The 10th DIFFERENTIAL GEOMETRY DAY
Wednesday the 14th of May 2014
(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.
(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).
(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
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
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.