# UCSB Differential Geometry Seminar 2021–2022

The talks are held on Zoom every Friday 3–4 PM (Pacific Time) unless otherwise noted. The Zoom link will be sent out to the members of the UCSB Math Department a few days before each talk. If you are not a member of the UCSB Math Department, but would like to attend one of our talks, please send me an email!
You can find other UCSB Math Colloquia, Conferences, Seminars & Events here.

## Fall 2021

### The Keller–Segel equations on curved planes

Abstract: The Keller–Segel equations provide a mathematical model for chemotaxis, that is the organisms (typically bacteria) in the presence of a (chemical) substance. These equations have been intensively studied on $$\mathbb{R}^n$$ with its flat metric, and the most interesting and difficult case is the planar, $$n = 2$$ one. Less is known about solutions in the presence of nonzero curvature.
In the talk, I will introduce the Keller–Segel equations in dimension 2, and then briefly recall a few relevant known facts about them. After that I will present my main results. First I prove sharp decay estimates for stationary solutions and prove that such a solution must have mass $$8 \pi$$. Some aspects of this result is novel already in the flat case. Furthermore, using a duality to the "hard" Kazdan–Warner equation on the round sphere, I prove that there are arbitrarily small perturbations of the flat metric on the plane that do not support a stationary solution to the Keller–Segel equations.
My last result is a curved version of the logarithmic Hardy–Littlewood–Sobolev inequality, which I use to prove a result that is complementary to the above ones, as it shows that the functional corresponding to the Keller–Segel equations is bounded from below only when the mass is $$8 \pi$$.

### The lower bound of the integrated Carathéodory–Reiffen metric and Invariant metrics on complete noncompact Kähler manifolds

Abstract: We seek to gain progress on the following long-standing conjectures in hyperbolic complex geometry: prove that a simply connected complete Kähler manifold with negatively pinched sectional curvature is biholomorphic to a bounded domain and the Carathéodory–Reiffen metric does not vanish everywhere. As the next development of the important recent results of D. Wu and S.T. Yau in obtaining uniformly equivalence of the base Kähler metric with the Bergman metric, the Kobayashi–Royden metric, and the complete Kähler–Einstein metric in the conjecture class but missing of the Carathéodory–Reiffen metric, we provide an integrated gradient estimate of the bounded holomorphic function which becomes a quantitative lower bound of the integrated Carathéodory–Reiffen metric. Also, without requiring the negatively pinched holomorphic sectional curvature condition of the Bergman metric on an $$n$$-dimensional complete noncompact Kähler manifold, we establish the equivalence of the Bergman metric, the Kobayashi–Royden metric, and the complete Kähler–Einstein metric of negative scalar curvature under a bounded curvature condition of the Bergman metric with some reasonable conditions which also imply nonvanishing Carathédoroy–Reiffen metric.

### On A Family Of Integral Operators On The Ball

Abstract: In this work, we transform the equation in the upper half space first studied by Caffarelli and Silvestre to an equation in the Euclidean unit ball $$\mathbb{B}^n$$. We identify the Poisson kernel for the equation in the unit ball. Using the Poisson kernel, we define the extension operator. We prove an extension inequality in the limit case and prove the uniqueness of the extremal functions in the limit case using the method of moving spheres. In addition we offer an interpretation of the limit case inequality as a conformally invariant generalization of Carleman's inequality.

### The global shape of universal covers

Abstract: If we start with a sequence of compact Riemannian manifolds $$X_n$$ shrinking to a point, take their universal covers $$\tilde{X}_n$$, and look at them from very far, how will they look like? It is well known that if there is a limiting shape $$\tilde{X}_n \to X$$, then $$X$$ is a nilpotent group with an invariant metric. On the other hand, the spaces $$\tilde{X}_n$$ are simply connected and one could (naively) expect $$X$$ to be simply connected as well. I will discuss how limits of simply connected spaces are usually simply connected and outline a proof of how in most cases $$X$$ is simply connected.

### Symmetry reduction in sub-Riemannian geometry with applications to quantum systems

Abstract: We consider a class of sub-Riemannian structures on Lie groups where the defining distribution is spanned by a set of right invariant vector fields. Such vector fields are determined by a $$K+P$$ Cartan decomposition of the corresponding Lie algebra, and, in particular, they span the $$P$$ part of the decomposition. We present a technique to calculate objects of interest in sub-Riemannian geometry such as geodesics and cut locus. The technique is based on recognizing that these problems admit a symmetry group mapping sub-Riemannian geodesics into sub-Riemannian geodesics. This group acts on the sub-Riemannian manifold properly but not freely and the associated orbit space is in general a stratified space and not a manifold. Nevertheless on the regular part of such orbit space, $$M_r$$ it is possible to define a Riemannian metric so that the Riemannian geodesics on $$M_r$$ correspond to classes of sub-Riemannian geodesics. on the original manifold. Such a symmetry reduction technique can be used not only to find sub-Riemannian geodesics but also for general problems of nonholonomic motion planning. We illustrate the technique with problems motivated by the control of quantum mechanical systems. These examples include in particular the minimum time optimal control of two level quantum systems.

### Mass rigidity for asymptotically locally hyperbolic manifolds with boundary

Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to -1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang–Chruściel–Herzlich mass integrals are well-defined for it, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present a recent result with L.-H. Huang, which characterizes ALH manifolds that minimize the mass integrals. The proof uses scalar curvature deformation results for ALH manifolds with nonempty compact boundary. Specifically, we show the scalar curvature map is locally surjective among either (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. As a direct consequence, we establish the rigidity of the known positive mass theorems.

Abstract: TBA

Abstract: TBA