# Dr Julian Scheuer

Lecturer

- scheuerj@cardiff.ac.uk
- 5.53, Abacws, Senghennydd Road, Cathays, Cardiff, CF24 4AG

## Overview

I am a german mathematician specialized in Analysis and Geometry. After completion of my undergraduate and doctoral education in Heidelberg, Germany, I have worked as a postdoc at various places in Germany and the United States.

## Biography

### Education

- July 2018: Habilitation in mathematics, University of Freiburg, Germany.
- Feb. 2013: PhD in mathematics under supervision of Prof. Dr. Claus Gerhardt, University of Heidelberg, Germany.
- Apr. 2005-Oct. 2009: Studies of Mathematics and Economics (Diplom), University of Heidelberg, Germany.

### Employments

- From July 2020: Lecturer for Analysis, School of Mathematics, Cardiff University.
- Apr. 2019-Sept. 2020: Visiting scholar at Columbia University, New York City, USA. (on leave from Freiburg and Cardiff)
- Apr. 2018-Sept. 2018: Interim professor (W2) at the institute for applied analysis, Universtiy of Ulm, Germany. (on leave from Freiburg)
- Apr. 2015-June 2020: Scientific assistant in the group of Prof. Dr. Ernst Kuwert and Prof. Dr. Guofang Wang, University of Freiburg, Germany.
- Jan. 2014-Mar. 2015: Postdoctoral research position (DFG) in the project "Curvature problems" in Prof. Claus Gerhardt's group, University of Heidelberg, Germany.
- Jan. 2012-Jan. 2014: Doctoral research position (DFG) in the project "Curvature problems" in Prof. Claus Gerhardt's group, University of Heidelberg, Germany.

## Publications

### 2022

- Chen, C., Guan, P., Li, J. and Scheuer, J. 2022. A fully-nonlinear flow and quermassintegral inequalities in the sphere. Pure and Applied Mathematics Quarterly 18(2), pp. 437-461. (10.4310/PAMQ.2022.v18.n2.a4)
- Roesch, H. and Scheuer, J. 2022. Mean curvature flow in null hypersurfaces and the detection of MOTS. Communications in Mathematical Physics 390, pp. 1149-1173. (10.1007/s00220-022-04326-9)
- Scheuer, J., Wang, G. and Xia, C. 2022. Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball. Journal of Differential Geometry 120(2), pp. 345-373. (10.4310/jdg/1645207496)
- Bryan, P., Ivaki, M. N. and Scheuer, J. 2022. Christoffel-Minkowski flows. Transactions of the American Mathematical Society

### 2021

- Scheuer, J. 2021. The Minkowski inequality in De Sitter space. Pacific Journal of Mathematics 314(2), pp. 425-449. (10.2140/pjm.2021.314.425)
- Scheuer, J. and Kuwert, E. 2021. Asymptotic estimates for the Willmore flow with small energy. International Mathematics Research Notices 2021(18), pp. 14252-14266. (10.1093/imrn/rnaa015)
- Bryan, P., Kröner, H. and Scheuer, J. 2021. Li-Yau gradient estimates for curvature flows in positively curved manifolds. Methods and Applications of Analysis 27(4), pp. 341-358. (10.4310/MAA.2020.v27.n4.a2)
- Langford, M. and Scheuer, J. 2021. Concavity of solutions to degenerate elliptic equations on the sphere. Communications in Partial Differential Equations 46(6), pp. 1005-1016. (10.1080/03605302.2020.1857404)
- Lambert, B. and Scheuer, J. 2021. Isoperimetric problems for spacelike domains in generalized Robertson-Walker spaces. Journal of Evolution Equations 21(1), pp. 377-389. (10.1007/s00028-020-00584-z)
- Bryan, P., Ivaki, M. N. and Scheuer, J. 2021. Parabolic approaches to curvature equations. Nonlinear Analysis: Theory, Methods and Applications 203, article number: 112174. (10.1016/j.na.2020.112174)
- Bryan, P., Ivaki, M. N. and Scheuer, J. 2021. Orlicz-Minkowski flows. Calculus of Variations and Partial Differential Equations 60, article number: 41. (10.1007/s00526-020-01886-3)

### 2020

- Bryan, P., Ivaki, M. N. and Scheuer, J. 2020. Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in Minkowski space and the cross curvature flow. In: Dearricott, O. et al. eds. Differential Geometry in the Large. Cambridge University Press, pp. 75-97., (10.1017/9781108884136.005)
- Scheuer, J. 2020. Minkowski inequalities and constrained inverse curvature flows in warped spaces. Advances in Calculus of Variations (10.1515/acv-2020-0050)
- Bryan, P., Ivaki, M. N. and Scheuer, J. 2020. Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds. Journal für die reine und angewandte Mathematik 2020(764), pp. 71-109. (10.1515/crelle-2019-0006)

