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Jeffrey S. Case

Publications and source records attributed to Jeffrey S. Case.

At least 55 records · Page 3Linked to original sources
arXiv2017

The $P^\prime$-operator, the $Q^\prime$-curvature, and the CR tractor calculus

We establish an algorithm which computes formulae for the CR GJMS operators, the $P^\prime$-operator, and the $Q^\prime$-curvature in terms of CR tractors. When applied to torsion-free pseudo-Einstein contact forms, this algorithm both gives an explicit factorisation of the CR GJMS operators and the $P^\prime$-operator, and shows that the $Q^\prime$-curvature is constant, with the constant explicitly given in terms of the Webster scalar curvature. We also use our algorithm to derive local formulae for the $P^\prime$-operator and $Q^\prime$-curvature of a five-dimensional pseudo-Einstein manifold. Comparison with Marugame's formulation of the Burns--Epstein invariant as the integral of a pseudohermitian invariant yields new insights into the class of local pseudohermitian invariants for which the total integral is independent of the choice of pseudo-Einstein contact form.

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arXiv2017

Boundary operators associated to the $σ_k$-curvature

We study conformal deformation problems on manifolds with boundary which include prescribing $σ_k\equiv0$ in the interior. In particular, we prove a Dirichlet principle when the induced metric on the boundary is fixed and an Obata-type theorem on the upper hemisphere. We introduce some conformally covariant multilinear operators as a key technical tool.

math.DG↗
arXiv2016

Some energy inequalities involving fractional GJMS operators

Under a spectral assumption on the Laplacian of a Poincaré--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order $2γ\in(0,2)$ or $2γ\in(2,4)$ and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption is necessary and sufficient for such an inequality to hold. We prove the energy inequalities by introducing conformally covariant boundary operators associated to the weighted conformal Laplacian and weighted Paneitz operator which generalize the Robin operator. As an application, we establish a new sharp weighted Sobolev trace inequality on the upper hemisphere.

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arXiv2016

The weighted $σ_k$-curvature of a smooth metric measure space

We propose a definition of the weighted $σ_k$-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted $σ_k$-curvature prescription problem is governed by a fully nonlinear second order elliptic PDE which is variational when $k=1,2$ or the smooth metric measure space is locally conformally flat in the weighted sense. Second, we show that, in the variational cases, quasi-Einstein metrics are stable with respect to the total weighted $σ_k$-curvature functional. We also discuss related conjectures for weighted Einstein manifolds.

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arXiv2016

A fully nonlinear Sobolev trace inequality

The $k$-Hessian operator $σ_k$ is the $k$-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the $k$-Hessian equation $σ_k(D^2u)=f$ with Dirichlet boundary condition $u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $-\int uσ_k(D^2u)$. We construct a natural boundary functional which, when added to the $k$-Hessian energy, yields as its critical points solutions of $k$-Hessian equations with general non-vanishing boundary data. As a consequence, we prove a sharp Sobolev trace inequality for $k$-admissible functions $u$ which estimates the $k$-Hessian energy in terms of the boundary values of $u$.

math.AP↗
arXiv2016

A notion of the weighted $σ_k$-curvature for manifolds with density

We propose a natural definition of the weighted $σ_k$-curvature for a manifold with density; i.e.\ a triple $(M^n,g,e^{-ϕ}\mathrm{dvol})$. This definition is intended to capture the key properties of the $σ_k$-curvatures in conformal geometry with the role of pointwise conformal changes of the metric replaced by pointwise changes of the measure. We justify our definition through three main results. First, we show that shrinking gradient Ricci solitons are local extrema of the total weighted $σ_k$-curvature functionals when the weighted $σ_k$-curvature is variational. Second, we characterize the shrinking Gaussians as measures on Euclidean space in terms of the total weighted $σ_k$-curvature functionals. Third, we characterize when the weighted $σ_k$-curvature is variational. These results are all analogues of their conformal counterparts, and in the case $k=1$ recover some of the well-known properties of Perelman's $\mathcal{W}$-functional.

math.DG↗
arXiv2016

A weighted renormalized curvature for manifolds with density

We introduce a scalar invariant on manifolds with density which is analogous to the renormalized volume coefficient $v_3$ in conformal geometry. We show that this invariant is variational and that shrinking gradient Ricci solitons are stable with respect to the associated $\mathcal{W}$-functional.

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arXiv2016

Extremal metrics for the ${Q}^\prime$-curvature in three dimensions

We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial steps are to show that the $P^\prime$-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green's function for $\sqrt{P^\prime}$.

