1,621 research outputs found
Minimal surfaces in properly projecting into
For all open Riemann surface M and real number we
construct a conformal minimal immersion
such that is positive and proper.
Furthermore, can be chosen with arbitrarily prescribed flux map.
Moreover, we produce properly immersed hyperbolic minimal surfaces with non
empty boundary in lying above a negative sublinear graph.Comment: 24 pages, 7 figures, to appear in Journal of Differential Geometr
On harmonic quasiconformal immersions of surfaces in
This paper is devoted to the study of the global properties of harmonically
immersed Riemann surfaces in We focus on the geometry of
complete harmonic immersions with quasiconformal Gauss map, and in particular,
of those with finite total curvature. We pay special attention to the
construction of new examples with significant geometry.Comment: 27 pages, 7 figures. Minor changues. To appear in Trans. Amer. Math.
So
Complete bounded embedded complex curves in C^2
We prove that any convex domain of C^2 carries properly embedded complete
complex curves. In particular, we exhibit the first examples of complete
bounded embedded complex curves in C^2Comment: To appear in J. Eur. Math. Soc. (JEMS
Approximation theory for non-orientable minimal surfaces and applications
We prove a version of the classical Runge and Mergelyan uniform approximation
theorems for non-orientable minimal surfaces in Euclidean 3-space R3. Then, we
obtain some geometric applications. Among them, we emphasize the following
ones:
1. A Gunning-Narasimhan type theorem for non-orientable conformal surfaces.
2. An existence theorem for non-orientable minimal surfaces in R3, with
arbitrary conformal structure, properly projecting into a plane.
3. An existence result for non-orientable minimal surfaces in R3 with
arbitrary conformal structure and Gauss map omitting one projective direction.Comment: 34 pages, 4 figure
Properness of associated minimal surfaces
We prove that for any open Riemann surface and finite subset there exist an infinite closed set
containing and a null holomorphic curve
such that the map is proper.
In particular, is a proper conformal minimal
immersion properly projecting into
for all Comment: 17 pages, 5 figure
Null Curves in and Calabi-Yau Conjectures
For any open orientable surface and convex domain there exists a Riemann surface homeomorphic to and a
complete proper null curve This result follows from a general
existence theorem with many applications. Among them, the followings: For any
convex domain in there exist a Riemann surface
homeomorphic to and a complete proper holomorphic immersion
Furthermore, if is a convex domain and is the
solid right cylinder then can be
chosen so that is proper. There exists a Riemann surface
homeomorphic to and a complete bounded holomorphic null immersion There exists a complete bounded CMC-1 immersion
For any convex domain
there exists a complete proper minimal immersion
with vanishing flux. Furthermore, if is a convex
domain and
then can be chosen so that is proper. Any of the above
surfaces can be chosen with hyperbolic conformal structure.Comment: 20 pages, 4 figures. To appear in Mathematische Annale
Every meromorphic function is the Gauss map of a conformal minimal surface
Let be an open Riemann surface. We prove that every meromorphic function
on is the complex Gauss map of a conformal minimal immersion
which may furthermore be chosen as the real part of a
holomorphic null curve . Analogous results are proved for
conformal minimal immersions for any . We also show
that every conformal minimal immersion is isotopic through
conformal minimal immersions to a flat one, and we identify
the path connected components of the space of all conformal minimal immersions
for any .Comment: J. Geom. Anal., to appear. Available on SpringerLink:
https://link.springer.com/article/10.1007%2Fs12220-017-9948-
A construction of complete complex hypersurfaces in the ball with control on the topology
Given a closed complex hypersurface
and a compact subset , we prove the existence of
a pseudoconvex Runge domain in such that and there is a
complete proper holomorphic embedding from into the unit ball of
. For , we derive the existence of complete properly
embedded complex curves in the unit ball of , with arbitrarily
prescribed finite topology. In particular, there exist complete proper
holomorphic embeddings of the unit disc into the
unit ball of .
These are the first known examples of complete bounded embedded complex
hypersurfaces in with any control on the topology.Comment: 20 pages, 3 figures. Main Theorem improved and proof simplified. To
appear in J. Reine Angew. Math. (Crelle's J.
Embedded minimal surfaces in
In this paper, we prove that every confomal minimal immersion of an open
Riemann surface into for can be approximated uniformly
on compacts by conformal minimal embeddings. Furthermore, we show that every
open Riemann surface carries a proper conformal minimal embedding into
. One of our main tools is a Mergelyan approximation theorem for
conformal minimal immersions to for any which is also
proved in the paper.Comment: Math. Z., in press. The official version is available on Springerlink
at http://link.springer.com/article/10.1007%2Fs00209-015-1586-
Complete minimal surfaces and harmonic functions
We prove that for any open Riemann surface and any non constant harmonic
function there exists a complete conformal minimal
immersion whose third coordinate function coincides with
As a consequence, complete minimal surfaces with arbitrary conformal
structure and whose Gauss map misses two points are constructed.Comment: 10 page
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