# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### Gradient Ricci Solitons

We present some recent development in the study of gradient shrinking Ricci solitons. We address some questions about their classification and their geometric and topological structure.

##### Regularity of absolutely minimizing Lipschitz extensions

I will present joint work with Lawrence C. Evans on the everywhere differentiability of absolutely minimizing Lipschitz extensions.

##### Counterexamples to Min-Oo's Conjecture

Consider a compact Riemannian manifold $M$ of dimension $n$ whose boundary $\partial M$ is totally geodesic and is isometric to the standard sphere $S^{n-1}$. A natural conjecture of Min-Oo asserts that if the scalar curvature of $M$ is at least $n(n-1)$, then $M$ is isometric to the hemisphere $S_+^n$ equipped with its standard metric. This conjecture is inspired by the positive mass theorem in general relativity, and has been verified in many special cases. I will present joint work with F.C. Marques and A. Neves which shows that Min-Oo's conjecture fails in dimension $n \geq 3$.

##### Lower Ricci Curvature, Convexity and Applications

We prove new estimates for tangent cones along minimizing geodesics in GH limits of manifolds with lower Ricci curvature bounds. We use these estimates to show convexity results for the regular set of such limits. Applications include the proofs of several conjectures dating back to the work of Cheeger/Colding and the ruling out of certain limit spaces, including the so called generalized trumpet spaces. We construct new examples which exhibit various new behaviors and show sharpness of the new theorems. This work is joint with Toby Colding.

##### Sharp gradient estimates for a class of elliptic equations

I will present some results related with existence and sharp regularity for solutions to a class of singular elliptic equations with gradient dependence for which solutions may exhibit a free boundary. Once we have obtained sharp regularity, further analysis of the free boundary may be carried out with nondegeneracy and refined gradient estimates. I will also show how to employ an asymptotic analysis obtaining some known results for the obstacle and cavity problems. Work done in collaboration with M. Montenegro and E. Teixeira.

##### Laplace eigenvalues via asymptotic separation of variables

We study the behavior of eigenvalues under geometric perturbations using a method that might be called asymptotic separation of variables. In this method, we use quasi-mode approximations to compare the eigenvalues of a warped product and another metric that is asymptotically close to a warped product. As one application, we shoe that the generic Euclidean triangle has simple Laplace spectrum. This is joint work with Luc Hillairet.

##### Rigidity of critical metrics in dimension four

The general quadratic curvature functional is considered in dimension four. It is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the moduli space of critical metrics. I'll show that a number of compact examples are infinitesimally rigid, and are therefore isolated as critical metrics. I'll also discuss solutions of the gauged linearized equation on several noncompact examples which are asymptotically locally Euclidean. This is joint work with Matt Gursky.

##### Differentiable rigidity with Ricci bounded below

We consider a closed hyperbolic manifold $(N,h)$ of dimension $n\geq 3$ and a manifold $(M,g)$ with a degre one map $f:M \to N$. We will show that if $Ricci_g \geq -(n-1)$ and $Vol (M,g) \leq (1+\epsilon) Vol (N,h)$, then the manifolds $M$ and $N$ are diffeomorphic. The proof relies on Cheeger-Colding theory of limits of Riemannian manifolds under lower Ricci curvature bound.

##### Metric flips with Calabi symmetry

I will discuss the metric behavior of the Kahler-Ricci flow on $\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus (m+1)})$, assuming that the initial metric satisfies the symmetry defined by Calabi. I will describe the Gromov-Hausdorff limit of the flow as time approaches the singular time and how the Kahler-Ricci flow can be continued. This is a joint work with Jian Song.

##### Initial time singularities for mean curvature flow

##### The logarithmic singularities of the Green functions of the conformal powers of the Laplacian

Green functions play an important role in conformal geometry. In this talk, we shall explain how to compute explicitly the logarithmic singularities of the Green kernels of the conformal powers of the Laplacian, including the Yamabe and Paneitz operators. The results are formulated in terms of explicit conformal invariants arising from the ambient metric of Fefferman-Graham. As applications we obtain a new characterization of locally conformally flat manifolds and a spectral-theoretic characterization of the conformal class of the round sphere.

##### Compactness and Quantization phenomena in conformal geometry: Some recent results and open problems.

I will briefly introduce the GJMS operators, Q-curvature and their basic properties, then I will describe the possible behaviours of a given sequence of metrics (on a closed Riemannian manifold or on a domain in $R^n$ or similar) having prescribed Q-curvatures which converge to a given function. If the sequence is not-precompact various blow-up phenomena appear, only in part well understood. I will complete the talk with a few open questions.

