Research area B: Spaces and Operators
Unit B1: Smooth, singular and rigid spaces in geometry
Research area C: Models and Approximations
Unit C4: Geometry-based modelling, approximation, and reduction
• Dynamical systems and irregular gradient flows The central goal of this project is to study asymptotic properties for gradient flows (GFs) and related dynamical systems. In particular, we intend to establish a priori bounds and related regularity properties for solutions of GFs, we intend to study the behaviour of GFs near unstable critical regions, we intend to derive lower and upper bounds for attracting regions, and we intend to establish convergence speeds towards global attrators. Special attention will be given to GFs with irregularities (discontinuities) in the gradient and for such GFs we also intend to reveal sufficient conditions for existence, uniqueness, and flow properties in dependence of the given potential. We intend to accomplish the above goals by extending techniques and concepts from differential geometry to describe and study attracting and critical regions, by using tools from convex analysis such as subdifferentials and other generalized derivatives, as well as by employing concepts from real algebraic geometry to describe domains of attraction. In particular, we intend to generalize the center-stable manifold theorem from the theory of dynamical systems to the considered non-smooth setting. Beside finite dimensional GFs, we also study GFs in their associated infinite dimensional limits. The considered irregular GFs and related dynamical systems naturally arise, for example, in the context of molecular dynamics (to model the configuration of atoms along temporal evoluation) and machine learning (to model the training process of artificial neural networks).
• CRC 1442: Geometry: Deformation and Rigidity - Geometric evolution equations Hamilton’s Ricci flow is a geometric evolution equation on the space of Riemannian metrics of a smooth manifold. In a first subproject we would like to show a differentiable stability result for noncollapsed converging sequences of Riemannian manifolds with nonnegative sectional curvature, generalising Perelman’s topological stability. In a second subproject, next to classifying homogeneous Ricci solitons on non-compact homogeneous spaces, we would like to prove the dynamical Alekseevskii conjecture. Finally, in a third subproject we would like to find new Ricci flow invariant curvature conditions, a starting point for introducing a Ricci flow with surgery in higher dimensions. online
Selected Publications of Christoph Böhm
$\bullet$ C. Böhm and R. A. Lafuente. The Ricci flow on solvmanifolds of real type. Adv. Math., 352:516–540, 2019.
$\bullet$ C. Böhm, R. Lafuente, and M. Simon. Optimal curvature estimates for homogeneous Ricci flows. Int. Math. Res. Not. IMRN, (14):4431–4468, 2019.
$\bullet$ C. Böhm and R. A. Lafuente. Immortal homogeneous Ricci flows. Invent. Math. 461-529, 2018.
$\bullet$ C. Böhm and R. A. Lafuente. Homogeneous einstein metrics on euclidean spaces are einstein solvmanifolds, 2018, 1811.12594.
$\bullet$ C. Böhm. On the long time behavior of homogeneous Ricci flows. Comment. Math. Helv., 90(3):543–571, 2015.
$\bullet$ C. Böhm and B. Wilking. Manifolds with positive curvature operators are space forms. Ann. of Math. (2), 167(3):1079–1097, 2008.
$\bullet$ C. Böhm and B. Wilking. Nonnegatively curved manifolds with finite fundamental groups admit metrics with positive Ricci curvature. Geom. Funct. Anal., 17(3):665–681, 2007.
$\bullet$ C. Böhm, M. Wang, and W. Ziller. A variational approach for compact homogeneous Einstein manifolds. Geom. Funct. Anal., 14(4):681–733, 2004.
$\bullet$ C. Böhm. Homogeneous Einstein metrics and simplicial complexes. J. Differential Geom., 67(1):79–165, 2004.
$\bullet$ C. Böhm. Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces. Invent. Math., 134(1):145–176, 1998.
$\bullet $ Christoph Böhm and Ramiro A. Lafuente. Homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds. Geom. Topol., 26(2):899–936, 2022. doi:10.2140/gt.2022.26.899.
$\bullet $ Christoph Böhm and Megan M. Kerr. Homogeneous Einstein metrics and butterflies. arXiv e-prints, July 2021. arXiv:2107.06609.
$\bullet $ Christoph Böhm and Ramiro A. Lafuente. Non-compact Einstein manifolds with symmetry. arXiv e-prints, July 2021. arXiv:2107.04210.
$\bullet $ Christoph Böhm and Ramiro A. Lafuente. The Ricci flow on solvmanifolds of real type. Advances in Mathematics, 352:516–540, August 2019. doi:10.1016/j.aim.2019.06.014.
$\bullet $ Christoph Böhm, Ramiro Lafuente, and Miles Simon. Optimal curvature estimates for homogeneous Ricci flows. International Mathematics Research Notices. IMRN, 2019(14):4431–4468, July 2019. doi:10.1093/imrn/rnx256.
$\bullet $ Christoph Böhm and Ramiro A. Lafuente. Homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds. arXiv e-prints, November 2018. arXiv:1811.12594.
$\bullet $ Christoph Böhm and Ramiro A. Lafuente. Real geometric invariant theory. arXiv e-prints, January 2017. arXiv:1701.00643.