# ND Student Geometry Working Seminar

This is a geometry centric, student organized seminar at Notre Dame, with an emphasis on work or work in progress.

We meet on Fridays 3pm. [Zoom link](https://notredame.zoom.us/j/96302373523?pwd=SWZMOTFuRCt2UEIrdGk0bStjc0M2Zz09)

Zoom info:
* Meeting ID: 963 0237 3523
* Passcode: 286777

Contact: [Shih-Kai Chiu](mailto://[e-mail])

## Fall 2020 Schedule

| Date | Speaker | Title |
| ----- | ------------------ | ------------------------------------------------------------------------------------ |
| 8/21 | Shih-Kai Chiu | A Liouville type theorem for harmonic 1-forms |
| 8/28 | Ethan Addison | Cohomology of Almost Complex Manifolds |
| 9/4 | Samuel Pérez-Ayala | Maximal Metrics for the Conformal Laplacian |
| 9/11 | Aaron Tyrrell | Renormalised Volume and Area in the Poincare-Einstein Setting |
| 9/18 | Aaron Tyrrell | Renormalised Area |
| 9/25 | Zhehui Wang | Liouville type and Bernstein type theorems for minimal graphs over unbounded domains |
| 10/2 | Xiaoxiao Li | Slowly Converging Yamabe Flows I |
| 10/9 | Xiaoxiao Li | Slowly Converging Yamabe Flows II |
| 10/16 | Ethan Addison | Generalized Poincaré-Type Kähler Metrics |
| 10/30 | Patrick Heslin | The geometry of ideal fluid motion |
| 11/6 | Patrick Heslin | A regularity property of the $L^2$ exponential map on the space of Volumorphisms |
| 11/13 | Ilya Marchenko | Regularity of Singular Sets of Solutions to Elliptic Equations |

## Abstracts, Notes and Resources

### 8/21 Shih-Kai Chiu

The famous Cheng-Yau gradient estimate implies that on a complete Riemannian manifold with nonnegative Ricci curvature, any harmonic function that grows sublinearly must be a constant. This is the same as saying the function is closed as a 0-form. We prove an analogous result for harmonic 1-forms. Namely, on a complete Ricci-flat manifold with Euclidean volume growth, any harmonic 1-form with polynomial sublinear growth must be the differential of a harmonic function. We prove this by proving an $L^2$ version of the "gradient estimate" for harmonic 1-forms. As a corollary, we show that when the manifold is Ricci-flat Kähler with Euclidean volume growth, then any subquadratic harmonic function must be pluriharmonic. This generalizes the result of Conlon-Hein.

### 8/28 Ethan Addison

Going back over a century to questions raised by Hirzebruch, there has been interest in extending the notion of the Hodge numbers $h^{p,q}$ to almost complex manifolds. After introducing the necessary preliminaries of almost complex geometry, we briefly discuss certain efforts to understand the relationship between almost complex structures and the classical de Rham Cohomology and then prioritize recent work by Cirici-Wilson which defines a Dolbeault Cohomology theory for almost complex manifolds, including some harmonic theory for these spaces as well. As an application, we will consider the case of nearly Kähler manifolds.

### 9/4 Samuel Pérez-Ayala

Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. Assume $[g]$ is a conformal class for which the Conformal Laplacian $L_g:= -\Delta_g + c_nR_g$ has at least two negative eigenvalues and $0\not \in \text{Spec}(L_g)$. If we define $\Lambda_2(M^n,[g])$ as the supremum of the second eigenvalue over generalized conformal metrics with unit volume, then we show that there is a nonnegative and nontrivial function $\bar u\in C^{\alpha_0}(M^n)\cap C^\infty(M^n\setminus\{\bar u=0\})$, $\alpha_0\in(0,1)$, such that $g_{\bar u}=\bar u^{\frac{4}{n-2}}g$ attains this supremum. As an application, and depending on the multiplicity of $\lambda_2(\bar u)$, we find either nodal solutions to a Yamabe type equation, or a harmonic map into a sphere.

### 9/11 Aaron Tyrrell

We will look at some results regarding the renormalised volume and area in the context of a Poincare-Einstein background metric.

### 9/18 Aaron Tyrrell

We will look at renormaised area and a renormalised area formula for 4-dimensional hypersurfaces of Poincare-Einstein manifolds.

### 9/25 Zhehui Wang

The classical Bernstein theorem asserts that any entire minimal graph over $\mathbb{R}^n$ with $2\leq n\leq 7$ is a hyperplane. In this talk, we begin with Liouville type theorems for minimal graphs over half-spaces with linear Dirichlet boundary value or constant Neumann boundary value. Then, we give a Bernstein type theorem for minimal graphs over convex cones (for any dimension). This talk is based on joint work with Guosheng Jiang and Jintian Zhu; and joint work with Nick Edelen.

