I am interested in the theory and applications of geometric evolution equations, such as the mean curvature flow, the inverse mean curvature flow and other fully nonlinear variants. A main motivation lies in their applicability to questions arising from convex geometry, such as generalized isoperimetric type inequalities or existence of prescribed curvature measures and also in questions arsing from general relativity, such as Penrose-type inequalities.
Peer-Reviewed Publications
- with P. Bryan and M. N. Ivaki: Constant rank theorems for curvature problems via a viscosity approach, Calc. Var. Partial Differ. Equ. 62, no. 3, art. 98, (2023), doi:10.1007/s00526-023-02442-5.
- with P. Bryan and M. N. Ivaki: Christoffel-Minkowski flows, Trans. Am. Math. Soc. 376, no. 4, p. 2373-2393, (2023), doi:10.1090/tran/8683.
- with C. Xia: Stability for Serrin's problem and Alexandroff's theorem in warped product manifolds, Int. Math. Res. Not., (2022), doi:10.1093/imrn/rnac294.
- Minkowski inequalities and constrained inverse curvature flows in warped spaces, Adv. Calc. Var. 15, no. 4, p. 735-748, (2022), doi:10.1515/acv-2020-0050.
- with C. Chen, P. Guan and J. Li: A fully-nonlinear flow and quermassintegral inequalities in the sphere, Pure Appl. Math Q. 18, no. 2, p. 437-461, (2022), doi:10.4310/PAMQ.2022.v18.n2.a4.
- with G. Wang and C. Xia: Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball, J. Differ. Geom. 120, no. 2, p. 345-373, (2022), doi:10.4310/jdg/1645207496.
- with H. Roesch: Mean curvature flow in null hypersurfaces and the detection of MOTS, Commun. Math. Phys. 390, no. 3, p. 1149-1173, (2022), doi:10.1007/s00220-022-04326-9.
- The Minkowski inequality in de Sitter space, Pac. J. Math. 314, no. 2, p. 425-449, (2021), doi:10.2140/pjm.2021.314.425.
- with E. Kuwert: Asymptotic estimates for the Willmore flow with small energy, Int. Math. Res. Not. 2021, no. 18, p. 14252-14266, (2021), doi:10.1093/imrn/rnaa015/5766440.
- with M. Langford: Concavity of solutions to degenerate elliptic equations on the sphere, Commun. Partial Differ. Equ. 46, no. 6, p. 1005-1016, (2021), doi:10.1080/03605302.2020.1857404.
- with B. Lambert: Isoperimetric problems for spacelike domains in generalized Robertson-Walker spaces, J. Evol. Equ. 21, no. 1, p. 377-389, (2021), doi:10.1007/s00028-020-00584-z.
- with P. Bryan and M. N. Ivaki: Orlicz-Minkowski flows, Calc. Var. Partial Differ. Equ. 60, no. 1, art. 41, (2021), doi:10.1007/s00526-020-01886-3.
- with P. Bryan and M. N. Ivaki: Parabolic approaches to curvature equations, Nonlinear Anal. 203, art. 112174, (2021), doi:10.1016/j.na.2020.112174.
- with P. Bryan and H. Kröner: Li-Yau gradient estimates for curvature flows in positively curved manifolds, Methods Appl. Anal. 27, no. 4, p. 341-358, (2020), doi:10.4310/MAA.2020.v27.n4.a2.
- with P. Bryan and M. N. Ivaki: Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in Minkowski space and the cross curvature flow, In: Differential Geometry in the large, LMS Lecture Note Series, p. 75-97, (2020), Cambridge University Press, doi:10.1017/9781108884136.005.
- with P. Bryan and M. N. Ivaki: Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds, J. Reine Angew. Math. 764, p. 71-109, (2020), doi:10.1515/crelle-2019-0006.
- with C. Xia: Locally constrained inverse curvature flows, Trans. Am. Math. Soc. 372, no. 10, p. 6771-6803, (2019), doi:10.1090/tran/7949. Also see the Addendum
- with H. Kröner: Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature, Math. Nachr. 292, no. 7, p. 1514-1529, (2019), doi:10.1002/mana.201700370.
