# Schedule of the Workshop on Geometric Flows

## Tuesday, November 22

09:00-10:00 |
Huy Nguyen: Singular Willmore spheres in R^{3} |

10:00-10:30 |
Coffee |

10:30-11:30 |
Simon Blatt: Analysis of O'Hara's knot energies |

11:30-12:30 |
Frederik Witt: The geometry of G_{2}-structures |

12:30-14:30 |
Lunch |

14:30-15:30 |
Hartmut Weiss: A stability theorem for the Dirichlet energy flow |

15:30-16:00 |
Coffee |

16:00-17:00 |
Johannes Nordström: Diffeomorphism types of compact G_{2}-manifolds |

17:00-18:00 |
Herbert Koch: Parabolic equations with rough initial data |

18:00- |
Welcome Reception |

## Wednesday, November 23

09:00-10:00 |
Jan Swoboda: The Yang-Mills gradient flow and its variants |

10:00-10:30 |
Coffee |

10:30-11:30 |
Karl-Theodor Sturm: Ricci bounds for metric measure spaces |

11:30-12:30 |
Burkhard Wilking: Applications of the Margulis Lemma |

12:30-14:30 |
Lunch |

14:30- |
Discussion Afternoon with Coffee at 16:00 |

## Thursday, November 24

09:00-10:00 |
Marc Nardmann: Existence of dominant energy metrics |

10:00-10:30 |
Coffee |

10:30-11:30 |
Nadine Große: The Yamabe equation on complete manifolds of finite volume |

11:30-12:30 |
Gregor Giesen: Ricci flow of incomplete surfaces |

12:30-14:30 |
Lunch |

14:30-15:30 |
Esther Cabezas-Rivas: How to produce a Ricci flow via Cheeger-Gromoll exhaustion |

15:30-16:00 |
Coffee |

16:00-17:00 |
Ernst Kuwert: Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds |

17:00-18:00 |
Christoph Böhm: The second best Einstein metric in higher dimensions |

## Friday, November 25

09:00-10:00 |
Reto Müller: A family of fourth-order gradient flows on three-manifolds |

10:00-10:30 |
Coffee |

10:30-11:30 |
Oliver Schnürer: Gauss curvature flows of entire graphs |

11:30-12:30 |
Miles Simon: Some local results for Ricci flows of manifolds with curvature bounded from below |

12:30-14:30 |
Lunch |

## Abstracts:

Simon Blatt (University of Warwick): Analysis of O'Hara's knot energies

All of us know how hard it can be to decide whether the cable spaghetti lying in front of us is really knotted or whether the knot vanishes into thin air after pushing and pulling at the right strings. In this talk, we approach this problem using gradient flows of a family of energies introduced by O'Hara in 1991-1994. We will see that this allows us to transform any closed curve into a special set of representatives - the stationary points of these energies - without changing the type of knot. We prove longtime existence and smooth convergence to stationary points for these evolution equations. Furthermore, I will mention some recent progress regarding the regularity of stationary points.

Esther Cabezas-Rivas (Universität Münster): How to produce a Ricci Flow via Cheeger-Gromoll exhaustion

We will talk about how to prove short time existence for the Ricci flow on open manifolds of nonnegative complex sectional curvature. We do not require upper curvature bounds. By considering the doubling of convex sets contained in a Cheeger-Gromoll convex exhaustion and solving the singular initial value problem for the Ricci flow on these closed manifolds, we obtain a sequence of closed solutions of the Ricci flow with nonnegative complex sectional curvature which subconverge to a solution of the Ricci flow on the open manifold. Furthermore, we find an optimal volume growth condition which guarantees long time existence, and we give an analysis of the long time behaviour of the Ricci flow. Finally, we construct an explicit example of an immortal nonnegatively curved solution of the Ricci flow with unbounded curvature for all time. This is joint work with Burkhard Wilking.

