分野別セミナー

＊いつもと開始時間・場所が異なるのでご注意ください。
概要：
Biharmonic maps are by definition critical maps of
the bienergy functional:
$$E_2(phi):= \frac12\int_M\vert\tau(phi)\vert^2\,v_g$$
for smooth maps $\phi$ from $(M^m,g)$ into $(N^n,h)$.
Here $\tau(phi)$ is the tension field of $phi$.
Harmonic maps are naturally biharmonic.
Biharmonic maps which are not harmonic are called proper.

So, our questions are:
(1) What are proper biharmonic maps?
(2) Can one extend theories of harmonic maps,
for example, regularity theorem, bubbling theorem to biharmonic maps?
(3) Can one extend theory of the integrable system of harmonic maps
to the one for biharmonic maps?

In my talk, I will report recent progresses about them.
Namely, after giving a survey of theory of biharmonic maps,
I will show the following:
(I) There exists a positive smooth function $f$ on
$S^1\times M^2$ such that the identity map of $S^1\times M^2$
under a conformal change by $f$to the product metric into
$S^1\times M^2$ with the original product metric is a proper
biharmonic map (a joint work with Prof. H. Naito, Nagoya Univ.).
(II) Some regularity theorem and bubbling theorem hold for
biharmonic maps (joint work with Prof. N. Nakauchi, Yamaguchi Univ.).
(III) There exists some theory of the integrable system
for biharmonic maps.