Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

where a > 0 is some constant and x ∈ R. Hypothesis (2) was removed by Caffarelli, Gidas and Spruck in [8]; this is important for applications. Such Liouville type theorems have been extended to general conformally invariant fully nonlinear equations by Li and Li ([24]–[27]); see also related works of Viaclovsky ([40]–[41]) and Chang, Gursky and Yang ([13]–[14]). The method used in [21], as well as in much of the above cited work, is the method of moving planes. The method of moving planes has become a very powerful tool in the study of nonlinear elliptic equations; see Aleksandrov [1], Serrin [38], Gidas, Ni and Nirenberg [21]–[22], Berestycki and Nirenberg [2], and others. In [30], Li and Zhu gave a proof of the above mentioned theorem of Caffarelli, Gidas and Spruck using the method of moving spheres (i.e. the method of moving planes together with the conformal invariance), which fully exploits the conformal invariance of the problem and, as a result, captures the solutions directly rather than going through the usual procedure of proving radial symmetry of solutions and then classifying radial solutions. Significant simplifications to the proof in [30] have been made in Li and Zhang [29]. The method of moving spheres has been used in [24]–[27]. Liouville type theorems for various conformally invariant equations have received much attention; see, in addition to the above cited papers, [23], [17], [15], [33], [42] and [43].

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=6&iss=2&rank=1

Remark on some conformally invariant integral equations: the method of moving spheres

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→ ∫

Conformal geometry, contact geometry, and the calculus of variations

Oxford University Press in Africa, 1927â80

Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and min-max schemes, jointly with the compactness result of [35].

/pdf/existence-of-conformal-metrics-with-constant-q-curvature-2a368t2pr2.pdf

Existence of conformal metrics with constant Q-curvature

We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.

http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p05.pdf

Prescribing symmetric functions of the eigenvalues of the Ricci tensor

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 Kahler 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.

Bach-flat asymptotically locally Euclidean metrics

We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2002/0010/0004/CAG-2002-0010-0004-a006.pdf

Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds

We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with a metric of positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. A special case of this deformation provides an alternative proof to the main result in Chang, Gursky & Yang, 2002. We also give a new conformally invariant condition for positivity of the Paneitz operator, generalizing the results in Gursky, 1999. From the existence results in Chang & Yang, 1995, this allows us to give many new examples of manifolds admitting metrics with constant Q-curvature.

/pdf/a-fully-nonlinear-equation-on-four-manifolds-with-positive-fvmy50xhyd.pdf

Jeff A. Viaclovsky

Papers

Conformal geometry, contact geometry, and the calculus of variations

Prescribing symmetric functions of the eigenvalues of the Ricci tensor

Bach-flat asymptotically locally Euclidean metrics

Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds

A Fully Nonlinear Equation on Four-Manifolds with Positive Scalar Curvature