Jeffrey S. Case
Assistant Professor
The Pennsylvania State University, Department of Mathematics

Contents:
Research Interests

I am a geometric analyst interested in global problems in Riemannian, Lorentzian, conformal, and CR geometry. My research has two primary focuses. First, I am studying conformally invariant boundary value problems and their relationship to problems involving conformally covariant pseudodifferential operators defined by scattering theory. Second, I am studying generalizations of useful conformally covariant objects to other geometries, especially smooth metric measure spaces and CR manifolds. In both cases, I am primarily interested in developing new sharp Sobolev inequalities and finding applications to geometric classification results in low dimensions.

Conformally invariant boundary value problems are PDEs on manifolds with boundary which are specified in terms of conformally covariant problems. Aspects of these problems, such as the number of homogeneous solutions and the number of negative eigenvalues, are conformally covariant. These problems have many applications to geometric and analytic questions on manifold boundary. For example:

  • (Sharp) Sobolev trace inequalities: Sharp norm inequalities for the embeddings H1(X)↪ H1/2(∂X) and H1(Xn+1)↪ L2n/(n-1)(∂X) follow from the work of Escobar on boundary value problems involving the conformal Laplacian. I proved the corresponding sharp norm inequalities for the embeddings H2(X)↪ H3/2(∂X) ⊕ H1/2(∂X) and H2(Xn+1)↪ L2n/(n-3)(∂X) ⊕ L2n/(n-1)(∂X). I am currently investigating higher-order versions of these inequalities; some problems related to this are available for student research projects.
  • Relations to fractional Laplacians: I primarily focus on the Dirichlet-to-Neumann (D-N) operators associated to conformally covariant boundary value problems involving the GJMS operators. These D-N operators are conformally covariant pseudodifferential operators with principal symbol a half-integral power of the Laplacian. On Poincaré–Einstein manifolds, Sun-Yung Alice Chang and I proved that these D-N operators agree with the fractional GJMS operators determined by scattering theory. I am currently working to refine these statements by removing the Poincaré–Einstein assumption; see this article for some progress for D-N operators determined by the Paneitz operator.
  • Fully nonlinear analogues: Yi Wang and I are currently studying fully nonlinear Dirichlet problems involving the σk-curvatures. In particular, we established a Dirichlet principle for these problems and made partial progress towards a sharp fully nonlinear Sobolev trace inequality. We are currently working to finish this proof, which includes studying a fully nonlinear boundary value problem with fully nonlinear Neumann boundary condition. We hope to give a classification of all solutions on balls in Euclidean space, prove uniqueness (modulo the conformal group) for solutions on all manifolds of dimension N=2k, and show nonuniqueness of solutions for solutions on manifolds of dimension N>2k.

Smooth metric measure spaces are Riemannian manifolds equipped with a smooth measure and, depending on the context, up to two real-valued geometric parameters. My early work on these spaces involved introducing a natural notion of conformal transformation and identifying the natural weighted analogues of the GJMS operators. These spaces arise in many situations, including the following:

  • (Sharp) Sobolev Inequalities: Smooth metric measure spaces were popularized by Bakry and Émery in their work on logarithmic Sobolev inequalities. Much work has been done since then in this setting. For example, I have studied the relationship to a family of sharp Gagliardo--Nirenberg inequalities. A student research project below asks to extend the latter relationship to study fast diffusion equations on manifolds.
  • Quasi-Einstein (or Weighted Einstein) Manifolds: Important special cases are gradient Ricci solitons, static metrics in relativity, and bases of warped product Einstein manifolds. These are the natural Einstein-type structures in smooth metric measure spaces.
  • Ricci Flow: The Ricci flow is a gradient flow on smooth metric measure spaces. This perspective leads to important monotone quanities, such as Perelman's nu-entropy. I have recently studied weighted analogues of the σk-curvatures and their properties near gradient Ricci solitons. In particular, these invariants provide a promising approach to studying geometric flows in higher dimensions via a fully nonlinear analogue of the nu-entropy.
  • Poincaré–Einstein Manifolds: Compactifications of Poincaré–Einstein manifolds are naturally smooth metric measure spaces. My work on D-N operators and sharp Sobolev trace inequalities generalizes to smooth metric measure spaces to yield realizations of general fractional GJMS operators as D-N operators with weights and to sharp weighted Sobolev trace inequalities, respectively. I expect much of my other work on manifolds with boundary to generalize to this setting.

CR manifolds are the abstract analogues of boundaries of domains in Cn. My research is focused on strictly pseudoconvex CR manifolds; there are strong analogies between such manifolds and conformal classes of Riemannian manifolds. The CR plurharmonic functions — functions which are locally the real part of a CR function — form a nontrivial CR invariant subspace of the space of smooth functions. Paul Yang and I introduced a CR invariant operator, the P'-operator, acting on these functions in dimension three which behaves like the Paneitz operator; the analoguous GJMS-type operator is now known in all dimensions. Among the nice properties of this operator and the associated Q'-curvature are the following:

Student Research Projects

Below is a list of topics for which I have available student research projects. These projects are suitable for graduate students or strong undergraduate students. Please contact me if you are interested in further details.

  • A Sharp Sobolev Trace Inequality: Conformally invariant proofs of sharp Sobolev inequalities are useful for later geometric applications. This project involves finding such a proof of a relatively high order sharp Sobolev trace inequality. Lots of (fun!) computations involved! Alternatively, we can write some Python code to do these computations.
    Prerequisites: Linear algebra, real analysis, and either patience or Python experience. Some Riemannian geometry and functional analysis (Sobolev spaces) useful, but not necessary.
  • "Primed" pseudohermitian invariants: These are a new class of pseudohermitian invariants defined in a similar way as is the Q'-curvature. Very few such invariants are known; we want to construct and study new such invariants.
    Prerequisites: Linear algebra. Some differential geometry is nice, but not necessary.
  • Construct Conformally Covariant Operators: I have constructed conformally covariant operators of low orders on smooth metric measure spaces, and given a formal construction for such operators of all orders. This project makes the formal construction more rigorous by finding the ambient metric for smooth metric measure spaces.
    Prerequisites: Riemannian geometry is very useful, but we could potentially make due with only familiarity with multivariable Taylor series
  • A Fast Diffusion Equation: The fast diffusion equation is ut=-Δuβ for some 0<β<1. There is a conformally invariant analogue of the fast diffusion equation which has an interesting relationship to sharp Gagliardo--Nirenberg inequalities. The goal here is to study its properties. In particular, there seems to be an interesting relationship to static metrics that is worth exploring.
    Prerequisites: A PDE course involving parabolic equations..

Notes

I have written some simple notes containing computations or explanations of various mathematical ideas I've tried to learn. I can't promise completeness or accuracy, though I hope they might be useful. If you have any comments, corrections, or suggestions, I would be happy to receive them.

  • Explicit computations deriving the expressions for the conformal change of the Bakry-Emery Ricci tensor and the weighted scalar curvature of a smooth metric measure space, from the perspective of the "auxiliary metric;" the terminology of this note is based upon my dissertation, and not the more standard terminology of smooth metric measure spaces I have since adopted.
  • A semi-formal write-up of my notes on the tractor calculus as it appears in conformal geometry, which I wrote as I learned the subject.
  • Some computations involving the fourth order Q-curvature: A formula for the first variation of the total Q-curvature in all dimensions and a formula relating the Q-curvature and the length of a particular type of tracefree symmetric (0,2)-tractor.
  • A description of the bigraded Rumin complex on three- and five-dimensional CR manifolds, including formulae for all operators in the complex.

Last updated: October 18, 2017