Gregor Giesen (University of Warwick): Ricci flow of incomplete surfaces

Since the initial value problem of a Ricci flow starting on an incomplete surface is not well-posed in general, we try to show that one can gain well-posedness requiring the metric to become complete instantaneously. While the issue of uniqueness is partly open, we state a general existence result and describe the asymptotic behaviour of the solution in most cases. This is joint work with Peter Topping.

Nadine Große (Universität Leipzig): The Yamabe equation on complete manifolds of finite volume

The well-known Yamabe problem asks whether on a given Riemannian manifold there is a conformal metric with constant scalar curvature. This is well understood for closed manifold, and has an affirmative answer which is obtained by studying the Yamabe equation, a nonlinear elliptic differential equation, arising from a variational problem given by the so-called Yamabe invariant. We study the existence of a solution of the Yamabe equation on complete manifolds with finite volume and positive Yamabe invariant. In order to circumvent the standard methods on closed manifolds which heavily rely on global (compact) Sobolev embeddings we approximate the solution by eigenfunctions of certain conformal complete metrics. This gives rise to a method which can e.g. be applied to the spinorial Yamabe problem or to reprove the existence of a solution in the closed case.

Herbert Koch (Universität Bonn): Parabolic equations with rough initial data

Maximal functions and square functions are central tools in real harmonic analysis. In recent years a harmonic analysis point of view became useful in studying linear elliptic problems, like the Kato square root problem, and nonlinear parabolic problems, like the Navier-Stokes equations, harmonic map heat flow, Willmore flow and surface diffusion. I will explain joint work with Tobias Lamm on nonlinear parabolic problems, the construction of function spaces and their relation to maximal functions, square functions and Carleson measures.

Ernst Kuwert (Universität Freiburg): Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold (M,g). Under the assumption that the sectional curvature K of M is strictly positive, we prove the existence of a smooth immersion f:S^{2}->M minimizing the L^{2} integral of the second fundamental form. Assuming instead that K <= 2 and that there is some point p in M with scalar curvature R_{g}(p) > 6, we obtain a smooth minimizer f:S^{2}->M for the functional given by the integrand 1/4*|H|^{2}+1, where H is the mean curvature. This is joint work with Andrea Mondino and Johannes Schygulla.

Reto Müller (Imperial College London): A family of fourth-order gradient flows on three-manifolds

We study the family of gradient flows of the quadratic functionals obtained by integrating |Rc|^{2} - 3a/8 R^{2} over a closed three-manifold, where Rc and R denote the Ricci and scalar curvature, respectively, and a is a nonnegative parameter. For a<1, we obtain short-time existence for the corresponding flows with a standard DeTurck trick, while for a>1 one cannot obtain short-time existence in general. Our main focus will be on the interesting boundary case a=1. This is joint work with Roberta Alessandroni and Zindine Djadli.

Marc Nardmann (Universität Hamburg): Existence of dominant energy metrics

Motivated by old problems in General Relativity, in particular the question whether the spatial topology of the universe can change with time in a physically reasonable way, I explain how Lorentzian metrics which satisfy the dominant energy condition can be constructed on a given manifold of possibly complicated topology.

Huy Nguyen (University of Warwick): Singular Willmore spheres in R^{3}

In this talk, we will discuss a classification of Willmore spheres in R^{3} with a certain bound on the number of singular points which extends a classification due to Bryant, in particular we show that such spheres have energy equal to a multiple of 4*pi. In particular, if the Willmore energy is equal to 8*pi, then the surface will be shown to be an inversion of a catenoid. We will then apply this result to the Willmore flow of spheres with energy less than or equal to 12*pi and show that either the Willmore flow converges to a round sphere or a rescaling of a singular point converges to a catenoid. This work is joint with Tobias Lamm.

