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Biharmonic curves on a surface
by
Stefano Montaldo
Dipartimento di Matematica, Cagliari,Italy
Coauthors: Renzo Caddeo, Paola Piu
Oral Communication
Since their first work on harmonic maps, J. Eells and
J.H. Sampson suggested the idea of
studying k-harmonic maps. If k=2, the idea is the
following. First define harmonic maps
f : (M, g) --> (N, h) between two
Riemannian manifolds as critical points of the energy
E(f)=\frac12\intM |df|2 dv. The corresponding
Euler-Lagrange
equation for the energy is given by the vanishing of the tension
field \tau(f).
Then define the bienergy of a map f by
E2(f)=\frac12\intM |\tau(f)|2 dv, and say that f is
biharmonic if it is a critical point of the bienergy.
In 1986 J.G. Ying derived the first variation formula of the bienergy showing that the Euler-Lagrange equation for E2 is \tau2(f)=J(\tau(f))=0, where J is the Jacobi operator of f.
In this talk we will restrict our attention to isometric immersions
f:I=[a, b] --> (M, g) from an interval I to a Riemannian
manifold. The image C=f(I) is the trace of a
curve in M and f is a parametrization of
C by arc length.
Note that C=f(I) is part of a geodesic of M if and only if
f is a harmonic map. Moreover,
from the biharmonic equation
if f is harmonic then it is biharmonic,
thus geodesics are a subclass of biharmonic curves.
The converse in not true and this makes the class of
biharmonic curves richer than that of geodesics.
It is then natural to ask what geometric properties
characterize biharmonic curves.
With this in mind, the aim of this talk is to present
some examples of nongeodesic biharmonic curves in a surface.
We first present some general results on the geometry of nongeodesic biharmonic curves on a surface. In the second part we concentrate on parametrized surfaces of revolution in three-dimensional Euclidean space. In particular, we will present some explicit examples of nongeodesic biharmonic curves in a surface of revolution with constant Gauss curvature.
Date received: April 5, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-24.