1,720,996 research outputs found
Stein Manifolds and Holomorphic Mappings
No full textThe main theme of this book is the homotopy principle for holomorphic mappings from Stein manifolds to the newly introduced class of Oka manifolds. This book contains the first complete account of Oka-Grauert theory and its modern extensions, initiated by Mikhail Gromov and developed in the last decade by the author and his collaborators. Included is the first systematic presentation of the theory of holomorphic automorphisms of complex Euclidean spaces, a survey on Stein neighborhoods, connections between the geometry of Stein surfaces and Seiberg-Witten theory, and a wide variety of applica
Runge approximation on convex sets implies the Oka property
No full textWe prove that the classical Oka property of a complex manifold Y, concerning the existence and homotopy classification of holomorphic mappings from Stein manifolds to Y, is equivalent to a Runge approximation property for holomorphic maps from compact convex sets in Euclidean spaces to Y
The parametric h-principle for minimal surfaces in R^n and null curves in C^n
No full textLet be an open Riemann surface. It was proved by Alarc{\'o}n and Forstneri{\v c}
that every conformal minimal immersion is isotopic to the real part of a holomorphic null curve
. We prove the following substantially stronger result in this direction:
for any , the inclusion of the space of real parts of nonflat null holomorphic immersions
into the space of nonflat conformal minimal immersions satisfies the parametric h-principle with approximation;
in particular, it is a weak homotopy equivalence. Analogous results hold for several other related maps.
For an open Riemann surface of finite topological type, we obtain optimal results by showing that the above
inclusion and several related maps are inclusions of strong deformation retracts; in particular, they are homotopy equivalences.
(Joint work with Finnur L{\'a}russon.)Non UBCUnreviewedAuthor affiliation: University of LjubljanaFacult
Proper superminimal surfaces of given conformal types in the hyperbolic four-space
Full text linkLet denote the hyperbolic four-space. Given a bordered Riemann surface,
, we prove that every smooth conformal superminimal immersion can be approximated uniformly on compacts in by proper conformal
superminimal immersions . In particular, contains properly
immersed conformal superminimal surfaces normalised by any given open Riemann
surface of finite topological type without punctures. The proof uses the
analysis of holomorphic Legendrian curves in the twistor space of
Embedding bordered Riemann surfaces in strongly pseudoconvex domains
Full text linkWe show that every bordered Riemann surface, , with smooth boundary
admits a proper holomorphic map into any bounded strongly
pseudoconvex domain in , , extending to a smooth map
which can be chosen an immersion if
and an embedding if . Furthermore, can be chosen to approximate a
given holomorphic map on compacts in and
interpolate it at finitely many given points in .Comment: In memory of Mihnea Coltoiu. To appear in Rev. Roumaine Math. Pures
Appl. Research was supported by the European Union (ERC Advanced grant HPDR,
101053085) and grants P1-0291, J1-3005, and N1-0237 from ARRS, Republic of
Sloveni
The Calabi-Yau property of superminimal surfaces in self-dual Einstein four-manifolds
Full text linkIn this paper, we show that if is an oriented four dimensional Einstein manifold which is self-dual or anti-self-dual then superminimal surfaces in of appropriate spin enjoy the Calabi-Yau property, meaning that every immersed surface of this type from a bordered Riemann surface can be uniformly approximated by complete superminimal surfaces with Jordan boundaries. The proof uses the theory of twistor spaces and the Calabi-Yau property of holomorphic Legendrian curves in complex contact manifolds.J. Geom. Anal., to appea
Recent developments on Oka manifolds
Full text linkIn this paper we present the main developments in Oka theory since the
publication of my book Stein Manifolds and Holomorphic Mappings (The Homotopy
Principle in Complex Analysis)}, Second Edition, Springer, 2017. We also give
several new results, examples and constructions of Oka domains in Euclidean and
projective spaces. Furthermore, we show that for the fibre
in a Stein family can degenerate to a non-Oka fibre, thereby answering a
question of Takeo Ohsawa. Several open problems are discussed.Comment: Dedicated to Jaap Korevaar in honour of his 100th birthday. Research
was supported by the European Union (ERC Advanced grant HPDR, 101053085) and
grants P1-0291, J1-3005, and N1-0237 from ARRS, Republic of Sloveni
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