Extremal metrics of zeta function determinants on 4-manifolds.

*(English)*Zbl 0842.58011In dimension two the Moser-Trudinger inequality is intimately related to the ratio of two zeta functions determinants for the scalar Laplacians of two conformal metrics. In particular, this inequality can be used to establish the existence of an extremum for this ratio within a given conformal class.

In the present paper the authors study a similar problem on a 4-manifold. The role of Laplacians on a surface is now taken by metrically defined, conformally covariant, symmetric, elliptic operators. If \(A_g\) denotes such an operator defined in terms of a given metric \(g\) then one is interested in the ratio \[ F[w] = \log {\text{det } A_{g_w} \over \text{det } A_g} \] where for each smooth function \(w : M \to \mathbb{R}\) \(g_w\) denotes the conformal metric \(e^{2w}g\). The above determinants are zeta function determinants. Basic examples of such operators are the conformal Laplacian, the square of the Dirac operator \(\not\kern-.3em\nabla\) acting on spinors and the Paneitz operator, which is a fourth order elliptic operator.

When \(A\) is a second order operator the ratio \(F(w)\) can be very explicitly described in terms of the Riemann tensor and the 4-dimensional Paneitz operator and it splits as a sum of three functionals of independent interest \[ F[w] = \gamma_1 F_1[w] + \gamma_2 F_2[w] + \gamma_3 F_3[w]\quad (\gamma_i \in \mathbb{R}). \] The Paneitz operator intervenes only in \(F_2\). The natural domain of \(F\) is \(W^{(2,2)} (M,g)\). In this form the functional \(F\) has an intrinsic meaning, independent of its origin as a ratio of determinants and thus it can be studied on its own.

The first results of this paper establish the existence of a maximum for \(F\) and a minimum for \(F_2\) provided some (conformally invariant) inequalities involving the coefficients \(\gamma_i\) are satisfied. The strategy adopted is to establish compactness results for the level sets of these functionals. A key ingredient is a sharp Moser inequality due to D. Adams. These results are then specialized to the case when \(F\) is the ratio of determinants of conformal Laplacians or \(\not\kern-.3em\nabla^2\).

Assuming a positivity assumption on the Paneitz operator the authors prove the uniqueness of the above extremals. This assumption turns out to “almost” necessary.

Provided the background is sufficiently symmetric one can also explicitly identify the extremal metric such as locally symmetric Einstein 4-manifolds. This requires an explicit understanding of the conformal inequalities which guarantee the existence of extremals. Section 3 of the paper is devoted to precise this issue.

In Section 4 of the paper, relying on the techniques which produced an extremal for \(F_2\), the authors provide a new proof of a sharp Moser inequality on \(S^n\) originally due to W. Beckner. These techniques make essential use of an \(n\)-dimensional generalization of the Paneitz operator, whose existence is known but whose explicit form remains mysterious for all non-conformally flat Riemannian manifolds of large dimension. For the conformally flat manifolds the Paneitz operator is simply a power of the Laplace-Beltrami operator.

The paper concludes with Section 5 which contains the proof of a priori estimates for a semilinear equation involving the Paneitz operator on a 4-manifold. As the authors point out, this should be compared with a similar estimate due to Schoen for the solutions of the Yamabe problem.

In the present paper the authors study a similar problem on a 4-manifold. The role of Laplacians on a surface is now taken by metrically defined, conformally covariant, symmetric, elliptic operators. If \(A_g\) denotes such an operator defined in terms of a given metric \(g\) then one is interested in the ratio \[ F[w] = \log {\text{det } A_{g_w} \over \text{det } A_g} \] where for each smooth function \(w : M \to \mathbb{R}\) \(g_w\) denotes the conformal metric \(e^{2w}g\). The above determinants are zeta function determinants. Basic examples of such operators are the conformal Laplacian, the square of the Dirac operator \(\not\kern-.3em\nabla\) acting on spinors and the Paneitz operator, which is a fourth order elliptic operator.

When \(A\) is a second order operator the ratio \(F(w)\) can be very explicitly described in terms of the Riemann tensor and the 4-dimensional Paneitz operator and it splits as a sum of three functionals of independent interest \[ F[w] = \gamma_1 F_1[w] + \gamma_2 F_2[w] + \gamma_3 F_3[w]\quad (\gamma_i \in \mathbb{R}). \] The Paneitz operator intervenes only in \(F_2\). The natural domain of \(F\) is \(W^{(2,2)} (M,g)\). In this form the functional \(F\) has an intrinsic meaning, independent of its origin as a ratio of determinants and thus it can be studied on its own.

The first results of this paper establish the existence of a maximum for \(F\) and a minimum for \(F_2\) provided some (conformally invariant) inequalities involving the coefficients \(\gamma_i\) are satisfied. The strategy adopted is to establish compactness results for the level sets of these functionals. A key ingredient is a sharp Moser inequality due to D. Adams. These results are then specialized to the case when \(F\) is the ratio of determinants of conformal Laplacians or \(\not\kern-.3em\nabla^2\).

Assuming a positivity assumption on the Paneitz operator the authors prove the uniqueness of the above extremals. This assumption turns out to “almost” necessary.

Provided the background is sufficiently symmetric one can also explicitly identify the extremal metric such as locally symmetric Einstein 4-manifolds. This requires an explicit understanding of the conformal inequalities which guarantee the existence of extremals. Section 3 of the paper is devoted to precise this issue.

In Section 4 of the paper, relying on the techniques which produced an extremal for \(F_2\), the authors provide a new proof of a sharp Moser inequality on \(S^n\) originally due to W. Beckner. These techniques make essential use of an \(n\)-dimensional generalization of the Paneitz operator, whose existence is known but whose explicit form remains mysterious for all non-conformally flat Riemannian manifolds of large dimension. For the conformally flat manifolds the Paneitz operator is simply a power of the Laplace-Beltrami operator.

The paper concludes with Section 5 which contains the proof of a priori estimates for a semilinear equation involving the Paneitz operator on a 4-manifold. As the authors point out, this should be compared with a similar estimate due to Schoen for the solutions of the Yamabe problem.

Reviewer: L.Nicolaescu (Ann Arbor)

##### MSC:

58E11 | Critical metrics |

58E30 | Variational principles in infinite-dimensional spaces |

58J05 | Elliptic equations on manifolds, general theory |

58J70 | Invariance and symmetry properties for PDEs on manifolds |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |

42B99 | Harmonic analysis in several variables |