Math @ Duke

Publications [#318282] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, R, Minimal Lagrangian submanifolds of KählerEinstein manifolds,
in Differential geometry and differential equations (Shanghai, 1985), Lecture Notes in Math., edited by Gu, C; Berger, M; Bryant, RL, vol. 1255
(1987),
pp. 112, Springer Verlag, ISBN 354017849X [MR88j:53061]
(last updated on 2018/09/22)
Author's Comments: Most Kähler manifolds do not contain any
submanifolds that
are simultaneously minimal and Lagrangian since the
combination of the
two conditions is equivalent to an overdetermined
system of PDE for the
submanifold that is generally incompatible.
However, in case the 2nmanifold M is
KählerEinstein, the situation
is different. I prove that, in this case, the overdetermined
system is
involutive. In fact, every realanalytic submanifold of
dimension n1 that
is subLagrangian (i.e., on which the Kähler form
vanishes) lies in
a circle of nmanifolds, each of which is minimal and
Lagrangian (these
will not generally be compact).
Quite recently, these minimal Lagrangian
manifolds have become a subject
of interest to physicists (in the physics literature this
comes under the
heading of `BPS states' in string theory). Works in this
area in physics
can be found by such authors as Vafa, Witten, Yau, and
Zaslow. On the mathematical
side, R. Schoen and J. Wolfson have worked in this
area, not to mention
R. Harvey and H. B. Lawson (in the Ricciflat case).
For some reason, I have been getting many
requests for reprints of this
article lately, perhaps because of it being referenced in
the physics literature.
However, I never received reprints for this article and it
predates my
use of TeX by many years; all I have is a typescript.
Since the article
has been published and is easily available, I am not
making photocopies
to mail out.


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