The 10th DIFFERENTIAL GEOMETRY DAY

at LUND

Wednesday the 14th of May 2014


ABSTRACTS

Stefano Montaldo (University of Cagliari, IT) Biharmonic Immersions and Related Topics In this talk I will give an overview on biharmonic immersions focusing, in particular, to biharmonic immersions into a sphere. In the last part of the talk, using the stress-energy tensor associated to the bienergy, I will introduce the notion of biconservative immersions, of which biharmonic immersions are a subclass. Bent Fuglede (University of Copenhagen, DK) Extensions of the Eells-Sampson theorem to harmonic maps between more general spaces The existence theorem of Eells-Sampson (1964) asserts that if $X$ and $Y$ are compact Riemannian manifolds without boundary, and if $Y$ has nonpositive sectional curvature, then every continuous map $\phi:X\to Y$ is homotopic to a harmonic map which has minimum energy in its homotopy class. This was obtained by using the heat equation method. The Eells-Sampson theorem was extended by a variational approach by Gromov-Schoen (1992), who allowed a suitable Riemannian polyhedron as target $Y$. It was further extended by N.J. Korevaar-Schoen (1993), who took for $Y$ any compact geodesic space of non-positive Alexandrov curvature. While keeping this geodesic space target, the compact source manifold $X$ was replaced in Eells-Fuglede (2001) by any compact "admissible" Riemannian polyhedron. A complete proof of this was obtained later by Fuglede (2008). Francis Burstall (University of Bath, UK) 1964 And All That: Eells-Sampson and After Inspired by a re-reading of the Famous Paper, I shall discuss its precursors, its achievements and its legacy. John C. Wood (University of Leeds, UK) Some constructions of harmonic maps, 50 years after Eells and Sampson In 1964, James Eells and Joseph Sampson ensured that harmonic maps would become an enduring part of mathematics by giving a general existence theorem for harmonic maps into manifolds of non-positive curvature. Joseph Sampson studied this case further, giving a uniqueness theorem for the harmonic maps in the case of non-positive curvature which strengthened that of Hartman. On the other hand, Jim Eells' main interest was in harmonic maps to other targets, in particular to positively curved ones, where no general existence theorem is available. I shall outline some results in the positively curved case, concentrating on (i) constructions of harmonic maps to the unitary group --- joint work with B. Simões and M.J. Ferreira; (ii) twistor theory for harmonic maps to the exceptional symmetric space G_2/SO(4) and connections with harmonic maps into the 6-sphere --- joint work with M. Svensson.