Second Variation of Volume and energy of vector fields. Stability of Hopf vector fields.
by
Olga Gil-Medrano
Universidad de Valencia, Spain.
Coauthors: Elisa Llinares-Fuster

Oral Communication

Given a Riemannian manifold (M, g), its tangent sphere bundle can be endowed with a natural Riemannian metric g^S, known as the Sasaki metric, involving g and its Levi-Civita connection Ñ. The set of unit vector fields \Cal X¹(M), if nonempty, is a submanifold of \Cal X(M).

For compact boundaryless M, two natural functionals can be defined in \Cal X¹(M): the volume F and the energy E. The volume of a vector field V is defined as the volume of V(M) in (T¹M, g^S) and its energy is defined as the energy of the map V: (M, g) ® (T¹M, g^S). The functionals admit the following expressions

F(V)= ó õ M  Ö   \operatornamedetLV    dvg ,       E(V) = n 2 Vol(M, g) + 1 2 ó õ M  ||ÑV||2 dvg,

where L_V=\operatornameId+(ÑV)^t øÑV . The relevant part of the energy is also known as the total bending B(V)=ò_M ||ÑV||²  dv_g.

The condition for a vector field to be a critical point of B has been computed in []. In [] we have computed the condition for a vector field to be a critical point of the volume that turns out to be equivalent to the corresponding submanifold of \Cal X¹(M) to be minimal.

Volume and energy can be obtained from a single functional

[`E]: \Cal X<sup>1</sup>(M)×\Cal M ® R where \Cal M is the manifold of all Riemannian metrics on M. Namely, we can define [`E](V, [g\tilde]) as the energy of the map V: (M, [g\tilde]) ® (T¹M, g^S). This functional is given by

E   (V, ~ g   ) = 1 2 ó õ M  \operatornametr(L[g\tilde]°LV) dvg,

where L_[g\tilde] = Ö{\operatornamedet(g^-1[g\tilde])}([g\tilde]^-1g) . If we represent by E_[g\tilde] the restriction of [`E] to the slice \Cal X<sup>1</sup>(M)×{ [g\tilde]}, it is clear that E=E<sub>g</sub>. Moreover, F(V)=[2/n][`E](V, V^* g^S). The condition for V Î \Cal X¹(M) to be a critical point of E_[g\tilde] has been computed by the first author in [].

The second variation of the total bending has been studied in []. The first result in this paper is to compute the second variation of the functionals E_[g\tilde]. We have used a completely different method to obtain

Theorem Let V be a critical point of E_[g\tilde] and let A be a tangent vector to \Cal X¹(M) at V then the Hessian of E_[g\tilde] is given by

(\Cal Hess E[g\tilde])V(A)= ó õ M  \operatornametr æ è L[g\tilde]ø((ÑA)t øÑA- ||A ||2 (ÑV)tøÑV) ö ø dvg.

In the particular case of [g\tilde] = g this provides a simpler proof of the result of Wiegmink quoted above.The second variation of F is a more involved problem and we have shown:

TheoremLet V be a unit vector field defining a minimal immersion, then the Hessian of F at V is given, for each A tangent to \Cal X¹(M) at V, by