Schedule for: 21w5064 - Geometric PDE and Applications to Problems in Conformal and CR Geometry (Online)

In this talk, we should briefly discuss how we can apply Gauss curvature flow to reprove the existence for prescribing Gauss curvature problem. The work is joint with X. Chen , M. Li and Z. Li.

We discuss some recent progress on compactness result and uniqueness result of conformally compact Einstein manifolds in all dimensions. This is a joint work with Alice Chang, Xiaoshang Jin and Jie Qing.

In this talk I will discuss the recently developed inner-outer gluing methods in constructing various Type II blow-up for some geometric flows, including harmonic map flows, 1/2-harmonic map flows, harmonic map flows with free boundary and porous-media flows.

I report on joint work with Alice Chang and Eden Prywes. A construction of Quasiconformal maps between two conformally related metrics in a positive Yamabe class metric on S^4. Another construction of a biLipschitz map from such a conformal class to the standard conformal class.

In contrast with the compact setting, less is known about the long-time existence and convergence of the Yamabe flow in noncompact settings. I will discuss in the case of asymptotically flat manifolds why long-time existence holds in general and why the flow converges if and only if the initial metric is Yamabe positive, based on decay rate estimates for the scalar curvature along the flow. This is joint work with Yi Wang.

The Levi forms of a CR structure is defined as a conformal class of hermitian metrics. We give a normal from for the Levi froms, in analogy with the normal form of conformal scale. As an application, we give a description of the logarithmic singularity of the Szegö kernel, which implies a local characterization of pseudo-Einstein structures in 3-dimensions in terms of the vanishing of the log singularity to the second order. This result can be seen as an analogy of the description of the Bergman kernel of Robin Graham for domains in $C^2$ and we explain the relation between the Bergman and Szegö kernel by using the deformation complex and Rumin complex on CR manifolds.

We consider the CR Yamabe equation with critical Sobolev ex-ponent on a closed contact manifold M of dimension 2n + 1. The problem of finding solutions with minimum energy has been resolved for all dimensions except for dimension 5 (n = 2). In this paper we prove the existence of minimum energy solutions in the 5-dimensional case when M is spin. The proof is based on a positive mass theorem built up through a spinorial approach. This is joint work with Hung-Lin Chiu.

We will introduce the notion of free boundary constant p-mean curvature (CPMC) surface in a 3-dimensional pseudohermitian manifold with boundary. For the domain bounded by the Pansu sphere in the 3-dimensional Heisenberg group, we will talk on examples of free boundary CPMC surfaces which are rotationally symmetric about the t-axis. This is a joint work with Shujing Pan and Jun Sun.

The non-abelian Hodge correspondence was established by Corlette-Donaldson-Hitchin-Simpson, it states that, on a compact K\"ahler manifold $(X, \omega )$, there is a one-to-one correspondence between the moduli space of semisimple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this talk, I will introduce our recent work on extending this correspondence to some \textcolor{red}{non-K\"ahler} case. This work is joint with Changpeng Pan and Chuanjing Zhan

We will present recent studies on the Ricci-flat Kaehler 4-manifolds in the collapsing setting. We will particularly introduce some structure theorems on their Gromov-Hausdorff limits, singualrity formations, and rescaling bubbles.

In the very simplest setting, the moduli space of all solutions to the Hitchin equations on a Riemann surface is a 4-dimensional hyperKaehler space of ALG type. The moduli space depends on certain parameters in the original equations, and the resulting ALG metric depends on these. A guiding conjecture due to Boalch asks whether all gravitational instantons (in particular, all ALG metrics) arise as gauge-theoretic moduli spaces.In this talk I will explain the background and explain a proof of this conjecture in this particular case

In this talk, we will prove positive mass theorems for ALF and ALG manifolds with model spaces $R^{n-1}\times S^1$ and $R^{n-2}\times T^2$ respectively in dimensions no greater than 7. Different from the compatibility condition for spin structure in Theorem 2 of V. Minerbe’s paper A mass for ALF manifolds, Comm. Math. Phys. 289 (2009), no. 3, 925–955 we show that some type of incompressible condition for $S^1$ and $T^2$ is enough to guarantee the nonnegativity of the mass. As in the asymptotically flat case, we reduce the desired positive mass theorems to those ones concerning non-existence of positive scalar curvature metrics on closed manifolds coming from generalize surgery to -torus. Finally, we investigate certain fill-in problems and obtain an optimal bound for total mean curvature of admissible fill-ins for flat product 2-torus $S^1(l_1)\times S^1(l_2)$. This talk is based on the paper joint with my Ph.D. students Peng Liu and Jintian Zhu, here is the link of the paper :http://arxiv.org/abs/2103.11289.

