VARIATIONAL CHARACTERIZATIONS OF THE TOTAL SCALAR CURVATURE AND EIGENVALUES OF THE LAPLACIAN

초록

For the dual operator s(g)'* of the linearization s(g)' of the scalar curvature function, it is well-known that if ker s(g)'* not equal 0, then s (g) is a nonnegative constant. Moreover, if the Ricci curvature does not vanish, then s (g) /(n - 1) is an eigenvalue of the Laplacian of the metric g. In this work, we give some variational characterizations for the space ker s(g)'*. To accomplish this, we introduce a fourth-order elliptic differential operator A and a related geometric invariant v. We prove that v vanishes if and only if ker s(g)'* not equal 0, and if the first eigenvalue of the Laplace operator is large compared to its scalar curvature, then v is positive and ker s(g)'* = 0. We calculate a lower bound for v in the case of ker s(g)'* = 0. We also show that if there exists a function which is A-superharmonic and the Ricci curvature has a lower bound, then the first nonzero eigenvalue of the Laplace operator has an upper bound.

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