Dr. Julian Scheuer

Albert-Ludwigs-Universität
Mathematisches Institut
Abteilung für Reine Mathematik
Arbeitsgruppe Analysis

Eckerstr. 1, Raum 206
79104 Freiburg i. Br.

E-Mail: julian.scheuer@math.uni-freiburg.de
Phone: +49-761-203-5551

Office hours: Tuesday, 8:00-10:00 and Thursday, 8:00-10:00.
Geometric Analysis and
Partial Differential Equations
Vita Research Teaching DFG-Project

Short Research Statement

In the past I have been interested in evolving hypersurfaces in ambient Riemannian manifolds, especially inverse curvature flows with general classes of speeds and also with geometric Neumann type boundary conditions. The main achievements in this area were generalizations of present results to a broader class of ambient spaces, curvature functions and homogenities. The strength of some convergence results for inverse curvature flows in Euclidean space could be improved.

In curvature flows estimates on the traceless second fundamental form have shown to be a useful tool to obtain convergence to spheres. Thus I am naturally interested in quantitative versions of the well known umbilicity theorem for hypersurfaces. This also includes pinching in terms of integral norms of the traceless second fundamental form or in terms of the Laplace-spectrum. Such question were addressed jointly with Julien Roth.

Recently, with Paul Bryan and Mohammad Ivaki, I have been focusing on differential Harnack estimates for curvature flows in Riemannian and Lorentzian manifolds. Such estimates are important to obtain uniform control on the flow speed and hence useful in the deduction of convergence results. Another application is the classification of ancient solutions to contracting flows on the sphere, which is, after partial results were achieved, research in progress. Also have a look at the
DFG-Project: Harnack inequalities for curvature flows and applications.

Publications and Preprints

Publications Preprints

My papers at arXiv.org

Talks

  • 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 at 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", jointly organized by the AEI, the 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.
  • Summer term 2012: Detailed presentation my Ph.D. thesis Non-scale-invariant inverse curvature flows in hyperbolic space in the group seminar on geometric analysis at the university of Heidelberg, Germany.
  • Conferences and Workshops

  • 21.08.-25.08.2017: Conference PADGE at the KU Leuven, Belgium.
  • 18.06.-24.06.2017: Conference Differential geometry, organised by W. Goemans, B. Opozda U. Simon, Banach Center Bedlewo, Poland
  • 06.06.-09.06.2017: Conference Advances in geometric analysis, organised by G. Huisken, R. Schoen, M. Struwe and T. Riviere, ETH Zürich, Switzerland
  • 19.09.-23.09.2016: Summer School Geometric Flows and the Geometry of Space-time, organised by V. Cortés, K. Kröncke and J. Louis, 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 organised by Rafael Lopez and Juncheol Pyo at the 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.