In this talk, we discuss various curvature flows for hypersurfaces in hyperbolic space and their applications to geometric inequalities.

The so called Yamabe problem in Conformal Geometry asks for a metric conformal to a given one and which has constant scalar curvature. When we focus on the Euclidean space in the presence of singularities (given by smooth submanifolds), the work of Schoen and Yau shows that to obtain a complete metric, the singular set must satisfy a dimensional restriction. Under this assumption, singular solutions exist and have been constructed. A quite recent notion of non-local curvature gives rise to a parallel study which weakens the geometric assumptions of positive scalar curvature giving rise to a non-local problem. In previous works, we covered the construction of solutions which are singular along (zero and positive dimensional) smooth submanifolds in this fractional setting. This was done through the development of new methods coming from conformal geometry and Scattering theory for the study of non-local ODEs. Due to the limitations of the techniques we used, the particular case of maximal possible dimension for the singularity was not covered. In this talk, we will focus on this specific dimension and we will construct and study singular solutions of critical dimension. This is a joint work with H. Chan.

I will report on joint work with Samuel Perez-Ayala in which we consider the problem of extremizing eigenvalues of the conformal laplacian in a fixed conformal class. This generalizes the problem of extremizing the eigenvalues of the laplacian on a compact surface. I will explain the connection of this problem to the existence of harmonic maps, and to nodal solutions of the Yamabe problem (first noticed by Ammann-Humbert).

Kirigami is the art of cutting paper to make it articulated and deployable, allowing for it to be shaped into complex two and three-dimensional geometries. The mechanical response of a kirigami sheet when it is pulled at its ends is enabled and limited by the presence of cuts that serve to guide the possible non-planar deformations. Inspired by the geometry of this art form, we ask two questions: (i) What is the shortest path between points at which forces are applied? (ii) What is the nature of the ultimate shape of the sheet when it is strongly stretched? Mathematically, these questions are related to the nature and form of geodesics in the Euclidean plane with linear obstructions (cuts), and the nature and form of isometric immersions of the sheet with cuts when it can be folded on itself. The talk is based on joint works with M. Lewicka and L. Mahadevan.

In this talk, I will present a result on the rigidity of local minimizers of the functional $\int \sigma_2+ \oint H_2$ among all conformally flat metrics in the Euclidean (n + 1)-ball. We prove the metric is flat up to a conformal transformation in some (noncritical) dimensions. We also prove the analogous result in the critical dimension n + 1 = 4. The main method is Frank-Lieb’s rearrangement-free argument. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. This is joint work with Jeffrey Case.

The boundary Yamabe problem consists in establishing if a given smooth compact Riemannian manifold with boundary can be conformally deformed to a scalar-flat manifold with boundary of constant mean curvature. In this talk I will present a recent result on compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are 4, 5 or 6. This work is in collaboration with Seunghyeok Kim and Juncheng Wei.

It is joint work with Boris Vertman (Oldenburg) and Jørgen Olsen Lye (Oldenburg). We study the convergence of the normalized Yamabe flow with positive Yamabe constant on a class of pseudo-manifolds that includes stratified spaces with iterated cone-edge metrics. We establish convergence under a low-energy condition. We also prove a concentration--compactness dichotomy, and investigate what the alternatives to convergence is.

I the talk we present a Holder estimate of complex Monge-Ampere equation on manifolds. The estimate follows from Kolodziej approach to solve complex Monge-Ampere with L^p bounded measure.

I will discuss a recent joint work with Fengbo Hang on improved Sobolev inequality on the sphere when certain moments vanish up to a given order. The 1st oder case was proved by Aubin’ about 40 years ago. Our new approach yields a characterization of the best constant for any order. It leads to an interesting extremal problem on the sphere. We are able to determine the constant explicitly in the second order case.

We introduce conformal and smooth invariants on oriented, compact four-manifolds with boundary and show that "positivity" conditions on these invariants will impose topological restrictions on underlying manifolds with boundary. We also establish conformally invariant rigidity theorems for Bach-flat four-manifolds with boundary under the assumptions on these invariants. It is noteworthy to point out that we rule out some examples arising from the study of closed manifolds in the setting of manifolds with umbilic boundary.

Motivated by the question from mathematical physics about the size of magnetic fields that support zero modes for the three dimensional Dirac equation, we study a certain conformally invariant spinor equation. We state some conjectures and present results in their support. Those concern, in particular, two novel Sobolev inequalities for spinors and vector fields. The talk is based on joint work with Michael Loss.

In this talk, I will provide a new lower bound for the relative volume inequality for conformally compact Einstein manifolds, as well as its applications in the rigidity theorem for CCE.