- Inverse curvature flows in Riemannian warped products, J. Funct. Anal. 276, no. 4, p. 1097-1144, (2019), doi:10.1016/j.jfa.2018.08.021.
- with P. Bryan and M. N. Ivaki: A unified flow approach to smooth, even L_p-Minkowski problems, Anal. & PDE 12, no. 2, p. 259-280, (2019), doi:10.2140/apde.2019.12.259.
- with J. Roth: Explicit rigidity of almost-umbilical hypersurfaces, Asian J. Math. 22, no. 6, p. 1075-1088, (2018), doi:10.4310/AJM.2018.v22.n6.a5.
- with P. Bryan and M. N. Ivaki: Harnack inequalities for evolving hypersurfaces on the sphere, Commun. Anal. Geom. 26, no. 5, p. 1047-1077, (2018), doi:10.4310/CAG.2018.v26.n5.a2.
- Isotropic functions revisited, Arch. Math. 110, no. 6, p. 591-604, (2018), doi:10.1007/s00013-018-1162-4.
- with B. Lambert: A geometric inequality for convex free boundary hypersurfaces in the unit ball, Proc. Am. Math. Soc. 145, no. 9, p. 4009-4020, (2017), doi:10.1090/proc/13516.
- with J. Roth: Pinching of the first eigenvalue for second order operators on hypersurfaces of the Euclidean space, Ann. Glob. Anal. Geom. 51, no. 3, p. 287-304, (2017), doi:10.1007/s10455-016-9535-z.
- The inverse mean curvature flow in warped cylinders of non-positive radial curvature, Adv. Math. 306, p. 1130-1163, (2017), doi:10.1016/j.aim.2016.11.003.
- with M. Makowski: Rigidity results, inverse curvature flows and Alexandrov-Fenchel-type inequalities in the sphere, Asian J. Math. 20, no. 5, p. 869-892, (2016), doi:10.4310/AJM.2016.v20.n5.a2.
- with B. Lambert: The inverse mean curvature flow perpendicular to the sphere, Math. Ann. 364, no. 3, p. 1069-1093, (2016), doi:10.1007/s00208-015-1248-2.
- Pinching and asymptotical roundness for inverse curvature flows in Euclidean space, J. Geom. Anal. 26, no. 3, p. 2265-2281, (2016), doi:10.1007/s12220-015-9627-1.
- Quantitative oscillation estimates for almost-umbilical closed hypersurfaces in Euclidean space, Bull. Aust. Math. Soc. 92, no. 1, p. 133-144, (2015), doi:10.1017/S0004972715000222.
Also see the Addendum
- Gradient estimates for inverse curvature flows in hyperbolic space, Geom. Flows 1, no. 1, p. 11-16, (2015), doi:10.1515/geofl-2015-0002.
- Non-scale invariant inverse curvature flows in hyperbolic space, Calc. Var. Partial Differ. Equ. 53, no. 1, p. 91-123, (2015), doi:10.1007/s00526-014-0742-9.
Preprints
- with X. Zhang: Stability of the Wulff shape with respect to anisotropic curvature functionals, (2023), arxiv:2308.15999.
- with P. Sahjwani: Stability of the quermassintegral inequalities in hyperbolic space, to appear in J. Geom. Anal., (2023), arxiv:2306.17610.
- with S. Blatt, G. Giacomin and A. Schikorra: A fractional Willmore-type energy functional - subcritical observations, to appear in Matrix Annals, (2023), arxiv:2306.16941.
- with W. Klingenberg and B. Lambert: A capillary problem for spacelike mean curvature flow in a cone of Minkowski space, (2023), arxiv:2304.02414.
- with E. Cabezas-Rivas: The quermassintegral preserving mean curvature flow in the sphere, to appear in Anal. & PDE, (2022), arxiv:2211.17040.
- Stability from rigidity via umbilicity, (2021), arxiv:2103.07178.
- with P. Bryan and M. N. Ivaki: On the classification of ancient solutions to curvature flows on the sphere, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., (2016), arxiv:1604.01694.
