# UCSB Differential Geometry Seminar 2021–2022

The talks are held every Friday 3–4 PM (Pacific Time) unless otherwise noted. The talks are held either in South Hall room 6635, or on Zoom, as indicated below.

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, then please email me.

You can find other UCSB Math Colloquia, Conferences, Seminars & Events here.

# Upcoming talks

## Winter 2022

### Asymptotics of finite energy monopoles on AC 3-manifolds

Abstract: I will report on my recent work on sharp decay estimates for critical points of the SU(2) Yang–Mills–Higgs energy functional on asymptotically conical (AC) 3-manifolds, generalizing classical results of Taubes in the 3-dimensional Euclidean space. In particular, I will explain how we prove the quadratic decay of the covariant derivative of the Higgs field of any critical point in this general context and, with an additional hypothesis on the link, we will also sketch the proof of the quadratic decay of the curvature by combining Bochner formulas with certain refined Kato inequalities with "error terms" and standard elliptic techniques. We deduce that every irreducible critical point converges along the conical end to a limiting configuration at infinity consisting of a reducible Yang–Mills connection and a parallel Higgs field. If time permits, I will mention a few open problems and future directions in this theory.

### Calabi–Yau Gauge Theory with Symmetries

Abstract: I will discuss the Calabi–Yau instanton and monopole equations for Calabi–Yau 3-folds, which are analogues of the classical anti-self-duality and Bogomol'nyi monopole equations in dimensions 4 and 3. I will speak about my recent work (arxiv.org/abs/2110.05439) describing the moduli-space of solutions, in the special case that the Calabi–Yau 3-fold admits a co-homogeneity one symmetry, and time permitting, report on the progress of a project to use these results to construct instantons Riemannian metrics with holonomy $$\mathit{G_2}$$.

### Some Remarks on Exact $$\mathit{G_2}$$-Structures on Compact Manifolds

Abstract: Closed $$\mathit{G_2}$$-Structures are utilized in the construction of every known $$\mathit{G_2}$$-Holonomy manifold, however, we have very little idea of when a compact manifold admits such a structure. Here I will survey what is known about closed $$\mathit{G_2}$$-Structures on compact manifolds and discuss my work concerning the existence of exact $$\mathit{G_2}$$-Structures on compact manifolds. Along the way I will prove several new results on soliton solutions of the Laplacian flow of $$\mathit{G_2}$$-Structures.

### Yamabe flow of asymptotically flat metrics

Abstract: In this talk, we will discuss the behavior of the Yamabe flow on an asymptotically flat (AF) manifold. We will first show the long-time existence of the Yamabe flow starting from an AF manifold. We will then discuss the uniform estimates on manifolds with positive Yamabe constant. This would allow us to prove convergence along the Yamabe flow. If time permits, we will talk about the behavior of the rescaled flow on manifolds with negative Yamabe constant. This is joint work with Eric Chen.

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# Past talks

## Winter 2022

### The Heat Kernel Method in $$RCD(K,N)$$ Spaces and Its Applications to Non-collapsed Spaces

Abstract: The intrinsic notion of Non-collapsed RCD spaces was defined by Gigli–De Philippis inspired by Colding’s volume convergence theorem on Ricci limit spaces. Moreover, Colding and Naber showed that if the top dimensional density of the reference measure is finite on a set of positive measure (hence almost everywhere) then the Ricci limit space is non-collapsed. The same was conjectured to hold on RCD spaces by Gigli–De Philippis. I will talk about the background of this conjecture and some related problems, as well as the proof of this conjecture by a “geometric flow” induced by the heat kernel, and I will explain how the heat kernel come into play. This is joint work with Brena–Gigli–Honda. If time permits I will also talk about another surprising connection between the aforementioned flow and non-collapsed spaces which is joint work with Honda.

## Fall 2021

### Special Lagrangians and Lagrangian mean curvature flow

Abstract: Building on conjectures of Richard Thomas and Shing-Tung Yau, together with the definition of Bridgeland stability conditions, a recent article by Dominic Joyce proposes to use Lagrangian mean curvature flow to "decompose" certain Lagrangian submanifolds into simpler volume minimizing Lagrangians called special Lagrangians. In this talk, I will report on joint work in progress with Jason Lotay to prove parts of Dominic Joyce's program for certain symmetric hyperKähler 4-manifolds. While the existence of the so-called Bridgeland stability conditions on Fukaya categories remains an important but difficult problem related to the existence of certain abstract algebraic structures of these categories, our work provides concrete geometric interpretations for many such algebraic notions.

### 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.

### 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.

### Towards hearing three-dimensional geometric structures

Abstract: The Laplace spectrum of a compact Riemannian manifold is defined to be the set of positive eigenvalues of the associated Laplace operator. Inverse spectral geometry is the study of how this set of analytic data relates to the underlying geometry of the manifold. A (compact) geometric structure defined to be a compact Riemannian manifold equipped with a locally homogeneous metric. Geometric structures played an important role in the study of two and three-dimensional geometry and topology. In dimension two, the only geometric structures are those of constant curvature and by a result of Berger, a surface of constant curvature is determined up to local isometry by its Laplace spectrum. In this work, we study the following question: "To what extent are the three-dimensional geometric structures determined by their Laplace spectra?" Among other results, we provide strong evidence that the local geometry of a three-dimensional geometric structure is determined by its Laplace spectrum, which is in stark contrast with results in higher dimensions. This is a joint work with Ben Schmidt (Michigan State University) and Craig Sutton (Dartmouth College).

### 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.

### 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 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.

### 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 are 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 $$8 \pi$$.