TRR 71: Geometric Partial Differential Equations
Final Report Abstract
The SFB/Transregio 71 Geometric Partial Differential Equations was devoted to problems which arise in a geometric context, either in differential geometry or in applications involving geometric modeling. With its two locations Freiburg and Tübingen, enhanced by a member from the University of Zürich, the project formed an important center for Geometric Analysis over the five years of funding. The project was unique within Germany to represent Geometric Measure Theory as a central topic. Results include the existence of minmax submanifolds and partitioning surfaces with controlled or prescribed topology, and a thorough analysis of the relation between integrability of mean curvature and higher rectifiability for varifolds. The study of sets of minimizers led to optimal rigidity results for planes and cylinders without conjugate points. The program brought together three groups studying the Willmore Functional from different perspectives: variational methods, integrable systems and numerical analysis. A main focus was on the conformally constrained problem, with results on the existence and regularity of minimizers, as well as the classification of minimizers, stable solutions and critical points. An existence theorem for minimizers with prescribed isoperimetric ratio relates to the Helfrich model for elastic cell membranes. On the numerical side a main task was the development of efficient and stable algorithms for the Willmore flow. Another issue was the modelling and discretization of anisotropic geometric functionals. A project on intrinsic geometry involved the analysis of curvature obstructions in the line of Gromov’s K-area. An interesting development was the study of fully nonlinear curvature flows in conformal geometry, with applications to geometric inequalities. A key feature of the SFB/TR 71 was the cooperation between analysis, numerical analysis and computation. This is strongly realized in work on Nonlinear Effects in Fluids, especially on the analysis and numerical analysis of partial differential equations on stationary and evolving surfaces, and on fluid structure interactions. One result in this direction is the existence of global weak solutions for the interaction of a Newtonian fluid with a linearly elastic Koiter shell. Another project studied the classical Euler equations, constructing a rather striking example of a continuous dissipative solution. The interplay between theory and numerics was also important in the project area on Geometry driven Dynamics. An effective description for quantum particles constrained to submanifolds was first derived and then transferred into a numerical method. A finite element discretization of Ricci curvature was developed and proved to be consistent, leading to simulations of Ricci flow. Singularities of Ricci flow also motivated an analysis of glueing constructions in the presence of lower curvature bounds. The 2009 application featured five young project leaders. By 2012 all of them received calls and accepted tenured positions at other universities. At the same time the SFB/Transregio 71 had significant impact on the scientific development at both its locations Freiburg and Tübingen.
Publications
- Genus bounds for minimal surfaces arising from min-max constructions, J. Reine Angew. Math. 644 (2010), 47–99
De Lellis, C. and Pellandini, F.
(See online at https://doi.org/10.1515/crelle.2010.052) - Lawson’s genus two surface and meromorphic connections, Math. Z. 274, Issue 3-4, 745–760
Heller, S.
(See online at https://doi.org/10.1007/s00209-012-1094-9) - Conformal maps from a 2-torus to the 4-sphere, J. Reine Angew. Math. 671 (2012), 1–30
Bohle, C., Leschke, K., Pedit, F. and Pinkall, U.
(See online at https://doi.org/10.1515/CRELLE.2011.156) - Decay estimates for the quadratic tilt-excess of integral varifolds, Arch. Ration. Mech. Anal. 204 (2012), no. 1, 1–83
Menne, U.
(See online at https://doi.org/10.1007/s00205-011-0468-1) - Isoperimetric inequalities for minimal surfaces in Riemannian manifolds: a counterexample in higher codimension, Calc. Var. Partial Differential Equations 45 (2012), 455–466
Bangert, V. and R öttgen, N.
(See online at https://doi.org/10.1007/s00526-011-0466-z) - Ln/2-Curvature Gaps of the Weyl Tensor, J. Geom. Anal.
Listing, M.
(See online at https://doi.org/10.1007/s12220-012-9356-7) - Runge-Kutta time discretization of parabolic differential equations on evolving surfaces, IMA J. Numer. Anal. 32 (2012), 394–416
Dziuk, G., Lubich, Ch., Mansour, D.
(See online at https://doi.org/10.1093/imanum/drr017) - W 2,2-conformal immersions of a closed Riemann surface into Rn, Comm. Anal. Geom. 20 (2012), no. 2, 313–340
Kuwert, E., Li, Y.
(See online at https://doi.org/10.4310/CAG.2012.v20.n2.a4) - Willmore minimizers with prescribed isoperimetric ratio, Arch. Ration. Mech. Anal. 203 (2012), 901–941
Schygulla, J.
(See online at https://doi.org/10.1007/s00205-011-0465-4) - A local discontinuous Galerkin approximation for systems with p-structure, IMA J. Num. Anal. (2013)
Diening, L., Kröner, D., Růžička, M. and Toulopoulos, I.