Other Publications
- Extrinsic curvature flows and applications, In: 2019-20 Matrix Annals, Matrix Book Series 4, Wood, de Gier, Praeger, Tao (eds.), Springer, p. 747-772, (2021), doi:10.1007/978-3-030-62497-2.
Theses
My papers at arXiv.org and mathscinet.ams.org.
- 16.06.2023: Go with the flow - Geometry in motion at the Night of Science Frankfurt, Germany.
- 21.02.2023: Stability from rigidity via umbilicity at the conference New Trends in Geometric and Variational PDEs at the University of Trento, Italy.
- 14.02.2023: Stability for the constant mean curvature problem in warped product spaces at the conference Paczki at the University of Warsaw, Poland.
- 13.01.2023: The quermassintegral preserving mean curvature flow in the sphere in the RMU Seminar Geometric Analysis at the University of Darmstadt, Germany.
- 10.01.2023: Alexandrov-Fenchel inequalities for convex free boundary hypersurfaces in a ball in the Oberseminar Geometrische Analysis at the GU Frankfurt, Germany.
- 01.12.2022: The quermassintegral preserving mean curvature flow in the sphere in the Geometric Analysis Seminar at the University of Xiamen (online), China.
- 30.06.2022: Alexandrov-Fenchel inequalities for convex free boundary hypersurfaces in a ball in the Privatissimum of the group "Convex and Discrete Geometry" at the TU Wien, Austria.
- 03.05.2022: Stability for the constant mean curvature problem in warped product spaces in the BIRS Workshop Integral and Metric Geometry, Oaxaca (online).
- 17.03.2022: Stability from rigidity via umbilicity in the Analysis Seminar at the University of Bath, UK.
- 10.03.2022: Stability from rigidity via umbilicity in the Analysis Seminar at the University of Warwick, UK.
- 25.02.2022: Curvature Energies: From Steiner to Willmore and beyond in the Pure Maths Colloquium at the University of Durham, UK.
- 14.02.2022: Stability from rigidity via umbilicity in the PDE Seminar at the University of Oxford, UK.
- 02.12.2021: The mean curvature flow in null hypersurfaces and the detection of MOTS at the conference Geometric PDEs in Freiburg - A conference in honour of the 60th birthday of Ernst Kuwert, University of Freiburg, Germany.
- 16.11.2021: The mean curvature flow in null hypersurfaces and the detection of MOTS in the Geometric Analysis Seminar at the University of Tennessee Knoxville, USA.
- 08.10.2021: Stability from rigidity via umbilicity in the 9th GAMM-Workshop on Analysis of PDEs at the University of Freiburg, Germany.
- 16.08.2021: Stability from rigidity via umbilicity at the conference Geometric Analysis at the Lakeside on the Frauneninsel, Chiemsee, Germany.
- 04.02.2021: A general approach to stability of the soap bubble theorem and related problems in the Geometry and Topology Seminar at the University of Durham (online), UK.
- 18.11.2020: Soap bubbles, the isoperimetric problem and the deformation of surfaces in the ILS at Cardiff University (online), Wales.
- 09.11.2020: A general approach to stability of the soap bubble theorem and related problems in the Differential Geometry Seminar at the University of Münster (online), Germany.
- 11.08.2020: Concavity of solutions to elliptic equations on the sphere in the Online Seminar Geometric Analysis organized by S. Blatt, P. Reiter, A. Schikorra and G. Wang.
- 22.06.2020: Concavity of solutions to elliptic equations on the sphere in the Cardiff Analysis Online Seminar at Cardiff University, Wales.
- 29.04.2020: Concavity of solutions to elliptic equations on the sphere in the Geometric Analysis Seminar at McGill University (online), Montreal, Canada.
- 25.02.2020: The Minkowski inequality in de Sitter space in the Geometric Analysis Seminar at Rutgers University, New Brunswick, USA.
- 07.02.2020: Concavity of solutions to elliptic equations on the sphere in the Geometry and Analysis Seminar at Columbia University, NYC, USA.
- 12.-15.01.2020: Inverse curvature flows and applications, a lecture series at the Early Career Researchers Workshop on Geometric Analysis and PDEs, Matrix Institute, University of Melbourne, Australia.
