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)
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
- 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
.