or equal to the one of the minimal Clifford torus.

 

The isoperimetric and Willmore problems, Lecture given by the author at the International Congress on Differential Geometry, in memory of Alfred Gray, September, 2000, Bilbao (Spain), pdf.

We review some of the methods used to study the isoperimetric problem in 3-dimensional Riemannian manifolds. We also give a new result about the topology of the isoperimetric regions in the positive curvature case and we prove the Willmore conjecture for tori in Euclidean 3-space  which are symmetric with respect to a point.