# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### On the Bombieri-De Giorgi-Giusti minimal graph and it applications

I will discuss refinements to the Bombieri-De Giorgi-Giusti minimal graph and three applications: 1. De Giorgi's Conjecture for Allen-Cahn equation; 2. Caffarelli-Berestycki-Nirenberg Conjecture for overdetermined problem on epigraphs; 3. Translating graphs of mean curvature flows.

##### A local regularity theorem for mean curvature flow with triple edges

We consider the evolution by mean curvature flow of surface clusters, where along triple edges three surfaces are allowed to meet under an equal angle condition. We show that any such smooth flow, which is weakly close to the static flow consisting of three half-planes meeting along the common boundary, is smoothly close with estimates. Furthermore, we show how this can be used to prove a smooth short-time existence result. This is joint work with B. White.

##### Direct methods of moving planes, moving spheres, and blowing-ups for the fractional Laplacian

Many conventional approaches on partial differential operators do not work on the nonlocal fractional operator. To overcome this difficulty arising from non-localness, Caffarelli and Silvestre introduce the extension method to reduced the problem into a local one in one higher dimensions, which has become a powerful tool in studying such nonlocal problems and has yielded a series of fruitful results. However, due to technical restrictions, sometimes one needs to impose extra conditions when studying the extended problems in higher dimensions, and these conditions may not be necessary if we investigate the original nonlocal problems directly.

##### A priori estimates for semistable solutions of semilinear elliptic equations

In this talk we will discuss semistable solutions of the boundary value problem $Lu+f(u)=0$ in $\Omega$ and $u=0$ on $\partial\Omega$, where $Lu:=\partial_i(a^{ij}u_j)$ is uniformly elliptic. By semistability we mean that the lowest Dirichlet eigenvalue of the linearized operator at u is nonnegative. The basic problem (which has a long history) is to obtain a priori $L^{\infty}$ bounds on a solution under minimal assumptions on $f(t)$. A basic and standard assumption is that $u>0$ in $\Omega$ and $f\in C^2$ is positive, nondecreasing, and superlinear at infinity, i.e. $f(0)>0$, $f' \geq 0$ and $f(t)/t$ tends to infinity as $t$ tends to infinity. For radially symmetric solutions, an $L^{\infty}$ bound for $u$ is known for $n\leq 9$. On the other hand there exists unbounded semistable solutions when $n\geq 10$ for $f(u)=e^u$.

##### A rigidity theorem with capacity and its applications

**Please note different day (Thursday) and time (4:30). **We will proof a rigidity theorem about subset in R^n that if a set is, in terms of capacity, like a cone at each of its boundary point in most of the scales, then it is either the whole space or half space. Its consequences to the regularity is also discussion from a local version of the theorem.

##### Scalar-flat Kahler ALE metrics on minimal resolutions

Scalar-flat Kahler ALE surfaces have been studied in a variety of settings since the late 1970s. All previously known examples have group at infinity either cyclic or contained in SU(2). I will describe an existence result for scalar-flat Kahler ALE metrics with group at infinity G, where the underlying space is the minimal resolution of C^2/G, for all finite subgroups G of U(2) which act freely on S^3. I will also discuss a non-existence result for Ricci-flat metrics on certain spaces, which is related to a conjecture of Bando-Kasue-Nakajima. If time permits, I will also present some new examples self-dual metrics on connected sums of complex projective planes. This is joint work with Michael Lock.

##### Variational stability of Kähler Ricci Solitons

I will explain the solution of the varational stability problem for compact Kähler Ricci Solitons.

##### Geometry of minimal surfaces in homogeneous spaces

**Please note special time. ** We will discuss global geometric properties of minimal surfaces and non compact constant mean curvatures surfaces in hyperbolic spaces H(3), H(2)xR and Heisenberg Riemannian space.

##### A borderline Sobolev inequality on the hyperbolic spaces

About a decade ago, Bourgain, Brezis and van Schaftingen established some borderline Sobolev embeddings on $\mathbb{R}^n$, which lent themselves to the proof of some Gagliardo-Nirenberg inequalities for differential forms on $\mathbb{R}^n$ (see e.g. work of Lanzani and Stein). In an attempt to understand the geometry underlying such estimates, we extend some of these results from the setting of $\mathbb{R}^n$ to the hyperbolic spaces. This is joint work with Sagun Chanillo and Jean van Schaftingen.

##### On the scattering operators for Kahler-Einstein manifolds with strictly pseudoconvex CR-infinity

I will talk about the positivity of scattering operators for Kahler-Einstein manifolds with strictly pseudoconvex CR-infinity which has positive Webster scalar curvature. The result is parallel to Guillarmou-Qing's positivity result for scattering operators for Poincare-Einstein manifolds. I will also give an energy identity between the boundary and the interior.

##### On the convergence of Kahler-Ricci flow on minimal models of general type

I will show that the Kahler-Ricci flow on a three dimensional minimal model of general type converges in the Gromov-Hausdorff topology to the unique singular Kahler-Einstein metric. The proof depends on an integral version of Cheeger-Colding-Tian theory and a diameter bound estimate to the singular Kahler-Einstein metric by Song. It is a joint work with Tian.