Differential geometry can be divided into classical and modern subfields. Surfaces and solids in the ambient space, as well as their geometric properties, are the subject of the classical theory. Examples of such objects include minimal surfaces, which appear in nature in the form of soap films. In contrast, modern differential geometry is founded on a theory that provides an intrinsic description of geometric objects, i.e., a description without reference to an ambient space. Key subfields in this context include, among others: Riemannian geometry, Kähler geometry, symplectic geometry, Lorentz and sub-Riemannian geometry.
Differential geometry is closely connected with other mathematical disciplines such as analysis, topology, algebraic geometry, and representation theory. It is also essential for applications within theoretical physics, with influences ranging from mechanics to special and general relativity, as well as string theory and cosmology.
At the Institute of Differential Geometry, the research groups cover, among others, the following branches of research:
Geometric evolution equations
A central area of research at the institute is geometric evolution equations, geometric partial differential equations, and geometric analysis. Geometric evolution equations are among the most exciting tools in modern differential geometry and have therefore been successfully applied in many areas, such as the proof of the Poincaré conjecture, Thurston’s geometrization conjecture, and the differentiable sphere theorem. The most important geometric evolution equations include the Ricci, Sasaki-Ricci, and Kähler-Ricci flows, mean curvature flow, Yamabe flow, harmonic heat flow, Willmore flow, and the Yang-Mills and spinor flows. The Lagrangian mean curvature flow is of particular significance both in the context of the Strominger-Yau-Zaslow and Thomas-Yau conjectures, as well as in the mirror symmetry of Calabi–Yau manifolds and the theory of minimal Lagrangian submanifolds.
Kähler and Sasaki geometry
In addition to the evolution equations already mentioned, the institute conducts projects in the areas of Kähler geometry and contact geometry, for example, studying Yamabe problems on the space of adapted contact metrics of a contact manifold. These questions can also be effectively investigated using geometric flow equations. Since singularities often occur even under simple initial conditions, the study of the resulting limiting cases is particularly relevant.
Hyperkähler geometry
A hyperkähler manifold is a Riemannian manifold (M, g) of dimension 2n whose holonomy group is contained in Sp(n). Hyperkähler manifolds are special cases of Kähler manifolds. Since their Ricci curvature vanishes, they are in particular special cases of Calabi–Yau manifolds.
On hyperkähler manifolds, there exist two anti-commuting complex structures I and J, which in turn define a third complex structure K := IJ. Together, I, J, and K generate, via the prescription L := xI + yJ + zK with (x, y, z) ∈ S², an S²-bundle of complex structures L on M. K3 surfaces are examples of hyperkähler manifolds. At the institute, various aspects related to hyperkähler manifolds are being investigated.
Twistor theory
Twistor theory essentially seeks to unify the fundamental mathematical properties of relativity and quantum mechanics. The foundations of twistor theory were developed by the British mathematician and physicist Roger Penrose.
Several important geometric structures can be constructed and studied via their twistor space, i.e., as the parameter space of (real) rational curves in a complex manifold. Naturally occurring examples of these geometries, which include hyperkähler metrics, are of great importance in several branches of mathematics and mathematical physics: for instance, Koecher varieties in representation theory, Hitchin’s moduli spaces in algebraic geometry and the theory of integrable systems, and gauge-theoretic moduli spaces of monopoles and instantons in mathematical physics.
Gauge theory
Another research area at the institute is gauge theory. Examples of gauge theories include electromagnetism, Yang–Mills theory, and the Yang–Mills–Higgs theory. Mathematically, gauge theories are formulated on principal bundles or vector bundles. Their solutions are connections that satisfy certain partial differential equations. The moduli spaces of these solutions often exhibit very interesting geometric structures. In Yang–Mills theory, this corresponds to the moduli space of instantons, and in Yang–Mills–Higgs theory to the moduli space of magnetic monopoles. Conversely, understanding these moduli spaces provides new insights into the dynamics of the relevant physical theory.
Contact and symplectic topology
Contact geometry has its roots in the Hamiltonian formalism of classical mechanics. It has since become important in many areas of modern mathematics, including symplectic geometry, differential geometry, topology, and dynamics. Contact manifolds occur as level sets of Hamiltonian functions on (even-dimensional) symplectic manifolds, so that methods from complex analysis and algebraic geometry can be fruitfully applied in their study. We are particularly interested in studying existence and classification problems, particularly in higher dimensions.
Foliations, flows and geometric topology
Dimension reduction is a powerful tool and was fundamental in Thurston’s Geometrisation Programme. In particular, whether a closed 3-manifold admits a foliation by minimal surfaces (called taut) is a fundamental question, as such a structure puts restrictions on the underlying topology in a similar way that a negatively (or non-positively curved metric) forces the manifold to be large (in terms of its fundamental group). We are interested in studying conditions under which such foliations exist, as well as related structures such as (pseudo-)Anosov flows.
