Research Interests

I am a geometric analyst interested in global problems in Riemannian, conformal, and CR geometry.  My research has XXX primary focuses.  First, I am interested in constructing and classifying conformal invariants (scalars, tensors, operators, etc.) on conformal manifolds (possibly with boundary or corner), on immersed submanifolds of conformal manifolds, on CR manifolds, and on smooth metric measure spaces.  Second, I am interested in applying these invariants to geometric and analytic problems; e.g. to constructing canonical representatives in a conformal class, establishing new sharp Sobolev inequalities, and better understanding the moduli spaces of Einstein-type structures.  Below I briefly summarize some of my results in these directions on conformal manifolds, smooth metric measure spaces, and CR manifolds.

Conformal manifolds are smooth manifolds (possibly with boundary or corner) equipped with a conformal class of manifolds.  A brief summary of my work in this area is as follows:

• Boundary operators for GJMS operators: Escobar established sharp norm inequalities for the embeddings H1(X)↪ H1/2(∂X) and H1(Xn+1)↪ L2n/(n-1)(∂X) using a conformally covariant boundary operator for 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) by introducing conformally covariant boundary operators associated to the Paneitz operator. Weiyu Luo and I proved the corresponding statement for embeddings of H3(X) in terms of conformally covariant boundary operators for the sixth-order GJMS operator. I also proved a sharp, conformally invariant analogue of these embeddings for Hk on n-dimensional Euclidean upper half space for all k. A key point, established by Alice Chang and myself, is that the associated Dirichlet-to-Neumann operators recover the fractional GJMS operators of half-integer order; my constructions of conformally covariant boundary operators on smooth metric measure spaces also lead to sharp Sobolev trace inequalities with weights and the recovery of fractional GJMS operators of general order as Dirichlet-to-Neumann operators.

• Conformally invariant integrals and Einstein manifolds: Ayush Khaitan, Yueh-Ju Lin, Aaron Tyrrel, Wei Yuan, and I recently described a systematic method for computing the integrals of many scalar Riemannian invariants in terms of a conformally invariant integral of a scalar Riemannian invariant.  In particular, this work allows us to compute the Euler characteristic of a compact Einstein manifold (resp. Poincaré–Einstein manifold) in terms of its value (resp. renormalized volume) and the integral of an explicit scalar conformal invariant.  We do this by introducing the new "straightening" approach to studying scalar invariants, which has the by-product of showing that there are scalar conformal invariants in dimensions eight and higher which are total divergences; in particular, the space of conformally invariant integrals of scalar Riemannian invariants generally has smaller dimension than the space of scalar conformal invariants.

• Conformally covariant polydifferential operators: Yi Wang and I studied fully nonlinear Dirichlet problems involving the σk-curvatures, establishing a Dirichlet principle for these operators and making progress towards a sharp fully nonlinear Sobolev trace inequality.  The main tool was a quartic Dirichlet form involving a multilinear, or "polydifferential", operator.  Yueh-Ju Lin, Wei Yuan, and I have developed a systematic approach to the construction of conformally covariant operators, and Zetian Yan and I have developed an approach to proving the formal self-adjointness of some of these operators.  I am very interested in studying how generally one can solve Yamabe-type problems for formally self-adjoint, conformally covariant, polydifferential operators.  Questions related to these operators have been and continue to be a fertile source of undergraduate research projects.

• Immersed submanifolds: Robin Graham, Tzu-Mo Kuo, and I constructed "extrinsic GJMS operators" on immersed submanifolds; these operators have the property that they have a nice factorization at minimal submanifolds of Einstein manifolds, a property which conjecturally classifies these operators.  Graham, Kuo, Tyrrell, Andrew Waldron, and I used this property to compute the renormalized area of two- and four-dimensional minimal submanifolds of Poincaré–Einstein manifolds, and for all even-dimensional minimal submanifolds under a conjectural classification of conformally invariant integrals of extrinsic scalar invariants, à la Alexakis. Khaitan, Lin, Tyrrell, Yuan, and I are working to adapt our straightening approach to this problem, thereby working around this conjectural classification of conformally invariant integrals.

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.

• 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 studied some weighted analogues of the σk-curvatures and their properties near gradient Ricci solitons.  My former student Ayush Khaitan systematically generalized this work by constructing ambient metrics for smooth metric measure spaces.  In particular, he constructed natural fully nonlinear analogue of the nu-entropy.

• Poincaré–Einstein Manifolds: Compactifications of Poincaré–Einstein manifolds are naturally smooth metric measure spaces. My aforementioned work on Dirichlet-to-Neumann operators and sharp Sobolev trace inequalities uses this perspective.  Khaitan's weighted ambient space allows one to easily construct weighted GJMS operators and should lead to a generalization of these sharp Sobolev trace inequalities to outside the setting of compactifications of Poincaré–Einstein manifolds.

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. I am studying three families of operators in this setting:

• P'-operators and Q'-curvatures: Paul Yang and I introduced the P'-operator and Q'-curvature in dimension three; they were generalized to all dimensions by Kengo Hirachi.  The key points are that the P'-operator is a CR covariant operator on CR pluriharmonic functions that shares many formal similarities with the critical GJMS operators, and the Q'-curvature is a pseudohermitian scalar invariant that, when evaluated at pseudo-Einstein contact forms, shares many formal similarities with the critical Q-curvature.  For example, Yang and I used the total Q'-curvature to classify the standard contact three-sphere in terms of certain CR invariants.  I am actively working on classifying CR invariant integrals of pseudo-Einstein contact forms; e.g. my work with Rod Gover and my work with Yuya Takeuchi helped show that this class of invariants is comparatively larger than their conformal analogues.

• Rumin complex: The Rumin complex is a contact invariant differential complex which computes De Rham cohomology, and the bigraded Rumin complex is a CR invariant differential multicomplex from which one can compute Kohn–Rossi cohomology (among other things).  I have given a systematic account of these complexes, stressing CR invariance, and proven relevant Hodge theorems.  I also proved similar results for the second page of the spectral sequence determined by the bigraded Rumin complex; these results, when applied to compact Sasaki manifolds, are analogous to results relating the Dolbeault and De Rham cohomology groups of compact Kähler manifolds.

• Q-curvature operators: I introduced Q-curvature operators on five-dimensional CR manifolds, analogous to a construction of Branson and Gover on conformal manifolds.  I am working to generalize this construction to all dimensions, and believe that these operators provide a promising approach to resolving 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.  This conjecture is especially important in light of the discovery of the Q'-curvatures and their nice behavior on pseudo-Einstein manifolds.

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.