Note on $L^2$ Harmonic Forms on a Complete Manifold

A note on quadratic forms

We extend an interesting theorem of Yuan [12] for two quadratic forms to three matrices. Let C1, C2, C3 be three symmetric matrices in <n×n , if max{xT C1x, xT C2x, xT C3x} ≥ 0 for all x ∈ <n , it is proved that there exist ti ≥ 0 (i = 1, 2, 3) such that ∑3 i=1 ti = 1 and ∑3 i=1 ti Ci has at most one negative eigenvalue.

A note on bilinear forms

Local Properties of Self-Dual Harmonic 2-forms on a 4-Manifold

We will prove a Moser-type theorem for self-dual harmonic 2-forms on closed 4manifolds, and use it to classify local forms on neighborhoods of singular circles on which the 2-form vanishes. Removing neighborhoods of the circles, we obtain a symplectic manifold with contact boundary we show that the contact form on each S1×S2, after a slight modification, must be one of two possibilities.

A Note on Hermitian Forms

In this note we effect a reduction of the theory of hermitian forms of two particular types (coefficients in a quadratic field or in a quaternion algebra with the usual anti-automorphism) to that of quadratic forms. The main theorem (§2) enables us to apply directly the known results on quadratic forms. This is illustrated in the discussion in §3 of a number of special cases. Let <£ be an arbit...

A Note on Complete Integrals

We present a theorem concerning the representation of solutions of a first-order partial differential equation in terms of a complete integral of the equation. We discuss the geometrical significance of that theorem.