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). Weiyu Luo and I proved the corresponding statement for embeddings of H3(X). I also proved a sharp, conformally invariant analogue of these embeddings for Hk on n-dimensional Euclidean upper half space for all k. Work on the curved analogue of this result is ongoing.
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 these three articles for some progress for D-N operators determined by weighted GJMS operators.
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. With Ana Claudia Moreira, we proved a nonuniqueness result for the associated Neumann problem in higher dimensions; see also work of Zhengyang Shan as an undergraduate. I am interesting in proving the sharp fully nonlinear Sobolev trace inequality indicated by this work.
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:
it provides a way to uniformize many CR three-manifolds;
it is closely related to a topological invariant in three- and five-dimensions.
This last point is related to the fact that the total Q'-curvature us a global secondary invariant of closed CR manifolds; i.e. it is independent of the choice of pseudo-Einstein contact form. Yuya Takeuchi and I constructed a large family of global secondary invariants which are not integrals of local CR invariants; this is in sharp contrast to the classification of global conformal invariants. We also conjecture that all global secondary invariants are a linear combination of the total Q'-curvature, our global secondary invariants, and the integral of a local CR invariant.
CR manifolds can also be regarded as odd-dimensional analogues of complex manifolds. If one forgets the complex structure, CR manifolds are contact manifolds, the odd-dimensional analogue of symplectic manifolds. I recently introduced the Rumin algebra as a "better" way to study the cohomology of contact and CR manifolds in analogy with the role of the de Rham algebra on complex manifolds.
I am actively working on understanding Q-curvature operators on CR manifolds. I currently believe that they give the best approach to studying the Lee Conjecture, which posits that a closed, strictly pseudoconvex, embeddable CR manifold admits a pseudo-Einstein contact form if and only if its real first Chern class vanishes.
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.
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 derivation of the factorization of the GJMS operators at Einstein metrics.
A complete description of the Q-curvature operators on three-dimensional CR manifolds, including their relation to the P'-operator and their role in constructing Q-flat contact forms.
A derivation of the Gauss–Bonnet–Chern formula in terms of the total Q-curvature and integrals of local conformal invariants in dimensions up to six.
A detailed derivation of the Merkulov model of an A∞-algebra with higher multiplications mk=0 for all k ≥ 4.