Research project is (co) funded by the Slovenian Research Agency.
UL Member: Faculty of Mathematics and Physics
Code: N1-0237
Project: Holomorphic Partial Differential Relations
Period: 1. 4. 2022 - 31. 3. 2025
Range per year: 4,4 FTE, category: A
Head: Franc Forstnerič
Research activity: Natural sciences and mathematics
Research Organisations, researchers and citations for bibliographic records
Project description:
The aim is to develop an emerging field of complex analysis and geometry focused on holomorphic partial differential relations (HPDR). Such a relation of order r≥0 is given by a subset of the manifold of r-jets of holomorphic maps between a pair of complex manifolds. The main question is when does a formal solution of an HPDR imply the existence of an honest analytic solution, and to classify solutions up to isotopies. This complex analytic analogue of Gromov’s h-principle is highly important but poorly understood. The project will focus on the following interrelated clusters of problems.
A. Oka theory concerns the existence of holomorphic maps (HPDRs of order zero) from Stein manifolds (closed complex submanifolds of complex Euclidean spaces) to other complex manifolds whose complex structure is sufficiently flexible. The main objects of interest are Oka manifolds which the PI introduced in the literature in 2009, and which have since become the central focus of Oka theory. (See the PI’s monograph Stein Manifolds and Holomorphic Mappings, Springer, 2011 & 2017.) In 2020 the subject received new subfield 32Q56 Oka principle and Oka manifolds in the mathematics classification MSC2020. Recently developed techniques and results give a promise of major new developments concerning the class of Oka manifolds, their position in complex geometry, and their applications. We shall investigate several problems in this area which are carefully described in the original ERC proposal.
B. Holomorphic directed systems are first order HPDRs given by analytic subsets in the manifold of 1-jets. One of the main examples are holomorphic contact systems. Such a system on a complex manifold X of necessarily odd dimension is given by a completely nonintegrable holomorphic hyperplane subbundle of the tangent bundle of X. We shall develop new techniques for constructing holomorphic integral (Legendrian) curves in such systems, focusing on problems of approximation, gluing, and Riemann-Hilbert modifications of such curves. Time and resources permitting, we shall also investigate the more challenging question of finding new analytic or geometric invariants which could be used to distinguish and classify holomorphic contact systems on certain model manifolds, with emphasis on complex Euclidean spaces. We have shown in 2017 that there exist nonstandard contact systems on Euclidean space, but nothing more is known at this time.
C. Minimal surfaces. Roughly speaking, a surface is minimal if it minimizes the area among nearby surfaces with the same boundary. They are extremely important objects in mathematics, physics and other fields. Any minimal surface is the image of a conformal harmonic map from a Riemann surface, and such a map also minimizes the internal tension energy. Minimal surfaces in Euclidean spaces Rn correspond to the HPDR given by a quadric subvariety of the complex Euclidean space. The punctured quadric with the origin removed is an Oka manifold and its convex hull is the whole space. This allows a successful treatment of the theory by Oka-theoretic methods combined with convex geometry (see the PI’s monograph Minimal Surfaces from a Complex Analytic Viewpoint, Springer, 2021). For minimal surfaces in more general Riemannian manifolds, there is an important relationship with holomorphic Legendrian curves in complex contact systems, given by Penrose twistor spaces and the Bryant correspondence. This relationship is most important on Einstein 4-manifolds, which are basic object in Einstein’s general relativity theory. By using this link, we have recently constructed bounded complete (super-)minimal surfaces in self-dual Einstein 4-manifolds, thereby solving the Calabi-Yau problem on this class of manifolds. Such surfaces may be thought of as two-dimensional black holes occupying a small space but having arbitrarily high energy. We believe that the Bryant correspondence offers many additional possibilities for effective constructions of minimal surfaces in a wider class of Riemannian manifolds, and we shall explore this direction. We shall also develop the hyperbolicity theory for minimal surfaces which we introduced in the literature in a recent preprint (March 2021), where we obtained precise estimates of derivatives and growth of minimal surfaces in the ball of the real Euclidean space.