##### On unique continuation for nonlinear elliptic equations

We will discuss the following issue: if two solutions of a nonlinear elliptic equation coincide in a small ball, do they necessarily coincide everywhere? The problem is fairly well understood in the linear setting, but it is open for most interesting nonlinear elliptic equations. We will analyze the difficulties of the problem and prove a result in arguably the simplest case in which one cannot linearize the equation a priori.

##### An Aronsson type approach to extremal quasiconformal mappings

Quasiconformal mappings $u:\Omega\to \Omega'$ between open domains in $R^n$, are $ W^{1,n}$ homeomorphisms whose dilation $K=|du|/ (det du)^1/n$ is in $L^\infty$. A classical problem in geometric function theory consists in finding QC minimizers for the dilation within a given homotopy class or with prescribed boundary data. In a joint work with A. Raich we study $C^2$ extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a 'localized' form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.

##### CR moduli spaces on a contact 3-manifold

We study low-dimensional problems in topology and geometry via a study of contact and Cauchy-Riemann ($CR$) structures. In particular, we consider various $CR$ moduli spaces on a contact 3-manifold. A contact structure is called spherical if it admits a compatible spherical $CR$ structure. We will talk about spherical contact structures and our analytic tool, an evolution equation of $CR$ structures. We argue that solving such an equation for the standard contact 3-sphere is related to the Smale conjecture in 3-topology. Furthermore, we propose a contact analogue of Ray-Singer's analytic torsion. This ''contact torsion'' is expected to be able to distinguish among ''spherical space forms'' $\{\Gamma\backslash S^{3}\}$ as contact manifolds.

##### Partial regularity of a minimizer of the relaxed energy for biharmonic maps

In 1999, Chang, Wang and Yang established the fundamental result on the partial regularity stationary biharmonic maps into spheres. Since then, the study of biharmonic maps has attracted much attention. In this talk, we will discuss some new result on the relaxed energy for biharmonic maps from an $m$-dimensional domain into spheres for an integer $m\geq 5$. We prove that the minimizer of the relaxed energy of the Hessian energy is biharmonic and smooth outside a singular set $\Sigma$ of finite $(m-4)$-dimensional Hausdorff measure. Moreover, when $m=5$, we also show that the singular set $\Sigma$ is $1$-rectifiable.

##### Geometrical variational problems in economics

The monopolist's problem of deciding what types of products to manufacture and how much to charge for each of them, knowing only statistical information about the preferences of an anonymous field of potential buyers, is one of the basic problems analyzed in economic theory. The solution to this problem when the space of products and of buyers can each be parameterized by a single variable (say quality $X$, and income $Y$) garnered Mirrlees (1971) and Spence (1974) their Nobel prizes in 1996 and 2001.

##### Minimal fillings and boundary rigidity - a survey

A Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined by the boundary distance function, that is the restriction of the distance function to the boundary. Loosely speaking, this means that the Riemannian metric can be recovered from measuring distances between boundary points only. The goal is to show that certain classes of metrics are boundary rigid (and, ideally, to suggest a procedure for recovering the metric). To visualize that, imagine that one wants to find out what the Earth is made of. More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the medium change from point to point). The speed of sound depends on the material.

##### Stability of hyperbolic manifolds with cusps under Ricci flow

We show that every hyperbolic manifold of finite volume and dimension greater or equal to 3 is stable under normalized Ricci flow, i.e. that every small perturbation of the hyperbolic metric flows back to the hyperbolic metric again. Here we do not need to make any decay assumptions on this perturbation. As we will see, the main difficulty in the proof comes from a weak stability of the cusps which has to do with the existence of certain cusp deformations. We will overcome this weak stability by using a new analytical method developed by Koch and Lamm.

##### Symmetries of extremal metrics on projective bundles and stability

In this talk, we will discuss some partial results in a project initiated by Vestislav Apostolov aiming to solve a conjecture about the specific form of an extremal Kahler metric on a projective bundle over a curve. The conjecture, on one hand, connects to the Donaldson-Uhlenbeck-Yau Theorem and, on the other hand relates the existence of extremal metrics to the stability of the vector bundle, which is a natural extension of Yau's conjecture in the case of Kaehler Einstein metric when the first Chern class of the Kaehler manifold is positive.