SHEAR-FREE RAY CONGRUENCES ON CURVED SPACE-TIMES. Abstract

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1 SHEAR-FREE RAY CONGRUENCES ON CURVED SPACE-TIMES PAUL BAIRD A shear-free ray congruence (SFR) on Minkowsi space is a family of null geodesics that fill our a region of space-time, with the property that Lie transport along each geodesic, of vectors in the screen space (a 2- dimensional space-like compliment), is conformal (without shear). They are the basis of the construction of solutions to the zero rest-mass field equations and arise from a complex analytic surface in the twistor space CP 3 ; they can be considered as the semi-riemannian analogue of an integrable complex structure on a Riemannian 4-manifold. In work with John C. Wood of a few years back, we showed how an SFR on Minkowski space corresponds to an evolving family of semi-conformal mappings on space-like slices. This latter perspective can be adapted to a curved setting: Let (M 3, g) be a Riemannian 3-manifold endowed with a unit vector field U tangent to a conformal foliation (which locally determines a semi-conformal mapping); find an evolution (g(t), U(t)) of the metric and the unit vector field which generates a space-time (M 4, f 2 dt 2 + g(t)) for some function f, in such a way that the null curves determined by U are both geodesic and shear-free. This turns out to be quite a tricky problem and the results are partial. This is joint work with Mohammad Wehbe.

2 HARMONIC GAUSS MAPS FRANCIS E. BURSTALL The modern theory of harmonic maps meets classical submanifold geometry in the notion of a Gauss map. The theorem of Ruh Vilms, for example, tells us that a surface in R 3 has constant mean curvature if and only if its Gauss map is a harmonic map to S 2. This leads to a fruitful interaction between classical and modern ideas, especially around the integrable systems approach to harmonic maps. Moreover, these ideas can be extended, if one is allowed sufficient flexibility in the notion of Gauss map, to treat Willmore surfaces; projectively minimal and Lie minimal surfaces; surfaces in 4-dimensional space-forms with holomorphic mean curvature and others. In my talk, I shall give an overview of these topics aimed at nonexperts.

3 ON HARMONICITY OF PRODUCTS OF HARMONIC FORMS D. KOTSCHICK From the point of view of the theory of harmonic maps, the study of harmonic differential forms, though important, is a simplistic special case. The underlying PDE is linear rather than non-linear and many issues that are interesting in the general theory are rather trivial in this case. Nevertheless, the study of harmonic forms does lead to interesting problems, some of which are non-linear. In this talk I shall survey some such problems arising from the question when wedge products of harmonic forms are again harmonic. Manifolds admitting a Riemannian metric for which this is the case are extremely special, and the harmonic forms for such a metric define interesting differential-geometric structures on the manifold.

4 THE QUATERNIONIC KP HIERARCHY AND CONFORMALLY IMMERSED 2-TORI IN THE 4-SPHERE IAN MCINTOSH The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing C with H, mutatis mutandis, in the standard construction of the KP hierarchy equations and solutions; it is equivalent to what is often called the Davey-Stewartson II hierarchy. I will summarise its relationship with the theory of quaternionic holomorphic 2-tori in HP 1 (which are equivalent to conformally immersed 2-tori in S 4 ). I will focus on explaining the relationship between different notions of spectral curve : the QKP spectral curve, which arises from an algebra of commuting differential operators; the (unnormalised) Floquet multiplier spectral curve for the related Dirac operator; and the curve parameterising Darboux transforms of a conformal 2-torus in S 4 (in the sense of Bohle, Leschke, Pedit and Pinkall). The concrete example provided by Hamiltonian stationary Lagrangian tori in C 2 will be used to help illustrate the general case.

5 AN EXPERIMENTAL STUDY OF GOLDSTEIN-PETRICH CURVES EMILIO MUSSO The interplay between integrable evolution equations and the motion of curves has been the focus of intense research, both in geometry and mathematical physics. In 1992, Goldstein and Petrich related the mkdv hierarchy to the motions of curves in 2-dimensional space forms. The second Goldstein-Petrich flow is defined by the mkdv equation k t + k sss k2 k ss = 0, where k(s, t) is the geodesic curvature of the evolving curve. Closed curves whose shape is invariant under the flow are referred to as Goldstein-Petrich curves. In this talk we give a brief overview of the construction of the Goldstein-Petrich flows. We use numerical methods to investigate global properties of Goldstein-Petrich curves in R 2. We show that, for every integer n 2 and for every l > 0, there exist a one-parameter family of simple, non congruent closed Goldstein-Petrich curves with length l and symmetry group Z n.

