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Plenary speaker

Franc Forstnerič

University of Ljubljana

Personal webpage

Biosketch

Franc Forstnerič received B.S. degree from University of Ljubljana (1980) and Ph.D. degree from University of Washington in Seattle (1985). He has been a Professor of Mathematics at the University of Ljubljana since 1997. During 1991-2000 he held a professor position at University of Wisconsin in Madison, USA. His research focuses on problems in complex analysis, complex geometry, and the theory of minimal surfaces. He is the author of 119 scientific papers and of a research monograph published by Springer-Verlag (2011 and 2017). He received Fulbright and Sloan scholarships, Vilas associates fellowship (UW-Madison, 1998-2000), and a prize of the Republic of Slovenia (1988). He is a member of Slovenian Academy of Sciences and Arts since 1999.

 

Research area

Franc Forstnerič works in the fields of complex analysis, complex geometry, and minimal surfaces. His early work focused on proper holomorphic maps and their boundary regularity. He showed that every bounded strongly pseudoconvex domain in a Stein manifold embeds properly holomorphically into the unit ball of CN for sufficiently big N; however, the map cannot be analytic up to the boundary in general even if the domain has real analytic boundary. He proved several rigidity and holomorphic extension theorems for such maps. One of his best known results is that every proper holomorphic map between a pair of balls, which is sufficiently smooth up to the boundary near a boundary point, is a rational map. His studies of group invariance properties of such maps marked the beginning of a major theory subsequently developed by J.-P. D’Angelo and collaborators. His early work also brought notable contributions to the Riemann-Hilbert boundary value problem with applications to polynomial hulls.

During his years at the UW-Madison in the 1990’s, the main focus of his work was the theory of holomorphic automorphisms of Euclidean spaces. In collaboration with J.-P. Rosay he provided a very useful version of the Andersén-Lempert theorem on the approximation of isotopies of biholomorphic maps between Runge domains in Cn by holomorphic automorphisms of Cn. This gave rise to numerous applications and led to the introduction by D. Varolin of a new class of Stein manifolds with the density property. These manifolds share many key properties with Euclidean spaces. For example, a recent result of Forstnerič and coauthors is that every such manifold of dimension >2n contains each Stein manifold of dimension n as a closed embedded complex submanifold.

After 2000 he focused on problems concerning the existence and homotopy classification of holomorphic maps from Stein manifolds into other complex manifolds (the h-principle or Oka principle). Together with J. Prezelj they provided sound foundations to results of M. Gromov on the h-principle for sections of elliptic holomorphic submersions over Stein spaces. In a series of works during 2005-9 he characterized the class of complex manifolds satisfying the Oka principle by a simple Runge type approximation property for holomorphic maps from convex sets in Euclidean spaces, and he obtained a number of nontrivially equivalent characterizations. This led him to introduce a new class of complex manifolds, the Oka manifolds, which have been intensively studied since then. In a related direction, he constructed holomorphic functions without critical points on any Stein manifold (or Stein space) and developed the h-principle for holomorphic submersions from Stein manifolds, with application to holomorphic foliations. These results led to numerous applications in complex geometry and are documented in his monograph Stein Manifolds and Holomorphic mappings (Springer, 2011 and 2017).

 

In the last decade, Forstnerič developed a lively collaboration with A. Alarcón and F. J. López from Granada in the classical theory of minimal surfaces in Euclidean spaces. They provided new constructions of proper minimal surfaces in Rn and in a certain natural class of domains therein, and they obtained major new results on the Calabi-Yau problem for minimal surfaces. Jointly with B. Drinovec Drnovšek they proved that every bordered Riemann surface admits an immersion into R3 as a bounded minimal surface with Jordan boundary which is complete, in the sense that the boundary of the surface is at infinite distance from the interior in the induced metric. They also showed that every meromorphic function on an open Riemann surface M is the Gauss map of a conformal minimal immersion on M into R3. Their collaboration has recently expanded (also with F. Lárusson) to the subject of holomorphic contact geometry.