## Priv.-Doz. Dr. Julian Scheuer## Albert-Ludwigs-Universität
Ernst-Zermelo-Str. 1, Raum 206 | Geometric Analysis and Partial Differential Equations |

Vita | Research | Teaching | DFG-Project |

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.

- 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. -
*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.

- with G. Wang and C. Xia:
*Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball*, (2018), arxiv:1811.05776. -
*Inverse curvature flows in Riemannian warped products*, (2017), to appear in J. Funct. Anal., arxiv:1712.09521. - with C. Xia:
*Locally constrained inverse curvature flows*, (2017), arxiv:1708.06125. - with P. Bryan and M. N. Ivaki:
*Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds*, (2017), arxiv:1703.07493. - with H. Kröner:
*Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature*, (2017), to appear in Math. Nachr., arxiv:1703.07087. - with P. Bryan and M. N. Ivaki:
*On the classification of ancient solutions to curvature flows on the sphere*, (2016), arxiv:1604.01694. - with P. Bryan and M. N. Ivaki:
*Harnack inequalities for evolving hypersurfaces on the sphere*, (2015), to appear in Comm. Anal. Geom., arxiv:1512.03374. - with J. Roth:
*Explicit rigidity of almost-umbilical hypersurfaces*, (2015), to appear in Asian J. Math., arxiv:1504.05749.

**My papers at arXiv.org**