### 2019

- Scheuer, J. and Xia, C. 2019. Locally constrained inverse curvature flows. Transactions of the American Mathematical Society 372(10), pp. 6771-6803. (10.1090/tran/7949)
- Kröner, H. and Scheuer, J. 2019. Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature. Mathematical News / Mathematische Nachrichten 292(7), pp. 1514-1529. (10.1002/mana.201700370)
- Scheuer, J. 2019. Inverse curvature flows in Riemannian warped products. Journal of Functional Analysis 276(4), pp. 1097-1144. (10.1016/j.jfa.2018.08.021)
- Roth, J. and Scheuer, J. 2019. Explicit rigidity of almost-umbilical hypersurfaces. Asian Journal of Mathematics 22(6), pp. 1075-1088. (10.4310/AJM.2018.v22.n6.a5)
- Bryan, P., Ivaki, M. N. and Scheuer, J. 2019. A unified flow approach to smooth, even Lp-Minkowski problems. Analysis & PDE 12(2), pp. 259-280. (10.2140/apde.2019.12.259)

### 2018

- Scheuer, J. 2018. Isotropic functions revisited. Archiv der Mathematik 110(6), pp. 591-604. (10.1007/s00013-018-1162-4)
- Bryan, P., Ivaki, M. N. and Scheuer, J. 2018. Harnack inequalities for evolving hypersurfaces on the sphere. Communications in Analysis and Geometry 26(5), pp. 1047-1077. (10.4310/CAG.2018.v26.n5.a2)

### 2017

- Roth, J. and Scheuer, J. 2017. Pinching of the first eigenvalue for second order operators on hypersurfaces of the Euclidean space. Annals of Global Analysis and Geometry 51(3), pp. 287-304. (10.1007/s10455-016-9535-z)
- Lambert, B. and Scheuer, J. 2017. A geometric inequality for convex free boundary hypersurfaces in the unit ball. Proceedings of the American Mathematical Society 145(9), pp. 4009-4020. (10.1090/proc/13516)
- Scheuer, J. 2017. The inverse mean curvature flow in warped cylinders of non-positive radial curvature. Advances in Mathematics 306, pp. 1130-1163. (10.1016/j.aim.2016.11.003)

### 2016

- Scheuer, J. 2016. Pinching and asymptotical roundness for inverse curvature flows in Euclidean space. Journal of Geometric Analysis 26(3), pp. 2265-2281. (10.1007/s12220-015-9627-1)
- Lambert, B. and Scheuer, J. 2016. The inverse mean curvature flow perpendicular to the sphere. Mathematische Annalen 364, pp. 1069-1093. (10.1007/s00208-015-1248-2)
- Makowski, M. and Scheuer, J. 2016. Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere. The Asian Journal of Mathematics 20(5), pp. 869. (10.4310/AJM.2016.v20.n5.a2)

### 2015

- Scheuer, J. 2015. Quantitative oscillation estimates for almost-umbilical closed hypersurfaces in euclidean space. Bulletin of the Australian Mathematical Society 92(1), pp. 133-144. (10.1017/S0004972715000222)
- Scheuer, J. 2015. Non-scale-invariant inverse curvature flows in hyperbolic space. Calculus of Variations and Partial Differential Equations 53(1-2), pp. 91-123. (10.1007/s00526-014-0742-9)
- Scheuer, J. 2015. Gradient estimates for inverse curvature flows in hyperbolic space. Geometric Flows 1(1) (10.1515/geofl-2015-0002)

## Teaching

### Academic Year 2021/22

#### Autumn Semester

MA3018: Measure Theory

For details on my teaching before I came to Cardiff please see my personal webpage.

### Research interests

All my papers (journal publications and preprints) are available at arXiv.org.

Here are reviews of my papers at mathscinet.ams.org.

### Funding

My research has partially been funded by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) within two projects on curvature flows:

- "Harnack inequalities for curvature flows and applications", 2016-2019, gepris:319506420
- "Quermassintegral preserving local curvature flows", 2019-2020, gepris:400729345

### Visibility

Over the past years I have presented my work at various places, including Harvard, Rutgers, Columbia (USA), McGill (Canada), Warwick (UK), Macquarie University Sydney (Australia) and Xiamen (China).

For a full list of talks and attended conferences see my personal webpage.

## Supervision

**PhD project:**

Currently I am supervising Ms. Prachi Sahjwani on her journey towards a PhD. Her project is called "Stability in physical systems governed by curvature quantities", funded by EPSRC.

**Undergraduate Project:**

I am supervising a year-4 undergraduate project on applied measure theory. We investigate the analytical properties of fractals (such as the Mandelbrot set) with tools from measure theroy and analysis.

**For interested prospective students:**

In general I am interested in supervising PhD students in the areas of:

- Partial differential equations, with possible applications in
- Geometric evolution equations (mean curvature flow, Ricci flow and their friends)
- Hypersurface geometry (Stability of soap bubbles, theory of convex bodies, isoperimetric inequalities)

- Lorentz geometry and general relativity

### Past projects

**Project leader**of the project "Harnack inequalities for curvature flows and applications", funded by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG), 2016-2019, gepris:319506420