Some Kind of Stabilities and Instabilities of Energies of Maps Between Kähler Manifolds

Abstract

We treat the variational problem of the energy of the map between two Riemannian manifolds. It is known that any holomorphic or anti-holomorphic map f: M → N between compact Kähler manifolds is stable for the variation f _t of f with fixed Kähler metrics compatible with the holomorphic structures. Is this also stable for the variation g _t of the metric g of M with fixed volume of M and fixed isometric map f? In this paper, we show that the answer is no if the dimension of M is no less than 3. This paper is a expositary note of Taniguchi and Udagawa (Characterizations of Ricci flat metrics and Lagrangian submanifolds in terms of the variational problem. To appear in Glasgow Math. J).

Appendix

We recall a weak form of results in [5]. Let $\mathbf{F_{2}^{0}}$ be the set of all smooth symmetric (0, 2)-tensors on M. Denote by $\mathbf{F_{2}^{0+}}$ the subset of all smooth positive definite symmetric (0, 2)-tensors. Define a function I on $\mathbf{F_{2}^{0}} \times \mathbf{F_{2}^{0+}}$ by

$$\displaystyle{ I(P,\eta ) =\int _{M}\sum _{i,j=1}^{m}P_{ ij}\eta ^{ij}\,d\mu _{\eta }\quad \mathrm{for\ }(P,\eta ) \in \mathbf{F_{ 2}^{0}} \times \mathbf{F_{ 2}^{0+}}. }$$

(3)

We normalize I so that it is invariant under the homothetic transformation. Next, fixing $P \in \mathbf{F_{2}^{0}}$, we define I(η) by

$$\displaystyle{ I(\eta ):= \frac{\,\int _{M}\sum _{i,j=1}^{m}P_{ ij}\eta ^{ij}\,d\mu _{\eta }\,} {\left (\int _{M}d\mu _{\eta }\right )^{\frac{m-2} {m} }} }$$

(4)

Theorem 3 (1st Variation Formula).

$\quad \delta I(\eta ) = \frac{1} {V ^{c+1}} \langle \langle V \left (\mathbf{p}\eta - P\right ) - cU\eta,\delta \eta \rangle \rangle,$

$$\displaystyle\begin{array}{rcl} \mathrm{where}\quad V =\int _{M}d\mu _{\eta },\quad \mathbf{p}& =& \frac{1} {2}\sum _{i,j=1}^{m}P_{ ij}\eta ^{ji},\quad U =\int _{ M}\mathbf{p} d\mu _{\eta },\quad c = \frac{m - 2} {m}, {}\\ < p,q >& =& \sum _{i,j,k,l=1}^{m}p_{ ij}\eta ^{jk}q_{ kl}\eta ^{li}, {}\\ \langle \langle p,q\rangle \rangle & =& \int _{M} < p,q > d\mu _{\eta },\quad (p,q \in \mathbf{F_{2}^{0}}). {}\\ \end{array}$$

Theorem 4.

δI(η) = 0 if and only if $P = 2\mathbf{p}\eta /m$ . Moreover p is constant if δI(η) = 0 and m≠2.

Theorem 5 (2nd Variation Formula).

Assume that δI(η) = 0. Then,

$$\displaystyle\begin{array}{rcl} \delta ^{2}I(\eta )& =& \frac{1} {V ^{c+1}}\left \{\frac{m - 2} {m^{2}V } \langle \langle \mathbf{p}\rangle \rangle \langle \langle \mathrm{trace}_{\eta }\delta \eta \rangle \rangle ^{2} -\frac{V } {m}\langle \langle \mathbf{p}(\mathrm{trace}_{\eta }\delta \eta )^{2}\rangle \rangle \ +\right. \\ & & \left.+\ \frac{2V } {m} \langle \langle \mathbf{p}\delta \eta,\delta \eta \rangle \rangle \right \}. {}\end{array}$$

(5)

In particular, if m = 1,2 and p is a non-negative(resp. non-positive) then δ ² I(η) ≥ 0 (resp. δ ² I(η) ≤ 0) holds.

Theorem 6.

Assume that m ≥ 3 and the critical point η of I is stable. Then p = 0.