Papers
Properly
embedded minimal planar domains, William H. Meeks III, Joaquín Pérez and Antonio Ros, Ann.
of Math. (to appear) pdf

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




Superficies
mínimas, A. Ros, Florentino García Santos: In Memoriam,
Universidad de Granada, pdf. The
only properly embedded minimal surfaces in the euclidean
3space are the plane, the helicoid, the catenoid and the Riemann minimal examples.


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 threemanifold. 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), 151155,
pdf.

We prove that a strictly stable
constantmeancurvature 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), 14291443,
pdf.

We study some global
properties of surfaces with nonzero constant mean curvature in the Euclidean
3space. 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), 179189, pdf


Stability of minimal and constant mean curvature surfaces with free
boundary, Matemática Contemporânea 35 (2009), 221240, pdf.

We prove that stable balance minimal surfaces with free boundary in a
centrally symmetric meanconvex region of R^3 are topological disks.
For surfaces with constant mean curvature and free boundary, we prove that
volumepreserving 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 3space 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) 301380. 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, 907926. 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
3manifolds. This family includes the case of minimal surfaces with finite
total curvature in R^{3} 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),
703723. pdf

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


Onesided
complete stable minimal surfaces, J. Diff. Geom, 74
(2006), 6992. pdf

We prove that there are not complete onesided
stable minimal surfaces in the euclidean 3space.
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 twosided index one minimal surfaces in nonnegatively 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), 145. 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  341. pdf.


Isoperimetric
inequalities in crystallography, J. Amer. Math. Soc. 17 (2004), 373388.

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, 20252047

Given a discrete group G of isometries in the 3space, we study among Ginvariant
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. GilMedrano and V. Miquel
ed., World Scientific, (2002) 235252,
pdf.

We study in a unified way several
properties of the Gauss map of a minimal surface in the Euclidean 3space. 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) 914, pdf.


Proof of the double bubble
conjecture, with M.
Hutchings, F. Morgan and M. Ritoré, Ann. of
Math. 155 (2002), 459489.

The standard double bubble
minimizes area among all compact surfaces in the Euclidean 3space 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) 153166.


Complete minimal surfaces with
index one and stable constant mean curvature surfaces, with F. J. López, Comment. Math. Helvet.
64 (1989) 3443.


On embedded complete minimal
surfaces of genus zero, with F. J. López, J. Diff. Geom. 33 (1991) 293300.


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) 147174, pdf.


The Gauss map of minimal
surfaces, Differential Geometry, Valencia 2001, Proceedings of the conference
in honour of Antonio M. Naveira,
O. GilMedrano and V. Miquel ed., World Scientific,
(2002) 235252, pdf.


Some
uniqueness and nonexistence theorems for Embedded minimal surfaces, with J. Pérez, Math. Ann. 295
(1993) 513525, pdf.


Compactness of spaces of properly
embedded minimal surfaces with finite total curvature, Indiana Univ. Math. J. 44 (1995) 139152.

Given a sequence of minimal
surfaces properly embedded in the euclidean
3space 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 twopiece property for compact
minimal surfaces in a threesphere, Indiana Univ. Math. J. 44 (1995) 841849.
If M is a compact minimal surface embedded in the unit 3sphere,
then any 2equator divides M just in two connected pieces.


Embedded minimal surfaces
: forces, topology and symmetries, Calculus of variations and PDE, 4 (1996) 469496.


The space of properly embedded
minimal surfaces with finite total curvature, with J. Pérez, Indiana Univ. Math. J. 45 (1996) 177204,
pdf.


Uniqueness of the Riemann minimal
examples, with W. Meeks III and J. Pérez,
Invent. Math. 131 (1998) 107132. pdf


The space of complete minimal
surfaces with finite total curvature as lagrangian submanifold, with J. Pérez, Trans. A.M.S. 351 (1999) 39353952, pdf.


A Plateau problem at infinity for
properly immersed minimal surfaces with finite total curvature, with C. Cosín, Indiana Univ. Math. J. 50 (2001), 847878.

We construct properly immersed
minimal surfaces in Euclidean 3space 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 MartinaFranca, Italy, summer 1999, Lecture Notes in Math., Springer, edited. by G. P. Pirola, 1775 (2002),
1566. more...


Properly
Embedded Minimal Annuli Bounded by a Convex Curve, with J. Pérez, Journal de l'Institut Mathématique
de Jussieu, 1 (2002) 293305, 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), 145. 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  341. pdf.


Onesided
complete stable minimal surfaces, J. Diff. Geom, 74 (2006),
6992. pdf

We prove
that there are not complete onesided stable minimal surfaces in the euclidean 3space. 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 twosided index one minimal
surfaces in nonnegatively 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) 3443.


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) 147174, pdf.


Stable constant mean
curvature tori and the isoperimetric problem in
threespace forms, with M. Ritoré, . Comment. Math. Helvet. 67 (1992) 293305, pdf.


Stability for hypersurfaces
of constant mean curvature with free boundary, with E. Vergasta, . Geometriae dedicata 56 (1995) 1933. 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) 391410, pdf.


On stability of capillary surfaces
in a ball, with R. Souam,
Pacific Math. J. 178 (1997) 345361.


Compact minimal hypersurfaces with index one in the real projective
space, with M. do Carmo
and M. Ritoré, Comment. Math.Helvet. 75 (2000)
247254, pdf.

The only (twosided) 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) 4549.


The isoperimetric and Willmore problems, Global Differential Geometry: the
mathematical legacy of Alfred Gray, M.
Fernandez & J. A: Wolf ed., Contemporay
Mathematics 288 (2001) 149161, 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
3dimensional 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 3space 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, 459489.
The standard double bubble minimizes area among all compact surfaces in the
Euclidean 3space 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, 20252047, pdf.

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


Isoperimetric
inequalities in crystallography, J. Amer. Math. Soc. 17 (2004), 373388.

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) 215220.


Compact hypersurfaces
with constant higher order mean curvatures, Revista Mat. Iberoamer. 3 (1987) 447453.


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) 280296.


Constant
mean curvature surfaces in a halfspace of R^{3 }with
boundary in the boundary of the halfspace, with H. Rosenberg, J. Diff. Geom. 44 (1996) 807817, pdf.


Lagrangian submanifolds
of C^n with conformal Maslov
form and the Withney sphere, with F. Urbano,
J. Math. Soc. Japan (1998) 203226, 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 3space. 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) 153166.


The Willmore
conjecture in the real projective space, Math. Research Letters 6 (1999) 487494, 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 3dimensional 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 3space
which are symmetric with respect to a point.