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arXiv2015

Extremal metrics for the $Q^\prime$-curvature in three dimensions

We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial steps are to show that the $P^\prime$-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green's function for $\sqrt{P^\prime}$.

math.DG↗
arXiv2015

The CR Paneitz Operator and the Stability of CR Pluriharmonic Functions

We give a condition which ensures that the Paneitz operator of an embedded three-dimensional CR manifold is nonnegative and has kernel consisting only of the CR pluriharmonic functions. Our condition requires uniform positivity of the Webster scalar curvature and the stability of the CR pluriharmonic functions for a real analytic deformation. As an application, we show that the real ellipsoids in $\mathbb{C}^2$ are such that the CR Paneitz operator is nonnegative with kernel consisting only of the CR pluriharmonic functions.

math.DG↗
arXiv2015

Boundary operators associated to the Paneitz operator

We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conformally covariant pseudodifferential operator. In the setting of Poincaré--Einstein manifolds, we show that these operators agree with the fractional GJMS operators of Graham and Zworski. We also use our operators to establish some new sharp Sobolev trace inequalities.

math.DG↗
arXiv2015

A Yamabe-type problem on smooth metric measure spaces

We describe and partially solve a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman's $ν$-entropy. This problem reduces in all dimensions on Euclidean space to the characterization of the minimizers of the family of Gagliardo--Nirenberg--Sobolev inequalities studied by Del Pino and Dolbeault. We show that minimizers always exist on a compact manifold provided the so-called weighted Yamabe constant is strictly less than its value on Euclidean space. We also show that strict inequality holds for a large class of smooth metric measure spaces, but we will also give an example which shows that minimizers of the weighted Yamabe constant do not always exist.

math.DG↗
arXiv2015

A Remark on the Kernel of the CR Paneitz operator

For CR structures in dimension three, the CR pluriharmonic functions are characterized by the vanishing of a third order operator. This third order operator, after composition with the divergence operator, gives the fourth order analogue of the Paneitz operator. In this short note, we give criteria under which the kernel of the CR Paneitz operator contains a supplementary space to the CR pluriharmonic functions.

math.DG↗
arXiv2014

On fractional GJMS operators

We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet-to-Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measure spaces. This gives a geometric interpretation of the Caffarelli--Silvestre extension for $(-Δ)^γ$ when $γ\in(0,1)$, and both a geometric interpretation and a curved analogue of the higher order extension found by R. Yang for $(-Δ)^γ$ when $γ>1$. We give three applications of this correspondence. First, we exhibit some energy identities for the fractional GJMS operators in terms of energies in the compactified Poincaré--Einstein manifold, including an interpretation as a renormalized energy. Second, for $γ\in(1,2)$, we show that if the scalar curvature and the fractional $Q$-curvature $Q_{2γ}$ of the boundary are nonnegative, then the fractional GJMS operator $P_{2γ}$ is nonnegative. Third, by assuming additionally that $Q_{2γ}$ is not identically zero, we show that $P_{2γ}$ satisfies a strong maximum principle.

math.DG↗
arXiv2013

A Paneitz-type operator for CR pluriharmonic functions

We introduce a fourth order CR invariant operator on pluriharmonic functions on a three-dimensional CR manifold, generalizing to the abstract setting the operator discovered by Branson, Fontana and Morpurgo. For a distinguished class of contact forms, all of which have vanishing Hirachi-$Q$ curvature, these operators determine a new scalar invariant with properties analogous to the usual $Q$-curvature. We discuss how these are similar to the (conformal) Paneitz operator and $Q$-curvature of a four-manifold, and describe its relation to some problems for three-dimensional CR manifolds.

math.DG↗
arXiv2013

A generalization of the Cai--Galloway splitting theorem to smooth metric measure spaces

We generalize the splitting theorem of Cai-Galloway for complete Riemannian manifolds with $\Ric\geq-(n-1)$ admitting a family of compact hypersurfaces tending to infinity with mean curvatures tending to $n-1$ sufficiently fast to the setting of smooth metric measure spaces. This result complements and provides a new perspective on the splitting theorems recently proven by Munteanu-Wang and Su-Zhang. We show that the mean curvature assumption in our result is sharp, which also provides an example showing that the assumption $R\geq-(n-1)$ in the Munteanu-Wang splitting theorem for expanding gradient Ricci solitons cannot be relaxed to $R>-n$. We also use our result to study a certain class of conformally compact quasi-Einstein metrics, giving, as generalizations of respective results of Cai-Galloway and Lee, necessary conditions for the boundary to be connected and for the bottom of the spectrum of the weighted Laplacian to be maximal.

math.DG↗
arXiv2012

Smooth metric measure spaces, quasi-Einstein metrics, and tractors

We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.

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