### 10/2, 10/9 Xiaoxiao Li

I'm going to talk about a result by Carlotto, Chodosh and Rubinstein. They characterize the rate of convergence of a converging volume-normalized Yamabe flow in terms of Morse theoretic properties of the limiting metric. If the limiting metric is an integrable critical point for the Yamabe functional (for example, this holds when the critical point is non-degenerate), then the flow converges exponentially fast. In general, they use a suitable Lojasiewicz-Simon inequality to prove that the slowest the flow will converge is polynomially. When the limit metric satisfies an Adams-Simon type condition there exist flows converging to it exactly at a polynomial rate.

### 10/16 Ethan Addison

Given a compact Kähler manifold $(M^n,\omega)$ with a smooth divisor $X$, we can define a complete Kähler metric $\varpi$ on the complement $M\setminus X$ modeled after the punctured disk cusp metric. These metric of Poincaré type have many nice properties, largely laid out by H. Auvray over the last decade. One of these properties, however, make their study in the Calabi Extremal Metric Program somewhat limited: extremal (resp. cscK) PT metrics on $M\setminus X$ induce extremal (resp. cscK) PT metrics on $X$. This was made fairly explicit by L. M. Sektnan in regards to blowing up extremal PT metrics, evincing the need for a slightly broader definition in order to get broader results. In this talk, we will introduce a generalization of PT metrics which take into account the automorphisms on $X$ to produce a gnarling effect on the metric, discussing some of their properties as well as recent progress towards proving a LeBrun-Simanca style openness result for these metrics.

### 10/30 Patrick Heslin

V. Arnold observed in his seminal paper that solutions of the Euler equations for ideal fluid motion can be viewed as geodesics of a certain right-invariant metric on the group of volume-preserving diffeomorphisms (known as *volumorphisms*), $D^s_\mu(M)$. In essence, this approach showcases the natural framework in which to tackle this infamous Cauchy problem from the so-called *Lagrangian* viewpoint. In their celebrated paper Ebin and Marsden provided the formulation of the above in the $H^s$ Sobolev setting. Here they proved that the space of volumorphisms can be given the structure of a smooth, infinite dimensional Hilbert manifold. They illustrated that, when equipped with a right-invariant $L^2$ metric, the geodesic equation on this manifold is a smooth ordinary differential equation. They then applied the classic iteration method of Picard to obtained existence, uniqueness and smooth dependence on initial conditions. In particular, the last property allows one to define a smooth exponential map on $D^s_\mu(M)$ in analogy with the classical construction in finite dimensional Riemannian geometry. Hence, the work of Arnold, Ebin and Marsden allows one to explore the problem of ideal fluid motion armed with tools from Riemannian geometry.

### 11/6 Patrick Heslin

In this talk we are concerned with a certain regularity property of exponential maps on the space of Sobolev $s$-regular volumorphisms of the 2-torus.

**Defintion:**
An exponential map on $D^s_\mu(\mathbb{T}^2)$ is said to exhibit *smoothing* if given initial data $u_0 \in T_eD^s_\mu(\mathbb{T}^2)$:
$$exp_e(u_0) \ \in \ D^{s+1}_\mu(\mathbb{T}^2) \ \Rightarrow \ u_0 \ \in \ T_eD^{s+1}_\mu(\mathbb{T}^2)$$

Kappeler, Loubet and Topalov observed this property in their study of the full group of (orientation preserving) diffeomorphisms of the 2-torus, equipped with various right-invariant $H^r$ metrics. Their goal was to show that, under certain restrictions, the arising exponential maps were $C^1$ diffeomorphisms in the sense of Frechet.

The earliest result of this nature however, to the best of the author's knowledge, is due to Constantin and Kolev where they used this smoothing property to show the existence of a $C^1$-Frech\'et exponential map on $C^\infty$ diffeomorphisms of the circle, equipped with a right-invariant $H^r$ metric, for $r \in \mathbb{Z}_{+}$, covering the case of the Camassa-Holm equation.

Most notably, in two dimensions, we cover the case of the $L^2$ exponential map, where the reduction of the geodesic equation to the Lie Algebra of $H^s$ vector fields yields the Euler equations.

### 11/13 Ilya Marchenko

We use pointwise Schauder estimates and a technique from free boundary problems to show that the singular set (the set of points where both the solution and its gradient vanish) of a solution to a homogeneous linear elliptic equation in $B_1\subset\mathbb{R}^n$ with appropriate regularity assumptions on the coefficients is contained in a countable union of $C^{1,\alpha}$ graphs. More precisely, the singular set can be expressed as a union of level sets based on the vanishing order of points in the singular set. Each of these level sets is made up of $j$-dimensional pieces for $j=1,\ldots,n-2$. We show that every such piece is contained in a countable union of $j$-dimensional $C^{1,\alpha}$ graphs.