Summary of recent research papers
Comments about the results obtained
in the period 20092012:
·
Firstly, we are going to summarize
the principal existence results obtained
in the period concerning vertical minimal graphs in the product space H^{n}×R, where H^{n} is the ndimensional hyperbolic space. This research
is a continuation of the study carried
out in a joint paper with Eric Toubiana, University
of Paris VII about minimal surfaces in
the three dimensional space H^{2}×R [SET7]. We were content to find
suitable barriers for the Dirichlet problem. We
remark that the starting point of this work is the discovery of the ndimensional minimal Scherk type graphs [SET8].
It is worth noticing that the Scherk type
surfaces (n=2) were constructed by
Barbara Nelli and Harold Rosenberg in their
fundamental paper [NR]. Actually, we
built with Eric Toubiana,
a minimal vertical graph in H^{n}×R on the
inside of a certain polyhedron admissible in H^{n},
which we call the first Scherk type minimal hypersurface taking infinite value on certain face of a
polyhedron and zero on the other faces. Furthermore, we construct a second Scherk type minimal graph which is a minimal vertical graph
in H^{n}×R over the
inside of a polyhedron with 2k sides in H^{n}, taking values +
∞ or  ∞ on adjacent faces [SET8].
Besides, we develop other related results in the same paper [SET8]. For
instance, using geometric barriers we obtain the solution of the Dirichlet
problem for the minimal equation in H^{n}×R
on a C^{0} convex domain of H^{n},
taking continuous boundary value data on the finite boundary and continuous
boundary value data on the asymptotic boundary. In fact, to obtain this result
we make use of a rotational Scherk hypersurface as barrier at a finite point. At a point of
the asymptotic boundary if the dimension is two we use a surface written in
[SE] and in arbitrary dimension we use a hypersurface
obtained with Perre Bérard
[BSE1]. We also some obtain existence results for minimal graphs on certain admissible non convex
domains [SET8]. We note that the same idea of these geometric
constructions can be applied to the situation where the ambient space is R^{n}^{ +1}, leading to Scherk
hypersurfaces and the related solution of the Dirichlet problem in Euclidean space [SET8]. Recall that
when the domain is of class C^{2}, Jenkins and Serrin
[JS2] showed that given a C^{2} boundary value data the mean convex
condition is the necessary and sufficient condition for the solvability of the Dirichlet problem for the minimal equation in Euclidean
space. When the environment is the product space M^{n}×R
where M is a Riemannian manifold, if the
domain is of classe C^{2} if the mean
curvature of the boundary of the domain
is bounded from below by a positive constant, then given continuous boundary data, the Dirichlet problem for the minimal equation was solved by J.
Spruck [Sp]. Summarizing: when the environment is H^{n}
×R, given continuous boundary value
data by using geometric barriersScherk
hypersurfaces, we solve the Dirichlet
problem for the minimal equation in C^{0} convex domains [SET8].
·
We prove in a joint work with Barbara Nelli, University
of Aquila, Italy, a vertical
halfspace theorem for mean curvature ½
surfaces in H^{2}×R [NSE]: We show that a
complete surface with mean curvature ½, properly immersed in a mean convex side
of a simply connected rotational surface
with mean curvature ½ is rotational. In fact, when the environment is H^{2}×R, the mean curvature is ½
and the end is an annulus of revolution, it is known that this end has an asymptotic development. This implies that it has an
exponential growth [SET3], [NSTUP]. Our result is
somehow an extension of the known halfspace theorem of Hoffman and Meeks [HT2] in the context of surfaces with constant
mean curvature ½ in H^{2}×R. The main
idea in the proof is quite simple and geometric. We argue by contradiction
using a one parameter family of rotational surfaces as barriers to ensure the result, by applying the maximum
principle on account of the knowledge of the geometric behavior of the
rotational ends (growth).
─ We
pause momentarily to point out two older works related with the theme:
1.
A half space theorem in the context
of special minimal type Weingarten surfaces in Euclidean space was accomplished
by Toubiana
and me [SET]. The proof arises also
from the idea of Hoffman and Meeks. It is an application of the maximum
principle working with a one parameter family of special rotational surfaces as
barriers. Since the family of rotational special surfaces has a nice geometric
behavior and since the mean curvature vector of the special rotational surface
has the “good” normal orientation, one can argue by contradiction to get the
result. To download an old version click here.
