Jeff A. Viaclovsky

Conformal geometry, contact geometry, and the calculus of variations

for metricsg in the conformal class of g0, where we use the metric g to view the tensor as an endomorphism of the tangent bundle and where σk d notes the trace of the induced map on the kth exterior power; that is, σk is the kth elementary symmetric function of the eigenvalues. The case k = 1,R = constant is known as the Yamabe problem, and it has been studied in great depth (see [11] and [17]). We let M1 denote the set of unit volume metrics in the conformal class [g0]. We show that these equations have the following variational properties. Theorem 1. If k 6= n/2 and (N, [g0]) is locally conformally flat, then a metric g ∈M1 is a critical point of the functional Fk : g 7→ ∫

Some properties of the Schouten tensor and applications to conformal geometry

The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a non-conformally invariant part, the Schouten tensor. A study of the kth elementary symmetric function of the eigenvalues of the Schouten tensor was initiated in an earlier paper by the second author, and a natural condition to impose is that the eigenvalues of the Schouten tensor are in a certain cone, Γ + k . We prove that this eigenvalue condition for k > n/2 implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of σ k -curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors.

Bach-flat asymptotically locally Euclidean metrics

We obtain a volume growth and curvature decay result for various classes of complete, noncompact Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual or Kähler metrics with zero scalar curvature, and metrics with harmonic curvature. Similar results were obtained for Einstein metrics in [And89], [BKN89], [Tia90], but our analysis differs from the Einstein case in that (1) we consider more generally a fourth order system in the metric, and (2) we do not assume any pointwise Ricci curvature bound.

Conformally invariant Monge-Ampère equations: Global solutions

In this paper we will examine a class of fully nonlinear partial differential equations which are invariant under the conformal group SO(n + 1, 1). These equations are elliptic and variational. Using this structure and the conformal invariance, we will prove a global uniqueness theorem for solutions in Rn with a quadratic growth condition at infinity.

Volume growth, curvature decay, and critical metrics

We make some improvements to our previous results in [TV05a] and [TV05b]. First, we prove a version of our volume growth theorem which does not require any assumption on the first Betti number. Second, we show that our local regularity theorem only requires a lower volume growth assumption, not a full Sobolev constant bound. As an application of these results, we can weaken the assumptions of several of our theorems in [TV05a] and [TV05b].

Twistor geometry and warped product orthogonal complex structures

The twistor space of the sphere S^{2n} is an isotropic Grassmannian that fibers over S^{2n}. An orthogonal complex structure on a subdomain of S^{2n} (a complex structure compatible with the round metric) determines a section of this fibration with holomorphic image. In this paper, we use this correspondence to prove that any finite energy orthogonal complex structure on R^6 must be of a special warped product form, and we also prove that any orthogonal complex structure on R^{2n} that is asymptotically constant must itself be constant. We will also give examples defined on R^{2n} which have infinite energy, and examples of non-standard orthogonal complex structures on flat tori in complex dimension three and greater.

Conformal symmetries of self-dual hyperbolic monopole metrics

We determine the group of conformal automorphisms of the self-dual metrics on n#CP due to LeBrun for n ≥ 3, and Poon for n = 2. These metrics arise from an ansatz involving a circle bundle over hyperbolic three-space H3 minus a finite number of points, called monopole points. We show that for n ≥ 3 connected sums, any conformal automorphism is a lift of an isometry of H3 which preserves the set of monopole points. Furthermore, we prove that for n = 2, such lifts form a subgroup of index 2 in the full automorphism group, which we show is a semi-direct product (U(1)×U(1))⋉D4, the dihedral group of order 8.

Anti-self-dual orbifolds with cyclic quotient singularities

An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with cyclic quotient singularities is proved. We present two applications of this theorem. The first is to compute the dimension of the deformation space of the Calderbank-Singer scalar-flat Kahler toric ALE spaces. A corollary of this is that, except for the Eguchi-Hanson metric, all of these spaces admit non-toric anti-self-dual deformations, thus yielding many new examples of anti-self-dual ALE spaces. For our second application, we compute the dimension of the deformation space of the canonical Bochner-Kahler metric on any weighted projective space

ON THE REGULARITY OF SOLUTIONS TO MONGE-AMPÈRE EQUATIONS ON HESSIAN MANIFOLDS

\mathbb{CP}^2_{(r,q,p)}

The Mass of The Product of Spheres

for relatively prime integers