(See online at https://doi.org/10.1093/imanum/drt040) - A new conformal invariant on 3-dimensional manifolds, Adv. Math. 249 (2013), 131–160
Ge, Y. and Wang, G.
(See online at https://doi.org/10.1016/j.aim.2013.09.009) - A relaxed partioning disk for strictly convex domains, PhD thesis (2013)
Ludwig, A.
- Area growth and rigidity of of surfaces without conjugate points, J. Differential Geom. 94 (2013), 367–385
Bangert, V. and Emmerich, P.
(See online at https://doi.org/10.4310/jdg/1370979332) - Dissipative continuous Euler flows, Inventiones Mathematicae 193 (2013), no. 2, 377–407
De Lellis, C. and Székelyhidi, L., Jr.
(See online at https://doi.org/10.1007/s00222-012-0429-9) - Estimation of the conformal factor under bounded Willmore energy, Mathematische Zeitschrift, 274 (2013), 1341–1383
Schätzle, R. M.
(See online at https://doi.org/10.1007/s00209-012-1119-4) - Effective Hamiltonians for constrained quantum systems, Memoirs of the AMS 1083, 99 pages, 2013
Teufel, S., Wachsmuth, J.
(See online at https://doi.org/10.1090/memo/1083) - Finite element methods for surface PDEs, Acta Numerica 22 (2013), 289–396
Dziuk, G., Elliott, C. M.
(See online at https://doi.org/10.1017/S0962492913000056) - Flows of constant mean curvature tori in the 3-sphere: the equivariant case, J. Reine Angew. Math. (2013)
Kilian, M., Schmidt, M. and Schmitt, N.
(See online at https://doi.org/10.1515/crelle-2013-0079) - Homology of finite K-area, Math. Z. 275 (2013), no. 1-2, 91–107
Listing, M.
(See online at https://doi.org/10.1007/s00209-012-1124-7) - Hyperbolic Alexandrov-Fenchel quermassintegral inequalities. J. Differential Geom.
Ge Yuxin, Wang, G. and Wu, Jie
(See online at https://doi.org/10.48550/arXiv.1304.1417) - Isoparametric finite element approximation of Ricci curvature, IMA J. Numer. Anal. 33 (2013), no. 4, 1265–1290
Fritz, H.
(See online at https://doi.org/10.1093/imanum/drs037) - Minimizers of the Willmore functional under fixed conformal class, J. Differential Geom. 93 (2013), 471–530
Kuwert, E. and Schätzle, R.
(See online at https://doi.org/10.4310/jdg/1361844942) - On problems related to an inequality of Andrews, De Lellis and Topping, IMRN, Int. Math. Res. Notices (2013) Vol. 2013, 4798–4818
Ge, Y., Wang, G. and Xia, C.
(See online at https://doi.org/10.1093/imrn/rns196) - Scalar conservation laws on moving hypersurfaces, Interfaces Free Bound. 15 (2013), 203–236
Dziuk, G., Kröner, D., Müller, T.
(See online at https://doi.org/10.4171/ifb/301) - Semiclassical approximations for Hamiltonians with operator-valued symbols, Commun. Math. Phys. 320 (2013), 821–849
Stiepan, H., Teufel, S.
(See online at https://doi.org/10.1007/s00220-012-1650-5) - The existence of embedded minimal hypersurfaces, Jour. Differential Geom. 95 (2013), no. 3, 355–388
De Lellis, C. and Tasnady, D.
(See online at https://doi.org/10.4310/jdg/1381931732) - Willmore-type regularization of mean curvature flow in the presence of a non-convex anisotropy. The graph setting: analysis of the stationary case and numerics for the evolution problem, Adv. Differential Equations 18 (2013), no. 3-4, 265–308
Pozzi, P. and Reiter, Ph.
(See online at https://doi.org/10.57262/ade/1360073018) - Energy quantization for Willmore surfaces and applications, Annals of Math. 180
Bernard, Y., and Rivière, T.
(See online at https://doi.org/10.4007/annals.2014.180.1.2) - Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds, Mathematische Annalen (2014)
Kuwert, E., Mondino, A. and Schygulla, J.
(See online at https://doi.org/10.1007/s00208-013-1005-3) - New examples of conformally constrained Willmore minimizers of explicit type (2014), Adv. Calc. Var.
Ndiaye, C. B., Schätzle, R. M.
(See online at https://doi.org/10.1515/acv-2014-0005) - Smoothing singularities of Riemannian metrics while preserving lower curvature bounds, PhD thesis (2014)
Schlichting, A.
(See online at https://doi.org/10.25673/4040) - Trapped Reeb orbits do not imply periodic ones, Inventiones mathematicae (2014)
Geiges, H., Röttgen N. and Zehmisch, K.
(See online at https://doi.org/10.1007/s00222-014-0500-9) - Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell, Arch. Rat. Mech. Anal. 211 (2014), 205–255
Lengeler, D., Růžička, M.
(See online at https://doi.org/10.1007/s00205-013-0686-9)