2.
We remark that the author and Rosenberg working with
the one parameter family of embedded Deluaunay surfaces proved a maximum principle
inside a Delaunay surface in Euclidean space that yields uniqueness and other
applications and generalizations [SER].
·
We generalize with Pierre Bérard, University of Grenoble,
France, a wellknown theorem of Lindelöf, investigating the maximum domain of
stability of minimal hypersurfaces of revolution, considering other environments different from the Euclidean
space. In the Euclidean space R^{3}, the vertical halfcatenoids are maximum domains of stability (Lindelöf’s Property). This is the Lindelöf theorem [Li]. We
outline a generalization and reinterpretation of this theorem with Pierre Bérard in the papers [BSE3], [BSE4]. In fact, we obtain in R^{n}^{
+1} a generalization of Lindelöf’s theorem in
the sense that we determine the maximum symmetric domains of stability. We also
determine the maximum symmetric domains of stability when the ambient space is
H^{2}×R or H^{3}. Surprisingly, we deduce that in R^{n}^{ +1}(n ≥ 3) the halfcatenoids are not
maximum domains of stability. Furthermore, we conclude that also in H^{2 }×
R and in H^{3}, the halfcatenoids are not maximum domains of stability. However, an
embedded catenoid cousin in H^{3} satisfies the Lindelöf’s Property [BSE4]. If the ambient is H^{n}×R,
these results are also established in arbitrary dimension. In the case of the
hyperbolic space H^{3}, we get an improvement of the related results
proved by H. Mori [M] and by M. Do CarmoM.Dacjzer [DoCD] about the index and stability of the family of catenoids in term of the parameter [BSE3]. In summary: In
the case of R^{3} the halfcatenoids are
maximum domains of stability (Lindelöf theorem), but
in the case of H^{2}×R
or H^{3 }the halfcatenoids are
not maximum domain of stability.
·
Moreover, we
study with Pierre Bérard some properties of minimal hypersurfaces in the product space H^{n}×R [BSE1]. In this paper we propose a
notion of total curvature in this environment relying with the index of the
Jacobi (stability) operator. We deduce roughly speaking that "total finite
total curvature implies finite index." However, the converse is not true,
as shown by the examples giving in the same paper. Particularly, we show that
certain problems are naturally posed and investigated in arbitrary
dimensions. In fact, in a paper with Eric Toubiana
we studied among other phenomena the minimal ('catenoids'), and constant mean curvature surfaces of revolution in H^{2}×R,
exhibiting an explicit formula that
has been very useful in the development
of the theory [SET3]. In the joint work with Pierre Bérard [BSE1],
using the description of [SET3] we show that the index (number
of negative eigenvalues of
the Jacobi operator) is 1 and we describe certain domains of stability of the Jacobi
operator, generalizing classical results
for the classical catenoids in R^{3}. We
establish the following general
result: Let M be a complete minimal surface in H^{2}×R. If
the integral of the intrinsic curvature of M is finite,
then the index M is
finite. The converse is not true,
due to the existence of translational stable surfaces
(that are vertical graphs) [SE] [SET7]. It turns out
that is quite natural to study
this class of surfaces (finite total curvature) because of the results obtained in [HR] (and, recently, also because the results in [HNSET]). When n ≥ 3 we deduce that the hypothesis of finiteness of the integral of A_{M} (complete minimal hypersurfaces in
H^{n}×R) implies
that the index of M is finite.
─ It is worth
noticing that together with P. Bérard we have studied hypersurfaces in H^{n} R of constant (non zero)
mean curvature H [BSE2], constructing new examples and doing some geometric applications. For instance, we construct in
these paper examples of hypersurfaces
of revolution and translation hypersurfaces
with non vanishing constant mean
curvature H. Among them, we get entire vertical graphs and therefore stable hypersurfaces.
We find examples of hypersurfaces
of constant mean curvature 0 <H <(n1)/n, which are complete vertical graphs over the exterior of an
equidistant hypersurface
of H^{n} taking infinite boundary value data (on the equidistant
hypersurface) and taking infinite asymptotic value data.
·
In a joint work with Maria Fernanda Elbert (UFRJ)
and Barbara Nelli, we construct examples of vertical graphs
of constant mean curvature H =
½in H^{2}×R over admissible exterior domains in H^{2} [ENSE]. Such embedded examples are vertical graphs
having a weak growth of a rotational end.
