Ngô Quốc Anh

February 22, 2015

The conditions (NN), (P), (NN+) and (P+) associated to the Paneitz operator for 3-manifolds

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 18:54

Of recent interest is the prescribed Q-curvature on closed Riemannian manifolds since it involves high-order differential operators.

In a previous post, I have talked about prescribed Q-curvature on 4-manifolds. Recall that for 4-manifolds, this question is equivalent to finding a conformal metric \widetilde g =e^{2u}g for which the Q-curvature of \widetilde g equals the prescribed function \widetilde Q? That is to solving

\displaystyle P_gu+2Q_g=2\widetilde Q e^{4u},

where for any g, the so-called Paneitz operator P_g acts on a smooth function u on M via

\displaystyle {P_g}(u) = \Delta _g^2u - {\rm div}\left( {\frac{2}{3}{R_g} - 2{\rm Ric}_g} \right)du

which plays a similar role as the Laplace operator in dimension two and the Q-curvature of \widetilde g is given as follows

\displaystyle Q_g=-\frac{1}{12}(\Delta\text{Scal}_g -\text{Scal}_g^2 +3|{\rm Ric}_g|^2).

Sometimes, if we denote by \delta the negative divergence, i.e. \delta = - {\rm div}, we obtain the following formula

\displaystyle {P_g}(u) = \Delta _g^2u + \delta \left( {\frac{2}{3}{R_g} - 2{\rm Ric}_g} \right)du.

Generically, for n-manifolds, we obtain

\displaystyle Q_g=-\frac{1}{2(n-1)} \Big(\Delta\text{Scal}_g - \frac{n^3-4n^2+16n-16}{4(n-1)(n-2)^2} \text{Scal}_g^2+\frac{4(n-1)}{(n-2)^2} |{\rm Ric}_g|^2 \Big)

and

\displaystyle {P_g}(u) = \Delta _g^2u + {\rm div}\left( { a_n {R_g} + b_n {\rm Ric}_g} \right)du + \frac{n-4}{2} Q_g u,

where a_n = -((n-2)^2+4)/2(n-1)(n-2) and b_n =4/(n-2).

Hence, for 3-manifolds, the Q-curvature of \widetilde g is now given as follows

\displaystyle Q_g=-\frac{1}{4}(\Delta\text{Scal}_g -\frac{23}{8} \text{Scal}_g^2 +2|{\rm Ric}_g|^2)

while the Paneitz operator P_g acts on a smooth function u on M via

\displaystyle {P_g}(u) = \Delta _g^2u + {\rm div}\left( -\frac{5}{4}{R_g} +4 {\rm Ric}_g \right)du - \frac{1}{2} Q_g u.

In a joint paper published in Calculus of Variations and Partial Differential Equations in 2004, see this link, Hang and Yang defined some new conditions called (NN), (P), (NN+) and (P+) associated to the Paneitz operator for 3-manifolds. To be precise, associated to the Paneitz operator, we define

\displaystyle \mathcal E(u) = \int_M uP_g(u) dv_g.

Definition. For any u \in H^2(M,g), we say:

  • (M,g) satisfies condition (NN) if whenver u(p)=0 for some p \in M then \mathcal E(u) \geqslant 0;
  • (M,g) satisfies condition (P) if whenver u(p)=0 for some p \in M then \mathcal E(u) > 0;
  • (M,g) satisfies condition (NN+) if whenver u(p)=0 for some p \in M and u \geqslant 0 then \mathcal E(u) \geqslant 0;
  • (M,g) satisfies condition (P+) if whenver u(p)=0 for some p \in M and u \geqslant 0 then \mathcal E(u) > 0.

The main use of the (NN) condition is that if (M,g) satisfies (NN) and there exists some u\in H^2(M,g) such that u(p)=0 for some p\in M and \mathcal E(u)=0 then

\displaystyle P_g(u) = {\rm const} \cdot \delta_p.

Therefore, if \ker P = 0 then u is simply a constant multiple of the Green function \mathbb G_p of the Paneitz operator with the pole p.

In future posts, I will explain these conditions in details.

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