Projects in Mathematics Muenster

Research area B: Spaces and Operators
Unit B1: Smooth, singular and rigid spaces in geometry

Further Projects
CRC 1442: Geometry: Deformation and Rigidity - B05: Scalar curvature in Kähler geometry In this project we propose to study the degeneration of Kähler manifolds with constant or bounded scalar curvature under a non-collapsing assumption. For Riemannian manifolds of bounded sectional curvature, this is the content of the classical Cheeger-Gromov convergence theory from the 1970s. For Riemannian manifolds of bounded Ricci curvature, definitive results were obtained by Cheeger-Colding-Naber in the past 10-20 years, with spectacular applications to the Kähler-Einstein problem on Fano manifolds. Very little is currently known under only a scalar curvature bound even in the Kähler case. We propose to make progress in two different directions: (I) Gather examples of weak convergence phenomena related to the stability of the Positive Mass Theorem for Kähler metrics and to Taubes' virtually infinite connected sum construction for ASD 4-manifolds. (II) Study uniqueness and existence of constant scalar curvature Kähler metrics on non-compact or singular spaces by using direct PDE methods. online

Research Interests

Research Interests

$\bullet$ Kähler geometry
$\bullet$ Einstein metrics, Ricci flow
$\bullet$ Geometry and analysis on singular spaces
$\bullet$ Elliptic PDEs, singular perturbations and asymptotics

Selected Publications

Hein HJ, Tosatti V Higher-order estimates for collapsing Calabi-Yau metrics. Cambridge Journal of Mathematics Vol. 8, 2020 online
Hein HJ, Sun S Compact Calabi-Yau manifolds with isolated conical singularities. Publications mathématiques de l'IHÉS Vol. 126, 2017 online
Hein HJ, LeBrun C Mass in Kähler Geometry. Communications in Mathematical Physics Vol. 347, 2016 online
Hein HJ, Naber A New logarithmic Sobolev inequalities and an epsilon-regularity theorem for the Ricci flow. Communications on Pure and Applied Mathematics Vol. 67, 2014 online
Hein HJ Gravitational instantons from rational elliptic surfaces. Journal of the American Mathematical Society Vol. 25, 2012 online

Current Publications

$\bullet$ Hans-Joachim Hein, Song Sun, Jeff Viaclovsky, and Ruobing Zhang. Nilpotent structures and collapsing Ricci-flat metrics on the K3 surface. J. Am. Math. Soc., 35(1):123–209, January 2022. doi:10.1090/jams/978.

$\bullet$ Ronan J. Conlon and Hans-Joachim Hein. Classification of asymptotically conical Calabi-Yau manifolds. arXiv e-prints, January 2022. arXiv:2201.00870.

$\bullet$ Hans-Joachim Hein, Song Sun, Jeff Viaclovsky, and Ruobing Zhang. Gravitational instantons and del Pezzo surfaces. arXiv e-prints, November 2021. arXiv:2111.09287.

$\bullet$ Xin Fu, Hans-Joachim Hein, and Xumin Jiang. Asymptotics of Kähler-Einstein metrics on complex hyperbolic cusps. arXiv e-prints, August 2021. arXiv:2108.13390.

$\bullet$ Hans-Joachim Hein, Rares Rasdeaconu, and Ioana Suvaina. On the classification of ALE Kähler manifolds. International Mathematics Research Notices, 2021(14):10957–10980, July 2021. doi:10.1093/imrn/rnz376.

$\bullet$ Hans-Joachim Hein and Valentino Tosatti. Smooth asymptotics for collapsing calabi-yau metrics. arXiv e-prints, February 2021. arXiv:2102.03978.

$\bullet$ Hans-Joachim Hein and Valentino Tosatti. Higher-order estimates for collapsing Calabi-Yau metrics. Cambridge Journal of Mathematics, 8:683–773, December 2020. doi:10.4310/cjm.2020.v8.n4.a1.