**Date:**- June 4th - 10th, 2006
**Organizers:**- Klaus Ecker, Berlin
- Michael Struwe, Zuerich
**Programme:**-
Nonlinear evolution equations are important tools in
differential geometry and mathematical physics.
One of the first examples was the harmonic map
heat flow which was used by Eells-Sampson to establish
the existence of harmonic maps into negatively curved
Riemannian manifolds.
Probably currently the most prominent example is the
Hamilton-Ricci flow, which has the aim of settling Thurstons'
geometrization conjecture, with recent major progress due
to Perelman.

A further example is Struwe's prescribed curvature flow approach to the famous Nirenberg problem which asks, which functions arise as the Gaussian curvature of a conformal metric on $S^2$. Lastly, we mention mean curvature flow of hypersurfaces which Huisken and Sinestrari recently used to prove some important topological classification results for 2-convex hypersurfaces in Euclidean space.

In general, nonlinear evolution equations develop concentration or singularities in finite or infinite time. A fine analysis of the behaviour of the solution near such points gives rise to evolution equations of reduced complexity that sometimes allow to continue the flow past singularities and/or permits to obtain a precise characterization of the longterm convergence behaviour of the flow. This behaviour is exemplified in the flows that we shall study in detail in this seminar:

- Singularity structure and asymptotic behaviour of mean curvature flow and Hamilton-Ricci flow
- The presribed curvature flow leading to the solution of Nirenberg's problem

We plan to have two lectures every morning and afternoon, altogether eight lectures by each organiser. The remaining time will be devoted to questions and discussions. For Wednesday afternoon an excursion is planned and on Friday afternoon there will be a final discussion session.

The topics of the lectures will be roughly as follows:

K. Ecker:- The mean curvature flow. Hamilton-Ricci flow. Special solutions such as homothetic and translating solutions. Survey of longterm existence and convergence results. Formation of singularities in finite time. (2-3 hours)
- The maximum principle. Local estimates for curvatures and their derivatives. Curvature blow-up at singularities. (2-3 hours)
- Local energy estimates. Monotonicity formulas. Rescaling near singularities and selfsimilarity of rescaling limits. Entropy and Harnack inequalities. (2-3 hours)
- Dimension reduction. Size and structure of the singular set. Asymptotic expansions near singularities. Continuation through singularities and long-time behaviour. (2-3 hours)

- The prescribed curvature flow. The normalized flow. Existence and regularity for all time. Curvature decay. (2-3 hours)
- Concentration-compactness results for families of conformal metrics on the sphere. Kazdan-Warner identity. Convergence for the Ricci-Hamilton flow on $S^2$. (2-3 hours)
- Concentration analysis for the prescribed scalar curvature flow. The shadow flow. (2-3 hours)
- Existence results for conformal metrics of prescribed scalar curvature. Morse theory. (2-3 hours)

**Prerequisites:**- Basic aquaintance with measure theory and functional analysis, Sobolev spaces, elliptic and parabolic pdes, basic differential geometry of hypersurfaces in Euclidean space and Riemannian geometry
**Literature:**-
S.-Y.A.Chang, P.C.Yang: Prescribing Guassian curvature on $S^2$,
Acta Math. 159 (1987) 215-259

B.Chow, D.Knopf, The Ricci flow: An Introduction, Mathematical Surveys and Monographs, Vol. 110. American Mathematical Society, Providence, RI, 2004.

K.Ecker: Regularity theory for mean curvature flow, Progress in nonlinear differential equations and their applications, Birkhaeuser (2004)

R.S.Hamilton: Formation of singularities in the Ricci flow, Surveys in Diff. Geom. 2 (1995), 7-136

G.Huisken, C.Sinestrari, Convexity estimates for mean curvature flow and singularities for mean convex surfaces, Acta Math. 183, 45-70 (1999)

A.Malchiodi, M.Struwe: The $Q$- curvature flow on $S^4$, preprint (2005)

G.Perelman: The entropy for the Ricci flow and its geometric applications, arXiv:math.DG/0211159v1 11 Nov 2002

M.Struwe: On the evolution of harmonic maps in higher dimensions, J. Diff. Geom. 28, 485-502 (1988)

M.Struwe: A flow approach to Nirenberg's problem, Duke Math.J. 128 (2005), 19-64

**Deadline for applications:**- April 1, 2006
- The seminars take place at the
Mathematisches Forschungsinstitut Oberwolfach.
The number of participants is restricted to 24.
The Institute covers accommodation and food.
Travel expenses cannot be reimbursed.
Applications including
- full name and address, including e-mail address
- present position, university
- name of supervisor of Ph.D. thesis
- a short summary of previous work and interest

should be sent as hard copy or by e-mail (.ps or .pdf file) to:

Prof. Dr. Gert-Martin Greuel

Universität Kaiserslautern

Fachbereich Mathematik

Erwin Schrödingerstr.

67663 Kaiserslautern, Germany

.

Mathematisches Forschungsinstitut Oberwolfach updated: March 11th, 2005