The tools of this paper are a combination of geometric barriers (rotational surfaces
of mean curvature ½) and elliptic theory, using the maximum principle.
·
We built with Maria
Fernanda Elbert
all minimal hypersurfaces of constant mean curvature in H^{n}×R, invariant by parabolic
screw motions [ESE]. Among these
examples we find several model stable hypersurfaces that are
entire vertical graphs, and other
invariant graphs which are not vertical but are
complete horizontal graphs of arbitrary
dimension. Some of these horizontal graphs are stable.
·
We study in the individual
article [SE2]
the horizontal
minimal equation in H^{2}×R [SE]. We deduce a Bernstein
type theorem and we
set an open Bernstein type problem
in the context of constant mean curvature ≤ ½. Moreover, we deduce for this equation a Radó
type result.
·
We study together with Laurent
Hauswirth, Barbara
Nelli and
Eric Toubiana minimal ends of finite total curvature immersed in H^{2}×R [HNIFT].
We establish the behavior of such an
end, making a full geometric description, determining the
horizontal section of this end
by intercepting it with a slice of H^{2}×R . This work is based on earlier
works as the pioneering study
done by L. Hauswirth
and H. Rosenberg
on finite total curvature minimal surfaces [HR]. It
is also aided by the theory of
harmonic applications developed by Z. Han,
L. Tan, A. Treiberg
and T. Wan [HTTW] and Y. N. Minsky [My]. Using the results mentioned before and other types of arguments, such as the Alexandrov reflection principle,
based on the maximum principle, one
can deduce a uniqueness Schoen type
theorem in the context of finite total
curvature surfaces in H^{2}×R. This result
characterizes the complete finite total
curvature minimal surface immersed in
H^{2}×R, with two different ends, each end asymptotic to a vertical plane, as
the model minimal surface independently
discovered by J. Pyo
[P] and F.
Morabito, M. Rodriguez
[MR].
·
We study in the
individual paper [SE3] a
class of horizontal
minimal equations in H^{n}×R,
involving a family of second order elliptic PDE's indexed by a parameter ε
in [0, 1]. When ε
= 0, we recover the horizontal minimal
equation which is
not a strictly elliptic EDP
in general. When ε
> 0, we obtain a strictly elliptic PDE that we call the ε horizontal minimal equation. We infer a priori estimates for the horizontal length and a priori
boundary gradient estimates that are quite general and quite
natural as we explained in the text. We also obtain a priori global gradient estimates in the presence of a strong constraint on
the horizontal length, which seems
to be natural
for this kind of PDE. This fact is somehow related to the following phenomenon: There
are no solutions to the
horizontal minimal equation over a bounded
strictly convex domain, which vanishes on the boundary of this domain and that are continuous up
to the boundary. This rather
surprising phenomenon, in dimension 2, follows from the asymptotic theorem
deduced in [SET7].
In arbitrary dimension
it follows from the generalization accomplished in [NET]. Furthermore, we deduce an existence result for the
ε horizontal minimal equation in the twodimensional
case, that combined together with
our uniform a priori estimates and elliptic theory yields an existence result
for the horizontal minimal equation (ε=0). The uniqueness of the
solutions obtained for the horizontal minimal equation is shown for admissible
boundary data satisfying an admissible
bounded slope condition.
This follows from the Radó type theorem mentioned above. We set in the context of elliptic quasilinear EDP´s several new (we
believe interesting) open problems.

We point out that in a joint work with Elias Marion Guio, we establish a priori
estimates for a prescribed mean curvature equation in hyperbolic space. In
fact, this paper is based on Elias Doctoral Thesis PUCRio, April 2003, under
my supervision. Click here.
·
Finally, in a joint work with
Barbara Nelli and Eric Toubiana, we obtain a characterization
of the ncatenoid in H^{n} R [NSET]. In fact ,we prove a Schoen type theorem [S] in the context of infinite total curvature. We remark
that the ncatenoid in H^{n}×R were constructed in [BSE1], when n
≥ 3. We also establish a maximum
principle for minimal surfaces lying in
a closed half space. Moreover, we infer a generalization of the Asymptotic Theorem proved if the dimension is two in the joint work with Eric Toubiana already cited above [SET7].