In this talk, we will discuss the problem of prescribing Webster scalar curvatures on compact strictly pseudo convex CR manifolds. In terms of the upper and lower solutions method and the perturbation theory of self-adjoint operators, we try to describe some sets of Webster scalar curvature functions which can be realized through pointwise CR conformal deformations and CR conformally equivalent deformations respectively from a given pseudo-Hermitian structure. This is a joint work with Yibin Ren and Weike Yu.

First I will introduce the equation of the deformed Hermitian-Yang Mills metric on the holomorphic line bundle of the Kahler manifold. Then I will introduce some existence results of this equation under some assumptions. I will also introduce the corresponding heat equation and some long time existence and convergence of the heat flow.

A famous vanishing theorem of due to Lichnerowicz states that if a closed spin manifold admits a Riemannian metric with positive scalar curvature, then it's a-hat genus equals to zero. In this talk we will describe some recent advanced generalizing this kind of results to other manifolds as well as foliations.

The renormalized volume of an asymptotically hyperbolic Einstein four-manifold is among its most important global invariants. We define renormalized volume for minimally bounded half-spaces, then prove a Gauss-Bonnet formula for the volume and compute its variation under variations of the minimal boundary. This is joint work with Matthew J. Gursky and Aaron J. Tyrrell.

We show some recent existence and classification results for conformal metrics in Euclidean spaces having prescribed constant Q-curvature and a singularity at the origin. While in dimension 2 this problem was studied and fully understood by Prajapat-Tarantello, in higher dimension new phenomena arise and several questions remain open. This is a joint work with A. Hyder and G. Mancini.

The gauge field equations known as the Yang-Mills equations are extremely important in both mathematics and physics, and their conformal invariance in dimension 4 is a critical feature for many applications. We show that there is a simple and elegant route to higher order equations in dimension 6 that are analogous and arise as the Euler-Lagrange equations of a conformally invariant action. The functional gradient of this action recovers the conformal Fefferman-Graham obstruction tensor when the gauge connection is taken to be the conformal Cartan (or tractor) connection. This also has importance for CR geometry through the Fefferman ambient metric. This is joint work with Larry Peterson and Callum Sleigh.

When one looks for radial solutions of an equation with fractional Laplacian, it is not generally possible to use usual ODE methods. If such equation has some conformal invariances, then one may rewrite it in Emden-Fowler (cylindrical) coordinates and to use the properties of the conformal fractional Laplacian on the cylinder, which is a fractional order Paneitz operator. After giving the necessary background, we will briefly consider two particular applications of this technique: 1. Symmetry breaking, non-degeneracy and uniqueness for the fractional Caffarelli-Kohn-Nirenberg inequality (joint work with W. Ao and A. DelaTorre). 2. Existence and regularity for fractional Laplacian equations with drift and a critical Hardy potential (joint with H. Chan, M. Fontelos and J. Wei).

It is well-known that conformally compact Einstein metrics give a tool to understand the conformal geometry of the boundary in terms of the Riemannian geometry of the interior. Using this philosophy we relate Dirac operators in the interior with the BGG operators of the boundary. In the flat case, this relates discrete series for the orthogonal group with the BGG operators of the boundary sphere.

Given a compact Riemannian manifold with no boundary $(M^n; g)$ endowed with a smooth metric g, one of the important objects of study is the Laplace-Beltrami operator and its eigenfunctions. That is $$-\Laplace u_k=\lambda_k u_k$$ The Courant nodal domain theorem asserts that the k-th eigenfunction has at most k nodal domains, where a nodal domain is a connected component of the set $\{x|u_k\not= 0\}$. Harold Donnelly and C. Fefferman initiated the study of local versions of this result with a goal to show that nodal domains cannot be long and narrow. This was related to a conjecture of S.-T. Yau on the length of the nodal set. The nodal set is the set $\{x| u_k(x)\not=0\}$. In this joint work with A. Logunov, E. Mallinikova and D. Mangoubi, we obtain an optimal bound for results of this type.

In this talk we discuss aspects of the CR analogue of the Deser—Schwimmer conjecture. One possible formulation is that any pseudohermitian scalar invariant whose integral is independent of the choice of pseudo-Einstein contact form is a linear combination of the Q-prime curvature, a local CR invariant, and a divergence. In dimension three, this statement was proved in the affirmative by Hirachi. In higher dimensions, the I-prime curvatures give counterexamples to this statement. We will describe the I-prime curvatures and some of their properties, including the proposal of a new CR analogue of the Deser—Schwimmer conjecture. This is based on joint works with Rod Gover and Yuya Takeuchi.