6 DIFFERENTIAL GEOMETRY OF LAGRANGIAN SUBMANIFOLDS AND HAMILTONIAN VARIATIONAL PROBLEMS YOSHIHIRO OHNITA In this talk I shall provide a survey on my recent works and their environs on differential geometry of Lagrangian submanifolds in specific symplectic Kähler manifolds. This talk is mainly based on my joint work with Hui Ma (associate professor of Tsinghua Univ.(Beijing) & visiting researcher of OCAMI). The minimality and stability of Lagrangian submanifolds in Kähler manifolds under Hamiltonian deformations was investigated first by Y. G. Oh about the beginning of 1990 s. It is fundamental and interesting as a geometric variational problem related to Lagrangian submanifolds. First we shall discuss several nice classes of Lagrangian submanifolds and their Hamiltonian stability problems in specific Kähler manifolds such as complex space forms (complex Euclidean spaces, complex projective spaces, complex hyperbolic spaces), Hermitian symmetric spaces etc. Especially we shall give attention to Lagrangian submanifolds in complex hyperquadrics, which is a compact Hermitian symmetric spaces of rank 2. The relationship between minimal Lagrangian submanifold in complex hyperquadrics and isoparametric hypersurfaces in spheres will be emphasized. Recently we gave a complete classification of compact homogeneous Lagrangian submanifolds in complex hyperquadrics and we determined the Hamilitonian stability of ALL compact minimal Lagrangian submanifold embedded in complex hyperquadrics which are obtained as the Gauss images of homogeneous isoparametric hypersurfaces in spheres. References [1] H. Ma and Y. Ohnita, On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres, Math.Z. 261 (2009), [2] H. Ma and Y. Ohnita, Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces, in preparation. Department of Mathematics, Osaka City University & Osaka City University Advanced Mathematical Institute, Sugimoto, Sumiyoshi-ku, Osaka, , JAPAN address: ohnita@sci.osaka-cu.ac.jp 1

7 RECENT RESULTS IN LOCALLY CONFORMALLY KÄHLER GEOMETRY LIVIU ORNEA After a brief introduction in the subject, I shall focus on the geometry and topology of LCK manifolds having an automorphic potential on a Kähler covering. I shall explain how these LCK structures can be deformed to Vaisman ones and apply this result to derive restrictions on the foundamental group. Moreover, I shall characterize LCK manifolds with automorphic potential in terms of transformation groups. Other results on transformation groups will also be mentioned. The talk is based on joint works with Andrei Moroianu and Misha Verbitsky.

8 GEOMETRY OF BIHARMONIC MAPS HAJIME URAKAWA, DIVISION OF MATHEMATICS, GRADUATE SCHOOL OF INFORMATION SCIENCES, TOHOKU UNIVERSITY Study of harmonic maps has been developed since the work of J. Eells and J.H. Sampson, in 1964, and the one of biharmonic maps was raised by J. Eells and L. Lemaire, in 1983, and begun by G.Y. Jiang, in In my talk, I would like to report the recent progress of our works about it as follows. 1. The first and second variation formulas of biharmonic maps. 2. Classication of biharmonic isoparametric hypersurfaces in the unit sphere. 3. Classications of biharmonic homogeneous hypersurfaces in the complex, and quaternionic projective spaces. 4. Chen, Caddeo, Montaldo and Piu s conjecture on biharmonic submanifolds of nonpositive curvature manifolds. 5. Conformal changes of Riemannian metrics and biharmonic maps. 6. Harmonic morphisms versus biharmonic morphisms. References [1] C.L. Bejan and H. Urakawa, Yang-Mills fields analogue of biharmonic maps, In: Topics in Almost Hermitian Geometry and Related Topics, World Scientific, (2005), [2] B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math., 17 (1991), [3] T. Ichiyama, J. Inoguchi and H. Urakawa Bi-harmonic maps and bi-yang-mills fields, Note di Matematica, (2009). [4] T. Ichiyama, J. Inoguchi and H. Urakawa, Classification and isolation phenomena of bi-harmonic maps and bi-yang-mills fields, Note di Matematica, (2009). [5] Jiang Guoying, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math., 7A, 7B, (4) (1986), ; English translation, Note di Matematica, (2009). [6] E. Loubeau and C. Oniciuc, The index of biharmonic maps in spheres, Compositio Math., 141 (2005), [7] C. Oniciuc, On the second variation formula for biharmonic maps into a sphere, Publ. Math. Debrecen, 61 (2002),

9 HARMONIC MAPS I HAVE KNOWN EXPLICITLY JOHN C. WOOD We will give a survey of those explicit constructions of harmonic maps from surfaces which have pleased the author over the years, culminating in the construction of M.J. Ferreira, B.A. Simões and the author of all harmonic maps from the 2-sphere to the unitary group in terms of freely chosen meromorphic functions. We will then discuss an extension of this construction by M. Svensson and the author to general uniton factorizations for the unitary group, showing how to find explicit formulae for harmonic maps into the orthogonal and symplectic groups and their symmetric quotients.