Finally, we draw several
conclusions from these results that suggest the strong influence of the asymptotic boundary in the geometry of the minimal surface or minimal hypersurface in H^{n}×R.
─ By the way, we would like to point out that we wrote two texts in
collaboration with Eric Toubiana about
applications of the
classical maximum principle to the theory of minimal and constant mean curvature.
The first text consists of several
applications to minimal
and constant mean curvature in both Euclidean and hyperbolic space. For instance, we solve an exterior Dirichlet problem for the minimal equation in the Euclidean
space. The construction uses some geometric estimates together with the Perron process. We also prove some existence results for
minimal graphs over a bounded annulus in the hyperbolic space. The assumptions lead to a geometric C^{1}a
priori estimates to ensure the result by applying the elliptic theory. Click here
to open the file.
─ The second is an expository
text in which we discuss several
analytic and geometric applications of the maximum principle in the hyperbolic space. We infer
symmetry and halfspace results in the hyperbolic space. Notably,
we demonstrate in the text the famous
theorem of Alexandrov
and we explain in detail the socalled Alexandrov
Reflection Principle. We carry
out a MolzonSerrin type theorem for a classical overdetermined elliptic problem in the hyperbolic space. We
also discuss the Perron process for vertical minimal
graphs in the hyperbolic space. Click here to open the file.
─ On the other hand, we refer to a paper written in
a joint work with Lucas Barbosa in which we apply
geometric and PDE methods to study constant mean curvature hypersurfaces
in the hyperbolic space. It is published
in Sémin. Théor. Spectr. Géom. 16,
4379, 1998. We explain throughout these paper the quasilinear
PDE techniques involved to obtain the existence and the uniqueness results. We
also give the geometric knowledge of the model surfaces in hyperbolic space
used as barriers to get the required a priori estimates. Click here.
The papers cited above can be downloaded in Ricardo Sa
EarpPreprints
References:
· [BSE1] P. Bérard and R. Sa Earp.
Minimal hypersurfaces in H^{n}×R, total curvature and
index, 2009. arXiv:
0808.3838v3 [Math. DG].
· [BSE2] P. Bérard and R. Sa Earp. Examples of Hhypersurfaces in H^{n}×R
and geometric applications. Matemática Contemporânea, 34, 1951, 2008. Escrito em homenagem
aos oitenta anos de Manfredo do Carmo.
· [BSE3] P. Bérard and
R. Sa Earp. Lindelöf’s
theorem for catenoids revisited. hal00407395v1, arXiv0907.4294v1.
· [BSE4] P. Bérard and R. Sa Earp. Lindelöf’s theorem for hyperbolic catenoids.
hal00429404v1. Proceedings of
the American Mathematical Society. 138, 36573657, 2010.
· [DoCD] M. do Carmo and M. Dajczer. Hypersurfaces in spaces of constant curvature. Transactions
of the American Mathematical Society, 277, 685–709, 1983.
[ ENSE] M. F. Elbert, N. Nelli and R. Sa Earp. Existence of vertical ends of mean curvature
½ in H^{2}×R. Transactions of the American Mathematical
Society, 364, 3, 1179119, 2012. DOI S00029947 (2011)053614.
· [ESE] M.
F. Elbert and R. Sa Earp. All
solutions of the CMCequation in H^{n}^{
}× R invariant by parabolic screw motion. Annali
di Matematica Pura ed Appplicata 193, 1, 103114, 2014. DOI 10.1007/s1023101202688.
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Order. Springer Verlag, 1983.
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and W. Meeks III. The Strong Halfspace Theorem for
Minimal Surfaces, Invent. Math. 101, No.1 373377, 1990.
· [HR] L. Hauswirth e H. Rosenberg, Minimal
Surfaces of Finite Total Curvature in H^{2}×R, Matemática
Contemporânea 31, 6580, 2006.
· [HSET] L. Hauswirth, R. Sa Earp and E. Toubiana. Associate and conjugate
minimal immersions in M^{2}×R. Tohôku
Math J, 60, 267286, 2008.
· [HNSET] L.
Hauswirth, B. Nelli, R. Sa Earp e Eric Toubiana. A Schoen theorem for minimal surfaces in H^{2}×R. ArXiv: 1111 0851.
· [HTTW] Z.C. Han, L.F. Tam, A. Treibergs,
T. Wan. Harmonic maps from the complex plane into surfaces with nonpositive curvature, Communications in Analysis and
Geometry, Vol 3, No. 1, 85–114, 1995.
