Research profile
My research interests are in geometric analysis, metric geometry, geometric group theory,
and geometric topology.
Current research topics include:
Activities
- Together with Alexander Lytchak, I am a PI of the project "Minimal surfaces in metric spaces II"
which is part of the DFG Priority Programme "Geometry at Infinity" (SPP 2026). - Together with Ursula Hamenstädt, Yevgeny Liokumovic , Andrea Mondino, Stefan Wenger and Robert Young
we are organizing the Trimester Program
"Metric Analysis" at the Hausdorff Research Institute for Mathematics:
Papers
- Rank Rigidity for CAT(0) spaces without 3-flats, preprint.
- Isoperimetric inequalities vs. upper curvature bounds, to appear in Geom. Topol. (2024), (with S. Wenger).
- CAT(0) spaces of higher rank II, Invent. Math. 235 (2024), no. 3, 709-743.
- CAT(0) spaces of higher rank I, Geom. Funct. Anal. 34 (2024), no. 2, 512-528.
- Six proofs of the Fary-Milnor theorem, Amer. Math. Monthly 131 (2024), no. 3, 239-251, (with A. Petrunin).
- CAT(0) spaces of higher rank, preprint (superseded).
- CAT(0) 4-manifolds are Euclidean, to appear in Geom. Topol. (2023), (with A. Lytchak and K. Nagano).
- Curvature bounds of subsets in dimension two, to appear in J. Differential Geom. (2023), (with A. Lytchak).
- Morse Quasiflats II, Adv. Math. 425 (2023), Paper No. 109075, (with J. Huang and B. Kleiner).
- Ricci curvature in dimension 2 , J. Eur. Math. Soc. (JEMS) 25 (2023), no. 3, 845-867, (with A. Lytchak).
- Morse Quasiflats I, J. Reine Angew. Math. 784 (2022), 53-129, (with J. Huang and B. Kleiner).
- The structure of minimal surfaces in CAT(0) spaces, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 11, 3521-3554.
- Improvements of upper curvature bounds, Trans. Amer. Math. Soc. 373 (2020), no. 10, 7153-7166, (with A. Lytchak).
- Metric-minimizing surfaces revisited, Geom. Topol. 23 (2019), no. 6, 3111-3139, (with A. Petrunin).
- Conformal deformations of CAT(0) spaces, Math. Ann. 373 (2019), no. 1-2, 155-163, (with A. Lytchak).
- Monotonicity of saddle maps, Geom. Dedicata 198 (2019), 181-188, (with A. Petrunin).
- Affine functions on Alexandrov spaces, Math. Z. 289 (2018), no. 1-2, 455-469, (with C. Lange).
- An obstruction to the smoothability of singular nonpositively curved metrics on
4-manifolds by patterns of incompressible tori, Geom. Funct. Anal. 25 (2015), no. 5, 1575-1587.
Lectures
- Dynamics in Nonpositive Curvature (Summer 2024)
- The Classical Plateau Problem (Winter 2023)