- 21.11.2019: Isoperimetric problems in Lorentzian manifolds in the Geometric Analysis Seminar at the University of Tennessee Knoxville, USA.
- 15.11.2019: The Minkowski inequality in de Sitter space in the GR Seminar at Columbia University, NYC, USA.
- 22.10.2019: Curvature flows and geometric inequalities in the PDE and Analysis Seminar at the University of Pittsburgh, USA.
- 01.10.2019: The Minkowski inequality in de Sitter space in the Differential Geometry Seminar at Harvard University, Cambridge, USA.
- 21.08.2019: New estimates for the Willmore flow and applications in the Geometric Analysis Seminar at McGill University, Montreal, Canada.
- 25.03.2019: Krümmungsflüsse und geometrische Anwendungen in the Kolloquium zur Geometrischen Analysis, University of Frankfurt, Germany.
- 13.02.2019: Harnack inequalities for evolving hypersurfaces in the Analysis Seminar at Macquarie University, Sydney, Australia.
- 07.02.2019: Alexandrov-Fenchel inequalities for convex free boundary hypersurfaces in a ball in the Australian-German Workshop on Differential Geometry in the Large at the University of Melbourne, Australia.
- 11.12.2018: The curvature flow approach to geometric inequalities at the Research Seminar Differential Geometry and Analysis at the University of Marburg, Germany.
- 06.12.2018: Harnack inequalities for evolving hypersurfaces in the Group Seminar on Partial Differential Equations at the University of Constance, Germany.
- 11.10.2018: Harnack inequalities for evolving hypersurfaces at the Geometry and Topology Seminar at the University of Durham, UK.
- 08.10.2018: The curvature flow approach to geometric inequalities in the Pure Maths Colloquium at the University of Durham, UK.
- 03.10.2018: Alexandrov-Fenchel inequalities for convex free boundary hypersurfaces in a ball in the KCL/UCL Geometry Seminar at King's college, London, UK.
- 13.09.2018: Harnack inequalities for evolving hypersurfaces at the Mini-Workshop on Differential Geometry at Hubei University, Wuhan, China.
- 06.09.2018: Harnack inequalities for evolving hypersurfaces in the Geometric Analysis Seminar at the University of Xiamen, China.
- 30.06.2018: Harnack inequalities for evolving hypersurfaces in the 40^th South German Colloquium on Differential Geometry at the University of Heidelberg, Germany.
- 18.06.2018: Harnack inequalities for evolving hypersurfaces in the Analysis Seminar at the University of Ulm, Germany.
- 08.03.2018: Locally constrained inverse curvature flows at the Workshop on Geometric Evolution Equations at the University of Regensburg, Germany.
- 12.02.2018: Locally constrained inverse curvature flows in the Geometry and Topology Seminar at the University of Toronto, Canada.
- 16.10.2017: Harnack inequalities for evolving hypersurfaces in the Differential Geometry Seminar at the University of Münster, Germany.
- 25.08.2017: The inverse mean curvature flow and convex free boundary hypersurfaces in the unit ball at the conference PADGE at the KU Leuven, Belgium.
- 19.06.2017: Harnack inequalities for evolving hypersurfaces at the conference Differential Geometry at the Banach Center, Bedlewo, Poland.
- 06.03.2017: Harnack inequalities for evolving hypersurfaces in the Séminaire de Géométrie Différentielle at the Institut Élie Cartan de Lorraine, Nancy, France.
- 08.12.2016: The inverse mean curvature flow and convex free boundary hypersurfaces in the unit ball in the Analysis Seminar at the University of Warwick, UK.
- 13.10.2016: Magical mathematics - Mathematical card tricks at the XV. Freiburg Math Days at the University of Freiburg, Germany. Lecture notes (in german).
- 17.06.2016: On the classification of ancient solutions to curvature flows on the sphere in the 38^th South German colloquium on Differential Geometry at the University of Mainz, Germany.
- 10.03.2016: Rigidity for weakly convex bodies in the sphere and applications in the Privatissimum of the group "Convex and Discrete Geometry" at the TU Wien, Austria.