· [JS2] H.
Jenkins e J. Serrin. The Dirichlet problem for the minimal surface equation
in higher dimensions. Journal fur die reine und angewandte Mathematik, 223, 170187,
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· [Li] L. Lindelöf. Sur les limites entre lesquelles le caténoïde
est une surface minimale. Math. Annalen, 2,
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of Constant mean curvature 1 surfaces in hyperbolic space. Indiana Univ. Math. J.. 47, 1998.
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} and their stability properties. Indiana Univ. Math. J., 30:787–794, 1981.
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· [MR] F. Morabito, M.
Rodriguez: Saddle towers and minimal knoids
in H^{2} × R; Journal of the Institute of
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· [My] Y. N. Minsky. Harmonic
maps, length, and energy in Teichmüller space, J.
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·[NR] B. Nelli
and H. Rosenberg. Minimal surfaces in H^{2} × R.
Bull. Braz.
Math. Soc. 33, 263292, 2002.
· [NSEST]. B. Nelli, R. Sa Earp,
W. Santos e E. Toubiana. Uniqueness of Hsurfaces in H^{2}×R, H≤ ½, with boundary
one or two parallel horizontal circles. Annals of Global Analysis and Geometry. 33,
No. 4, 307321, 2008.
· [NSE] B. Nelli and R. Sa Earp. A
halfspace theorem for Mean Curvature H=1/2 in H^{2}×R.
J. Math. Anal. Appl. 365, 167170, 2010.
· [NSET]
B. Nelli, R. Sa Earp and E. Toubiana. Maximum
Principle and Symmetry for Minimal Hypersurfaces in H^{n}× R. ArXiv: 12112439, 2012. To appear in Annali della
Scuola Normale Superiore di Pisa, Classe di Scienze. DOI: 10.2422/20362145.201211_004.
· [P] J. Pyo. New complete embedded minimal surfaces in H^{2}×R.
Ann. Global Anal. Geom. 40, 2, 167–176,
2011.
· [SER] R. Sa Earp and H. Rosenberg.
Differential
Geometry, Pitman Monographs and Surveys in Pure and Applied Mathematics. K.
Tenenblat; B.
Lawson, Eds 52, 123148, 1991.
· [SET] R. Sa Earp and E. Toubiana. Sur les surfaces de Weingarten spéciales de type minimale. Boletim da Sociedade Brasileira de Matemática, 26, No. 2, 129148, 1995.
· [SET3] R. Sa Earp and E. Toubiana. Screw motion surfaces
in H^{2}×R and S^{2}×R . Illinois J. Math., 49, No. 3, 13231362, 2005.
· [SET5] R. Sa Earp and E. Toubiana. Existence and uniqueness of minimal surfaces. Asian J.
Math., 4, No. 3, 669694, 2000.
· [SET7] R. Sa Earp and E. Toubiana.
An asymptotic theorem for minimal surfaces and existence results for minimal
graphs in H^{2}×R. Mathematische Annalen,
342, No. 2, 309331, 2008.
· [SET8] R. Sa Earp and E. Toubiana. Minimal
graphs in H^{n}× R and R^{n+1}. Annales Institut Fourier 60, 23732402, 2010.
· [SET9] R. Sa
Earp and E. Toubiana. Introduction à la Geométrie
Hyperbolique et aux Surfaces de Riemann. Segunda edição, Cassini Eds, Paris (com
Eric Toubiana). Edição revista e ampliada. Enseignement des mathématiques, No 27. ISBN
2842250850, 2009.
· [SE] R. Sa Earp. Parabolic and hyperbolic screw
motion surfaces in H^{2}×R. J. Austr.
Math. Soc. 85, 113143, 2008.
· [SE2] R. Sa Earp. Uniqueness of minimal surfaces
whose boundary is a horizontal graph and some Bernstein problems in H^{2}×R.
Mathematische Zeitschrift,
273, 1, 211217, 2013. DOI: 10.1007/s0020901210014.
· [SE3] R. Sa Earp. Uniform a
priori estimates for a class of horizontal minimal equation. hal00699216, arXiv: 1205.4375,
2012.
· [S] R. Schoen. Uniqueness,
symmetry and embeddedness of minimal surfaces.
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