Johannes Nordström (Imperial College London): Diffeomorphism types of compact G_{2}-manifolds

G_{2} is an exceptional case in the classification of Riemannian holonomy groups. Compact manifolds with holonomy G_{2} have been constructed by Joyce by resolving flat orbifolds, and by Kovalev from twisted connected sums of asymptotically cylindrical Calabi-Yau manifolds. Still very little is known about the question of which smooth 7-manifolds admit G_{2} holonomy metrics, and the global properties of the space of such metrics on a fixed manifold. I will describe joint work with Alessio Corti, Mark Haskins and Tommaso Pacini that produces more examples by a generalisation of Kovalev's construction, where we can compute invariants that determine the diffeomorphism types. In particular, we find many examples of different G_{2} metrics on the same smooth manifold.

Oliver Schnürer (Universität Konstanz): Gauss curvature flows of entire graphs

We study entire graphs in Euclidean space that evolve with normal velocity equal to a power of the Gauss curvature. Mild restrictions on the initial data ensure that smooth solutions exist for all positive times. For initial data close to cones, we obtain stability results. This is joint work with John Urbas.

Miles Simon (Universität Magdeburg): Some local results for Ricci flows of manifolds with curvature bounded from below

We prove some local estimates for solutions to Ricci flow which satisfy (locally): a) curvature operator is bounded from below by minus one; b) volume of a ball is bounded from below by a constant v. We explain some consequences.

Karl-Theodor Sturm (Universität Bonn): Ricci Bounds for Metric Measure Spaces

We present the concept of generalized lower Ricci curvature bounds for metric measure spaces (M,d,m), introduced by Lott, Villani and the author. These curvature bounds are defined in terms of optimal transportation, more precisely, in terms of convexity properties of the relative entropy Ent(. | m) regarded as function on the Wasserstein space of probability measures on the given space M. For Riemannian manifolds, Curv(M,d,m) ≥ K if and only if Ric_M ≥ K on M. Other important examples covered by this concept are Finsler manifolds and Alexandrov spaces. One of the main results is that these lower curvature bounds are stable under (e.g. measured Gromov-Hausdorff) convergence. Moreover, we introduce a curvature-dimension condition CD(K,N) being more restrictive than the curvature bound Curv(M,d,m) ≥ K. For Riemannian manifolds, CD(K,N) is equivalent to Ric_M ≥ K and dim(M) ≤ N. Condition CD(K,N) implies sharp versions of the Brunn-Minkowski inequality, of the Bishop-Gromov volume comparison theorem and of the Bonnet-Myers theorem. Extension of this curvature concept to discrete spaces and infinite dimensional spaces will be indicated, e.g. for the Wiener space Curv(M,d,m)=1.

Jan Swoboda (MPI Bonn): The Yang-Mills gradient flow and its variants

We review our construction of a Morse homology theory for the Yang-Mills gradient flow in two dimensions and its relation to Weber's heat flow homology. We discuss compactness and Morse-Smale transversality for the perturbed flow, which invokes a novel L^{2} local slice theorem due to Mrowka-Wehrheim. Finally, we show how a modified Yang-Mills functional leads to an "elliptic Yang-Mills flow" for which a Floer type homology theory is currently under construction. This is joint work with Remi Janner.

Hartmut Weiss (LMU München): A stability theorem for the Dirichlet energy flow

In a joint paper with Frederik Witt, we introduced a natural energy functional on the space of G_{2}-forms on a 7-manifold and studied its negative gradient flow. In this talk I will discuss a stability result for this flow near a torsion-free G_{2}-structure.

Burkhard Wilking (Universität Münster): Applications of the Margulis Lemma

We (joint work with V. Kapovitch) establish a Margulis Lemma for manifolds with lower Ricci curvature bound. In this talk I will focus on some of the applications of the Margulis Lemma rather than the proof of it.

Frederik Witt (Universität Münster): The geometry of G_{2}-structures

In this talk, I will give an introduction to G_{2}-structures on seven manifolds and develop the theory of holonomy G_{2}-manifolds from a variational point of view.