- 17.12.2015: The inverse mean curvature flow perpendicular to the sphere at the conference Geometric Aspects on Capillary Problems and Related Topics at the IEMath, Granada, Spain.
- 19.11.2015: A rigidity result in the sphere and applications to fully nonlinear curvature flows in the Oberseminar Differentialgeometrie at the Leibniz Universität Hannover, Germany.
- 23.04.2015: Explicit rigidity for almost-umbilical hypersurfaces in the seminar "Topics in Geometric Analysis", AEI, FU Berlin and the University of Potsdam, Germany.
- 09.03.2015: Quantitative oscillation estimates for almost-umbilical hypersurfaces at the Journée de Géométrie at the University Paris-Est Marne-la-Vallée, France.
- 27.11.2014: The inverse mean curvature flow perpendicular to the sphere in the Group Seminar on Partial Differential Equations at the University of Constance, Germany.
- 24.10.2014: Gradient estimates for inverse curvature flows in hyperbolic space in the Group Seminar on Geometric Analysis at the University of Heidelberg, Germany.
- 07.07.2014: Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere in the Group Seminar on Geometric Analysis at the University of Freiburg, Germany.
- 22.06.2013: Non-scale-invariant inverse curvature flows in hyperbolic space in the 36^th South German Colloquium on Differential Geometry at the University of Constance, Germany.
- 10.07.-14.07.2023: Geometric flows and applications, at ICMS Edinburgh, Scotland.
- 26.06.2023: FRActional curvature in FRAnkfurt, organized at Goethe-Universität Frankfurt, Germany.
- 20.02.-22.02.2023: New Trends in Geometric and Variational PDEs, at the University of Trento, Italy.
- 13.02.-17.02.2023: Paczki at the University of Warsaw, Poland.
- 02.05.-06.05.2022: Integral and Metric Geometry, BIRS Workshop, Oaxaca (online).
- 29.11.-03.12.2021: Geometric PDEs in Freiburg - A conference in honour of the 60th birthday of Ernst Kuwert, University of Freiburg, Germany.
- 16.08.-20.08.2021: Geometric analysis at the lakeside, Fraueninsel, Chiemsee, Germany.
- 12.01.-17.01.2020: Early career researchers workshop on geometric analysis and PDE at Matrix Institute Creswick, University of Melbourne, Australia.
- 03.06.-07.06.2019: Geometric Analysis and General Relativity. A conference in honour of Gerhard Huisken's 60th birthday at ETH Zürich, Switzerland.
- 04.02.-08.02.2019: Australian-German Workshop on Differential Geometry in the Large at Matrix Institute Creswick, University of Melbourne, Australia.
- 23.07.-27.07.2018: Geometric PDEs in Freiburg 2018, jointly organised with Armin Schikorra and Guofang Wang, university of Freiburg, Germany.
- 05.03.-08.03.2018: Workshop on geometric evolution equations at the university of Regensburg, Germany.
- 21.08.-25.08.2017: Conference PADGE at the KU Leuven, Belgium.
- 18.06.-24.06.2017: Conference Differential geometry at the Banach Center Bedlewo, Poland
- 06.06.-09.06.2017: Conference Advances in geometric analysis at ETH Zürich, Switzerland
- 19.09.-23.09.2016: Summer School Geometric Flows and the Geometry of Space-time at the University of Hamburg, Germany.
- 26./27.07.2016: Analysis and Numerics In Curvature Energies-Meeting, jointly organised with Patrick Dondl and Armin Schikorra, university of Freiburg, Germany.
- 16.02.-19.02.2016: Winter School Geometric evolution equations at the University of Regensburg, Germany.
- 14.12.-17.12.2015: Conference on Geometric aspects on capillary problems and related topics at IEMath, Granada, Spain.
- 06.09.-12.09.2015: Oberwolfach Seminar Singularity analysis for geometric flows by Simon Brendle and Gerhard Huisken.
- 02.03.-05.03.2015: Conference Flow(er)s and Friends at the University of Frankfurt, Germany.
- 11./12.07.2014: 37^th south German colloquium on differential geometry at the University of Ulm, Germany.
- 21.09.-23.09.2011: Conference Geometric evolution equations at the University of Constance, Germany.